Av(123, 3412)
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Generating Function
4x49x3+10x25x+1(2x1)(x1)4
Counting Sequence
1, 1, 2, 5, 13, 33, 80, 185, 411, 885, 1862, 3853, 7881, 15993, 32284, ...
Implicit Equation for the Generating Function
(2x1)(x1)4F(x)+4x49x3+10x25x+1=0
Recurrence
a(0)=1
a(1)=1
a(2)=2
a(3)=5
a(4)=13
a(n+1)=2a(n)+(n1)(n22n+6)6,n5
Explicit Closed Form
111n6n36+2n+1

This specification was found using the strategy pack "Point Placements" and has 32 rules.

Found on January 18, 2022.

Finding the specification took 5 seconds.

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Created with Raphaël 2.1.4
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Copy 32 equations to clipboard:
F0(x)=F1(x)+F2(x)F1(x)=1F2(x)=F3(x)F3(x)=F4(x)F8(x)F4(x)=F5(x)+F9(x)F5(x)=F1(x)+F6(x)F6(x)=F7(x)F7(x)=F5(x)F8(x)F8(x)=xF9(x)=F10(x)F10(x)=F11(x)F8(x)F11(x)=F12(x)+F24(x)F12(x)=F13(x)+F16(x)F13(x)=F14(x)+F5(x)F14(x)=F15(x)F15(x)=F11(x)F8(x)F16(x)=F17(x)F6(x)F17(x)=F18(x)+F5(x)F18(x)=F19(x)+F6(x)F19(x)=F20(x)+F21(x)+F23(x)F20(x)=0F21(x)=F22(x)F8(x)F22(x)=F19(x)+F6(x)F23(x)=F18(x)F8(x)F24(x)=F25(x)+F26(x)F25(x)=F0(x)+F4(x)F26(x)=F27(x)F6(x)F27(x)=F28(x)+F6(x)F28(x)=F20(x)+F29(x)+F31(x)F29(x)=F30(x)F8(x)F30(x)=F28(x)+F6(x)F31(x)=F27(x)F8(x)