PermPal Database

Av(1234, 1324, 2143, 2413, 3142)

Generating Function

\frac{2 x - 1}{2 x^{4} + x^{3} - x^{2} + 3 x - 1}

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Counting Sequence

1, 1, 2, 6, 19, 55, 156, 444, 1269, 3629, 10374, 29650, 84743, 242211, 692288, ...

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Implicit Equation for the Generating Function

(2 x^{4} + x^{3} - x^{2} + 3 x - 1) F (x) - 2 x + 1 = 0

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Recurrence

a (0) = 1

a (1) = 1

a (2) = 2

a (3) = 6

a (n + 4) = 2 a (n) + a (n + 1) - a (n + 2) + 3 a (n + 3), n \geq 4

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Explicit Closed Form

\frac{2068379 (((2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - \frac{5139}{101}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + (- \frac{427 2^{\frac{2}{3}} \sqrt{3} \sqrt{571}}{3232} - \frac{5139 2^{\frac{2}{3}}}{3232}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + \frac{496 \sqrt{571} \sqrt{3}}{101}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} - \frac{479640 2^{\frac{1}{3}} (\sqrt{3} + \frac{9 \sqrt{571}}{571}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{101} + \frac{419685 (\sqrt{3} - \frac{369 \sqrt{571}}{3997}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{404} + \frac{282240 \sqrt{571}}{101}) \sqrt{48 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296} + \frac{1169408 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{101} - \frac{8840724480}{101} + ({(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} (\frac{70804 \sqrt{3}}{101} - \frac{8292 \sqrt{571}}{101}) + \frac{255808 (\sqrt{3} - \frac{33 \sqrt{571}}{3997}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{101}) \sqrt{(19 \sqrt{571} \sqrt{3} + 827) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + (128 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} - 5248 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 25696 2^{\frac{1}{3}}}) (- \frac{3394838200320 {(\frac{\sqrt{48 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (9 \sqrt{571} \sqrt{3} - 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 7296}}{192} - \frac{19 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{60480} + \frac{(- 85 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - 256 {(340 + 12 \sqrt{571} \sqrt{3})}^{\frac{1}{3}}) \sqrt{(57 \sqrt{571} \sqrt{3} + 2481) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 384 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 77088 2^{\frac{1}{3}}}}{1935360} + \frac{\sqrt{(10849 \sqrt{571} \sqrt{3} + 472217) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 73088 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 14672416 2^{\frac{1}{3}}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{215040} - \frac{1}{8})}^{- n}}{20479} + ((((\frac{5767671 2^{\frac{2}{3}}}{655328} + \frac{375471 2^{\frac{2}{3}} \sqrt{3} \sqrt{571}}{655328}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + (- \frac{96129 \sqrt{571} \sqrt{3} 2^{\frac{1}{3}}}{20479} + \frac{4726167 2^{\frac{1}{3}}}{20479}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - \frac{478416 \sqrt{571} \sqrt{3}}{20479}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} - \frac{383652045 (\sqrt{3} - \frac{122387 \sqrt{571}}{1217943}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{81916} + \frac{522327960 2^{\frac{1}{3}} (\sqrt{3} + \frac{1091 \sqrt{571}}{207273}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479} - \frac{271152000 \sqrt{571}}{20479}) \sqrt{48 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296} - \frac{634988544 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{20479} + ((\frac{703674 \sqrt{571}}{20479} - \frac{3429426 \sqrt{3}}{20479}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - \frac{28559136 (- \frac{9423 \sqrt{571}}{297491} + \sqrt{3}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479}) \sqrt{(19 \sqrt{571} \sqrt{3} + 827) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + (128 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} - 5248 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 25696 2^{\frac{1}{3}}} - \frac{67639320 (\sqrt{571} \sqrt{3} - \frac{200421}{8947}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{20479} - \frac{3686582108160}{20479} - \frac{75237120 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - \frac{56529}{311}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479}) {(- \frac{\sqrt{48 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (9 \sqrt{571} \sqrt{3} - 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 7296}}{192} - \frac{19 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{60480} + \frac{(- 85 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - 256 {(340 + 12 \sqrt{571} \sqrt{3})}^{\frac{1}{3}}) \sqrt{(57 \sqrt{571} \sqrt{3} + 2481) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 384 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 77088 2^{\frac{1}{3}}}}{1935360} + \frac{\sqrt{(10849 \sqrt{571} \sqrt{3} + 472217) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 73088 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 14672416 2^{\frac{1}{3}}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{215040} - \frac{1}{8})}^{- n} + (((- \frac{299171 ((- \frac{27087 I}{299171} - \frac{371233 \sqrt{3}}{299171}) \sqrt{571} + I \sqrt{3} - \frac{12349017}{299171}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{1310656} - ((\frac{66994 \sqrt{3}}{20479} + \frac{381 I}{20479}) \sqrt{571} + I \sqrt{3} - \frac{2888118}{20479}) 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - \frac{593000 \sqrt{3} (\frac{35611 \sqrt{571}}{74125} + I)}{20479}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} - \frac{2451645 ((I \sqrt{3} - \frac{155261}{7783}) \sqrt{571} - \frac{216081 I}{7783} + \frac{1257913 \sqrt{3}}{7783}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{163832} - \frac{32760 ((I \sqrt{3} + \frac{1832}{13}) \sqrt{571} - \frac{21123 I}{13} - \frac{155312 \sqrt{3}}{13}) 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479} + \frac{3632247360 I}{20479} - \frac{179726400 \sqrt{571}}{20479}) \sqrt{48 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296} - \frac{159744 \sqrt{3} (I \sqrt{571} + \frac{207273}{26}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{20479} + (((\frac{7130547}{20479} - \frac{49299 I \sqrt{3}}{20479}) \sqrt{571} - \frac{59596983 \sqrt{3}}{20479} - \frac{950715 I}{20479}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + \frac{220080 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} ((I \sqrt{3} + \frac{45393}{4585}) \sqrt{571} - \frac{217551 I}{4585} - \frac{4654221 \sqrt{3}}{4585}) 2^{\frac{2}{3}}}{20479}) \sqrt{(19 \sqrt{571} \sqrt{3} + 827) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + (128 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} - 5248 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 25696 2^{\frac{1}{3}}} + \frac{757591380 ((- \frac{8947 I}{66807} + \frac{8947 \sqrt{3}}{200421}) \sqrt{571} + I \sqrt{3} - 1) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{20479} - \frac{3686582108160}{20479} - \frac{6837747840 ((- \frac{311 I}{18843} - \frac{311 \sqrt{3}}{56529}) \sqrt{571} + I \sqrt{3} + 1) 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479}) {(- \frac{I \sqrt{48 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296}}{192} + \frac{19 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{60480} + \frac{(85 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 256 {(340 + 12 \sqrt{571} \sqrt{3})}^{\frac{1}{3}}) \sqrt{(57 \sqrt{571} \sqrt{3} + 2481) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 384 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 77088 2^{\frac{1}{3}}}}{1935360} - \frac{\sqrt{(10849 \sqrt{571} \sqrt{3} + 472217) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 73088 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 14672416 2^{\frac{1}{3}}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{215040} - \frac{1}{8})}^{- n} + {(\frac{I \sqrt{48 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296}}{192} + \frac{19 \sqrt{768 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 154176 + (57 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 2481 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{60480} + \frac{(85 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 256 {(340 + 12 \sqrt{571} \sqrt{3})}^{\frac{1}{3}}) \sqrt{(57 \sqrt{571} \sqrt{3} + 2481) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 384 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 77088 2^{\frac{1}{3}}}}{1935360} - \frac{\sqrt{(10849 \sqrt{571} \sqrt{3} + 472217) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 73088 2^{\frac{2}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 14672416 2^{\frac{1}{3}}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{215040} - \frac{1}{8})}^{- n} (((\frac{299171 ((- \frac{27087 I}{299171} + \frac{371233 \sqrt{3}}{299171}) \sqrt{571} + I \sqrt{3} + \frac{12349017}{299171}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{1310656} + 2^{\frac{1}{3}} ((\frac{381 I}{20479} - \frac{66994 \sqrt{3}}{20479}) \sqrt{571} + I \sqrt{3} + \frac{2888118}{20479}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + \frac{593000 \sqrt{3} (- \frac{35611 \sqrt{571}}{74125} + I)}{20479}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + \frac{2451645 ((I \sqrt{3} + \frac{155261}{7783}) \sqrt{571} - \frac{216081 I}{7783} - \frac{1257913 \sqrt{3}}{7783}) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{163832} + \frac{32760 2^{\frac{1}{3}} ((I \sqrt{3} - \frac{1832}{13}) \sqrt{571} - \frac{21123 I}{13} + \frac{155312 \sqrt{3}}{13}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479} - \frac{3632247360 I}{20479} - \frac{179726400 \sqrt{571}}{20479}) \sqrt{48 \sqrt{3} \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}} + (- 9 \sqrt{571} \sqrt{3} + 255) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + 1536 2^{\frac{1}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} - 7296} + \frac{159744 \sqrt{3} (I \sqrt{571} - \frac{207273}{26}) \sqrt{256 2^{\frac{1}{3}} (\sqrt{571} \sqrt{3} - 41) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 51392 + (19 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} + 827 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}}{20479} + (((\frac{7130547}{20479} + \frac{49299 I \sqrt{3}}{20479}) \sqrt{571} - \frac{59596983 \sqrt{3}}{20479} + \frac{950715 I}{20479}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} - \frac{220080 {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} ((I \sqrt{3} - \frac{45393}{4585}) \sqrt{571} - \frac{217551 I}{4585} + \frac{4654221 \sqrt{3}}{4585}) 2^{\frac{2}{3}}}{20479}) \sqrt{(19 \sqrt{571} \sqrt{3} + 827) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}} + (128 2^{\frac{2}{3}} \sqrt{3} \sqrt{571} - 5248 2^{\frac{2}{3}}) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}} + 25696 2^{\frac{1}{3}}} - \frac{757591380 ((- \frac{8947 I}{66807} - \frac{8947 \sqrt{3}}{200421}) \sqrt{571} + I \sqrt{3} + 1) 2^{\frac{2}{3}} {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{2}{3}}}{20479} - \frac{3686582108160}{20479} + \frac{6837747840 2^{\frac{1}{3}} ((- \frac{311 I}{18843} + \frac{311 \sqrt{3}}{56529}) \sqrt{571} + I \sqrt{3} - 1) {(3 \sqrt{571} \sqrt{3} + 85)}^{\frac{1}{3}}}{20479}))}{120051316732832671334400}

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This specification was found using the strategy pack "Point Placements" and has 72 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

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\begin{aligned} F_{0} (x) & = F_{1} (x) + F_{2} (x) \\ F_{1} (x) & = 1 \\ F_{2} (x) & = F_{3} (x) \\ F_{3} (x) & = F_{4} (x) F_{5} (x) \\ F_{4} (x) & = x \\ F_{5} (x) & = F_{16} (x) + F_{6} (x) \\ F_{6} (x) & = F_{1} (x) + F_{7} (x) \\ F_{7} (x) & = F_{8} (x) \\ F_{8} (x) & = F_{4} (x) F_{9} (x) \\ F_{9} (x) & = F_{10} (x) + F_{13} (x) \\ F_{10} (x) & = F_{1} (x) + F_{11} (x) \\ F_{11} (x) & = F_{12} (x) \\ F_{12} (x) & = F_{10} (x) F_{4} (x) \\ F_{13} (x) & = F_{14} (x) + F_{7} (x) \\ F_{14} (x) & = F_{15} (x) \\ F_{15} (x) & = F_{13} (x) F_{4} (x) \\ F_{16} (x) & = F_{17} (x) + F_{2} (x) \\ F_{17} (x) & = F_{18} (x) + F_{19} (x) + F_{57} (x) \\ F_{18} (x) & = 0 \\ F_{19} (x) & = F_{20} (x) F_{4} (x) \\ F_{20} (x) & = F_{21} (x) + F_{28} (x) \\ F_{21} (x) & = F_{22} (x) + F_{25} (x) \\ F_{22} (x) & = F_{23} (x) \\ F_{23} (x) & = F_{24} (x) F_{4} (x) \\ F_{24} (x) & = F_{1} (x) + F_{4} (x) \\ F_{25} (x) & = F_{26} (x) \\ F_{26} (x) & = F_{27} (x) F_{4} (x) \\ F_{27} (x) & = F_{4} (x) \\ F_{28} (x) & = F_{29} (x) + F_{51} (x) \\ F_{29} (x) & = F_{18} (x) + F_{19} (x) + F_{30} (x) \\ F_{30} (x) & = F_{31} (x) F_{4} (x) \\ F_{31} (x) & = F_{2} (x) + F_{32} (x) \\ F_{32} (x) & = F_{18} (x) + F_{33} (x) + F_{50} (x) \\ F_{33} (x) & = F_{34} (x) F_{4} (x) \\ F_{34} (x) & = F_{35} (x) + F_{37} (x) \\ F_{35} (x) & = F_{36} (x) + F_{4} (x) \\ F_{36} (x) & = F_{26} (x) \\ F_{37} (x) & = F_{32} (x) + F_{38} (x) \\ F_{38} (x) & = 2 F_{18} (x) + F_{39} (x) + F_{48} (x) \\ F_{39} (x) & = F_{4} (x) F_{40} (x) \\ F_{40} (x) & = F_{41} (x) + F_{44} (x) \\ F_{41} (x) & = F_{42} (x) \\ F_{42} (x) & = F_{4} (x) F_{43} (x) \\ F_{43} (x) & = F_{4} (x) \\ F_{44} (x) & = F_{45} (x) \\ F_{45} (x) & = 2 F_{18} (x) + F_{39} (x) + F_{46} (x) \\ F_{46} (x) & = F_{4} (x) F_{47} (x) \\ F_{47} (x) & = F_{32} (x) \\ F_{48} (x) & = F_{4} (x) F_{49} (x) \\ F_{49} (x) & = F_{32} (x) \\ F_{50} (x) & = F_{2} (x) F_{4} (x) \\ F_{51} (x) & = 2 F_{18} (x) + F_{48} (x) + F_{52} (x) \\ F_{52} (x) & = F_{4} (x) F_{53} (x) \\ F_{53} (x) & = F_{54} (x) + F_{55} (x) \\ F_{54} (x) & = F_{42} (x) \\ F_{55} (x) & = F_{56} (x) \\ F_{56} (x) & = 2 F_{18} (x) + F_{46} (x) + F_{52} (x) \\ F_{57} (x) & = F_{4} (x) F_{58} (x) \\ F_{58} (x) & = F_{59} (x) + F_{64} (x) \\ F_{59} (x) & = F_{2} (x) + F_{60} (x) \\ F_{60} (x) & = F_{18} (x) + F_{33} (x) + F_{61} (x) \\ F_{61} (x) & = F_{4} (x) F_{62} (x) \\ F_{62} (x) & = F_{2} (x) + F_{63} (x) \\ F_{63} (x) & = F_{61} (x) \\ F_{64} (x) & = F_{65} (x) + F_{71} (x) \\ F_{65} (x) & = F_{66} (x) \\ F_{66} (x) & = F_{4} (x) F_{67} (x) \\ F_{67} (x) & = F_{62} (x) + F_{68} (x) \\ F_{68} (x) & = F_{65} (x) + F_{69} (x) \\ F_{69} (x) & = F_{70} (x) \\ F_{70} (x) & = F_{4} (x) F_{68} (x) \\ F_{71} (x) & = F_{70} (x) \end{aligned}

Av(1234, 1324, 2143, 2413, 3142)

This specification was found using the strategy pack "Point Placements" and has 72 rules.

Proof Tree

System of Equations