A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

0, 0, 0, 1, 3, 5, 5, 0, -13, -34, -55, -55, 0, 144, 377, 610, 610, 0, -1597, -4181, -6765, -6765, 0, 17711, 46368, 75025, 75025, 0, -196418, -514229, -832040, -832040, 0, 2178309, 5702887, 9227465, 9227465, 0, -24157817, -63245986, -102334155, -102334155

Offset: 0

Author

Paul Curtz, May 04 2008

Keywords

• sign

Comments

Obeys also the recurrence a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5), so the sequence is identical to its fifth differences (cf. A135356). a(n) = A138110(0,n): if A138110 is interpreted as an array with five rows, this is the top row.

The first differences are represented by A100334(n-1).

The 2nd differences are represented by A103311(n).

The 3rd differences are essentially represented by -A138003(n-2).

The 4th differences are represented by -A105371(n).

A102312 contains the absolute values of the terms which occur in pairs, for example a(5)=a(6)=5=A102312(1), a(10)=a(11)= -55 = -A102312(2).

Inverse BINOMIAL transform yields two zeros followed by A105384. - R. J. Mathar, Jul 04 2008

Links

• Index entries for linear recurrences with constant coefficients, signature (3, -4, 2, -1).

Crossrefs

Cf. A138003, A103311, A105371.

Programs

Mathematica

CoefficientList[Series[x^3/(1-3x+4x^2-2x^3+x^4),{x,0,45}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{0,0,0,1},45] (* Harvey P. Dale, Jun 22 2011 *)

Formula

O.g.f.: x^3/(1-3x+4x^2-2x^3+x^4). - R. J. Mathar, Jul 04 2008

Extensions

Edited and extended by R. J. Mathar, Jul 04 2008