Av(1234, 1243, 3214)
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Generating Function
2x5−x4−3x3−2x2−2x+12x5−2x3−x2−3x+1
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Counting Sequence
1, 1, 2, 6, 21, 73, 250, 861, 2967, 10220, 35203, 121263, 417710, 1438865, 4956391, ...
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Implicit Equation for the Generating Function
(2x5−2x3−x2−3x+1)F(x)−2x5+x4+3x3+2x2+2x−1=0
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Recurrence
a(0)=1
a(1)=1
a(2)=2
a(3)=6
a(4)=21
a(5)=73
a(n)=a(n+2)+a(n+3)2+3a(4+n)2−a(n+5)2,n≥6
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Explicit Closed Form
{1n=07337RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−n+141583+5657RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−n+241583−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−n+341583−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−n+441583+(3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)3−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+5517)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)−n+141583+(3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)−n+241583+(3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−296)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)−n+341583+((−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+296)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)2+(−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1))RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+5221)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−n+141583+((−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+296)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−n+241583+(((3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−296)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+(−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−2017RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+5221)RootOf(2Z5−2Z3−Z2−3Z+1,index=4)−n+141583+3650RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−n41583+(3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)4−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2−1820RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−1810)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)−n41583+((−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+296)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)3+(−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1))RootOf(2Z5−2Z3−Z2−3Z+1,index=2)2+(−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)3+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−296)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)3−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−1958)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−n41583+(((3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−296)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)2+((3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−296)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)2+(3640RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2−592RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2−2017RootOf(2Z5−2Z3−Z2−3Z+1,index=1))RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+(−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)2+(−296RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2−2017RootOf(2Z5−2Z3−Z2−3Z+1,index=1))RootOf(2Z5−2Z3−Z2−3Z+1,index=3)−2017RootOf(2Z5−2Z3−Z2−3Z+1,index=1)2+59)RootOf(2Z5−2Z3−Z2−3Z+1,index=4)−n41583+((((−3640RootOf(2Z5−2Z3−Z2−3Z+1,index=4)+296)RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+296RootOf(2Z5−2Z3−Z2−3Z+1,index=4)+2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)+(296RootOf(2Z5−2Z3−Z2−3Z+1,index=4)+2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+2017RootOf(2Z5−2Z3−Z2−3Z+1,index=4)−5221)RootOf(2Z5−2Z3−Z2−3Z+1,index=2)+((296RootOf(2Z5−2Z3−Z2−3Z+1,index=4)+2017)RootOf(2Z5−2Z3−Z2−3Z+1,index=1)+2017RootOf(2Z5−2Z3−Z2−3Z+1,index=4)−5221)RootOf(2Z5−2Z3−Z2−3Z+1,index=3)+(2017RootOf(2Z5−2Z3−Z2−3Z+1,index=4)−5221)RootOf(2Z5−2Z3−Z2−3Z+1,index=1)−5221RootOf(2Z5−2Z3−Z2−3Z+1,index=4)+59)RootOf(2Z5−2Z3−Z2−3Z+1,index=5)−n41583otherwise
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This specification was found using the strategy pack "Point Placements" and has 80 rules.

Found on January 18, 2022.

Finding the specification took 10 seconds.

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Copy 80 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F4(x)F5(x)F4(x)=xF5(x)=F19(x)+F6(x)F6(x)=F1(x)+F7(x)F7(x)=F8(x)F8(x)=F4(x)F9(x)F9(x)=F10(x)+F11(x)F10(x)=F1(x)+F4(x)F11(x)=F12(x)+F7(x)F12(x)=F13(x)+F14(x)+F18(x)F13(x)=0F14(x)=F15(x)F4(x)F15(x)=F16(x)+F17(x)F16(x)=F4(x)F17(x)=F18(x)F18(x)=F4(x)F7(x)F19(x)=F2(x)+F20(x)F20(x)=F13(x)+F21(x)+F66(x)F21(x)=F22(x)F4(x)F22(x)=F23(x)+F34(x)F23(x)=F24(x)+F7(x)F24(x)=F13(x)+F14(x)+F25(x)F25(x)=F26(x)F4(x)F26(x)=F11(x)+F27(x)F27(x)=F24(x)+F28(x)F28(x)=F13(x)+F29(x)+F30(x)+F33(x)F29(x)=0F30(x)=F31(x)F4(x)F31(x)=F17(x)+F32(x)F32(x)=F33(x)F33(x)=F24(x)F4(x)F34(x)=F35(x)+F48(x)F35(x)=F36(x)F36(x)=F37(x)F4(x)F37(x)=F38(x)+F41(x)F38(x)=F2(x)+F39(x)F39(x)=F40(x)F40(x)=F2(x)F4(x)F41(x)=F35(x)+F42(x)F42(x)=2F13(x)+F43(x)+F47(x)F43(x)=F4(x)F44(x)F44(x)=F45(x)+F46(x)F45(x)=F39(x)F46(x)=F47(x)F47(x)=F35(x)F4(x)F48(x)=2F13(x)+F49(x)+F54(x)F49(x)=F4(x)F50(x)F50(x)=F51(x)+F52(x)F51(x)=F39(x)F52(x)=F53(x)F53(x)=F20(x)F4(x)F54(x)=F4(x)F55(x)F55(x)=F56(x)+F58(x)F56(x)=F20(x)+F57(x)F57(x)=2F13(x)+F49(x)+F53(x)F58(x)=F48(x)+F59(x)F59(x)=2F13(x)+F60(x)+F61(x)+F65(x)F60(x)=0F61(x)=F4(x)F62(x)F62(x)=F63(x)+F64(x)F63(x)=F53(x)F64(x)=F65(x)F65(x)=F4(x)F48(x)F66(x)=F4(x)F67(x)F67(x)=F68(x)+F74(x)F68(x)=F2(x)+F69(x)F69(x)=F13(x)+F40(x)+F70(x)F70(x)=F4(x)F71(x)F71(x)=F72(x)+F73(x)F72(x)=F12(x)+F4(x)F73(x)=F39(x)+F57(x)F74(x)=F20(x)+F75(x)F75(x)=F13(x)+F49(x)+F53(x)+F76(x)F76(x)=F4(x)F77(x)F77(x)=F78(x)+F79(x)F78(x)=F12(x)+F28(x)F79(x)=F42(x)+F59(x)