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)
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
KEYWORD
sign,easy
AUTHOR
Creighton Dement, Apr 02 2005
STATUS
approved