A048788 -id:A048788 - cp's OEIS frontend

A137408 Triangular sequence from coefficients of a switched even -odd polynomial recursion: odd:p(x,n)=2xp(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2x)p(x, n - 1) - p(x, n - 2);.

1, 0, 2, -1, 2, -4, 0, -4, 4, -8, 1, -6, 16, -16, 16, 0, 6, -16, 40, -32, 32, -1, 12, -44, 88, -128, 96, -64, 0, -8, 40, -128, 208, -288, 192, -128, 1, -20, 100, -296, 592, -800, 832, -512, 256, 0, 10, -80, 328, -800, 1472, -1792, 1792, -1024, 512, -1

Offset: 1

Roger L. Bagula, Apr 14 2008

A048788 gives the row sums: {1, 2, -3, -8, 11, 30, -41, -112, 153, 418, -571}

			{1},
{0, 2},
{-1, 2, -4},
{0, -4, 4, -8},
{1, -6, 16, -16,16},
{0, 6, -16, 40, -32, 32},
{-1, 12, -44, 88, -128, 96, -64},
{0, -8, 40, -128, 208, -288, 192, -128},
{1, -20, 100, -296, 592, -800, 832, -512, 256},
{0,10, -80, 328, -800,1472, -1792, 1792, -1024, 512},
{-1, 30, -200, 784, -2048, 3872, -5568, 5888, -4864, 2560, -1024}

Cf. A048788.

Clear[p, x, a] p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = 2*x; p[x_, n_] := p[x, n] = If[Mod[n, 2] == 1, 2*x*p[x, n - 1] - p[x, n - 2], (1 - 2*x)*p[x, n - 1] - p[x, n - 2]]; Table[ExpandAll[p[x, n]], {n, 0, 10}]; a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a]

p(x,-1)=0;p(x,0)=1;p(x,1]=2*x; odd:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);

A142880 a(n) = 7*a(n-3) - a(n-6).

Original entry on oeis.org

0, 1, 2, 3, 8, 13, 21, 55, 89, 144, 377, 610, 987, 2584, 4181, 6765, 17711, 28657, 46368, 121393, 196418, 317811, 832040, 1346269, 2178309, 5702887, 9227465, 14930352, 39088169, 63245986, 102334155, 267914296, 433494437, 701408733, 1836311903

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 28 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-1).

Cf. A119016, A002965, A002530, A048788.

a[0] = 0; a[1] = 1;
a[n_] := a[n] = If[Mod[n, 3] == 1, 2*a[n - 1] + a[n - 2], If[Mod[n, 3] == 0, a[n - 1] + a[n - 2], 2*a[n - 1] - a[n - 2]]];
Table[a[n], {n, 0, 50}]
LinearRecurrence[{0,0,7,0,0,-1},{0,1,2,3,8,13},40] (* Harvey P. Dale, Jul 17 2021 *)

G.f.: -x*(1+x)*(x^3 - 2*x^2 - x - 1) / ( 1 - 7*x^3 + x^6 ).
a(3n) = A033888(n).
a(3n+1) = A033890(n).
a(3n+2)= A033891(n).
a(n) = 2*a(n-1) + a(n-2) if n == 1 (mod 3).
a(n) = a(n-1) + a(n-2) if n == 0 (mod 3).
a(n) = 2*a(n-1) - a(n-2) if n == 2 (mod 3).

cp's OEIS Frontend

A137408 Triangular sequence from coefficients of a switched even -odd polynomial recursion: odd:p(x,n)=2xp(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2x)p(x, n - 1) - p(x, n - 2);.

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A142880 a(n) = 7*a(n-3) - a(n-6).

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cp's OEIS Frontend

A137408 Triangular sequence from coefficients of a switched even -odd polynomial recursion: odd:p(x,n)=2*x*p(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2*x)*p(x, n - 1) - p(x, n - 2);.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A142880 a(n) = 7*a(n-3) - a(n-6).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A137408 Triangular sequence from coefficients of a switched even -odd polynomial recursion: odd:p(x,n)=2xp(x, n - 1) - p(x, n - 2); even:p(x,n)=(1 - 2x)p(x, n - 1) - p(x, n - 2);.