A104237 Expansion of (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4)/((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).

1, -2, 5, -11, 26, -53, 104, -198, 375, -700, 1299, -2401, 4432, -8167, 15038, -27676, 50925, -93686, 172337, -316999, 583078, -1072473, 1972612, -3628226, 6673379, -12274288, 22575967, -41523709, 76374044, -140473803, 258371642

Offset

0,2

Comments

A floretion-generated sequence involving tribonacci numbers. Formula for the g.f. provided by Alec Mihailovs. See sequence A104187 for the sequence generated without using a cyclic transformation (i->j, j->k, k->i), i.e. 1lesforrokseq (refer to FAMP Code).

Floretion Algebra Multiplication Program, FAMP Code: 1lesforcycrokseq[A*B} with A = - .5'ii' + .5'jj' + .5'kk' + .5e and B = + 'kj'. 1vesforcycrokseq[A*B] = A000004. ForType: 1A.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (-2,0,0,0,2,0,0,1).

Formula

G.f.: (1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)).

a(n) = -2*a(n-1) + 2*a(n-5) + a(n-8) for n>7. - Colin Barker, May 21 2019

a(n) = (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*A056594(n-1) - 3*(-1)^n*(2*n+3) - 1). - G. C. Greubel, Jul 08 2022

Mathematica

LinearRecurrence[{-2, 0, 0, 0, 2, 0, 0, 1}, {1, -2, 5, -11, 26, -53, 104, -198}, 40] (* Harvey P. Dale, May 07 2016 *)

Prog

(PARI) Vec((1 - x + x^2)*(1 + x + x^2 - x^3 + 2*x^4) / ((1 - x)*(1 + x)^2*(1 + x^2)*(1 + x - x^2 + x^3)) + O(x^40)) \\ Colin Barker, May 21 2019
(SageMath)
@CachedFunction
def A000073(n):
    if (n<3): return (n//2)
    else: return A000073(n-1) + A000073(n-2) + A000073(n-3)
def A104237(n): return (1/4)*(2*(-1)^n*(A000073(n+5) + A000073(n+4)) - 2*i^(n-1)*(n%2) - 3*(-1)^n*(2*n+3) + 1)
[A104237(n) for n in (0..60)] # G. C. Greubel, Jul 08 2022

Crossrefs

Cf. A000073, A056594, A104187.

Sequence in context: A127075 A053429 A266879 * A085945 A365322 A371662

Adjacent sequences: A104234 A104235 A104236 * A104238 A104239 A104240

Keyword

sign,easy

Author

Creighton Dement, Apr 02 2005

Status

approved