A113067 Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); invert transform gives signed version of tetrahedral numbers A000292.

0, -1, 4, -11, 28, -72, 188, -493, 1292, -3383, 8856, -23184, 60696, -158905, 416020, -1089155, 2851444, -7465176, 19544084, -51167077, 133957148, -350704367, 918155952, -2403763488, 6293134512, -16475640049, 43133785636, -112925716859, 295643364940, -774004377960

Offset: 0

Author

Creighton Dement, Oct 13 2005

Keywords

• easy
• sign

Comments

Invert(a(n)) gives (0, -1, 4, -10, 20, -35, ...) = A000292 (with alternating signs).

Binomial(a(n)) gives (0, -1, 2, -2, 4, -7, 10, ...) = A094686 (with alternating signs, from 2nd term).

Floretion Algebra Multiplication Program, FAMP Code: 2basei[C*F]; C = - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki'; F = + .5'i + .5'ii' + .5'ij' + .5'ik'

References

• Creighton Dement, Floretion Integer Sequences (work in progress).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-4,-5,-4,-1).

Crossrefs

Cf. A000292, A113067, A113068, A094686, A001906, A109961, A290890.

Programs

Mathematica

-x/((x^2+x+1)*(x^2+3*x+1)) + O[x]^30 // CoefficientList[#, x]& (* Jean-François Alcover, Jun 15 2017 *)

PARI

concat(0, Vec(-x / ((1 + x + x^2)*(1 + 3*x + x^2)) + O(x^30))) \\ Colin Barker, May 11 2019

SageMath

[((lucas_number1(n,3,1)-lucas_number1(n,1,1)))/(-2) for n in range(1,32)] # Zerinvary Lajos, Jul 06 2008

Formula

a(n) + a(n+1) + a(n+2) = (-1)^n *A001906(n+2) = (-1)^n*F(2n+4).

a(n) + 3*a(n+1) + 3*a(n+2) + a(n+3) = ((-1)^(n+1))*A109961(n+2).

(|a(n)|) = A290890(n) for n >= 0, this being the p-INVERT of (1,2,3,4,...), where p(S) = 1 - S^2. - Clark Kimberling, Aug 21 2017

a(n) = -4*a(n-1) - 5*a(n-2) - 4*a(n-3) - a(n-4) for n > 3. - Colin Barker, May 11 2019

2*a(n) = (-1)^n*A001906(n+1) - A049347(n). - R. J. Mathar, Sep 20 2020