OFFSET
0,1
COMMENTS
If p is prime, p divides a(p).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1).
FORMULA
a(n) = a(n-4) + a(n-5), with a(0)=5, a(1)=a(2)=a(3)=0.
a(n) = (x_1)^n + (x_2)^n + (x_3)^n + (x_4)^n + (x_5)^n where (x_i) 1 <= i <= 5 are the roots of x^5=x+1.
G.f.: (5 - x^4)/(1 -x^4 -x^5). - Colin Barker, Jun 16 2013
a(0) = 5 and a(n) = n*Sum_{k=1..floor(n/4)} binomial(k,n-4*k)/k for n > 0. - Seiichi Manyama, Mar 04 2019
From Aleksander Bosek, Mar 06 2019: (Start)
a((s+5)*n + m) = Sum_{j=0..n} binomial(n-j,j)*a(s*n+j+m) for all s > 0, m > 0.
a(m) = Sum_{j=0..n} (-1)^(n-j)*binomial(n-j,j)*a(m+n+4*j) for all m > 0. (End)
MAPLE
seq(coeff(series((x^4-5)/(x^5+x^4-1), x, n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Mar 06 2019
MATHEMATICA
LinearRecurrence[{0, 0, 0, 1, 1}, {5, 0, 0, 0, 4}, 60] (* Harvey P. Dale, Oct 03 2016 *)
PROG
(GAP) a:=[5, 0, 0, 0, 4];; for n in [6..60] do a[n]:=a[n-4]+a[n-5]; od; Print(a); # Muniru A Asiru, Mar 06 2019
(PARI) my(x='x+O('x^60)); Vec((5-x^4)/(1-x^4-x^5)) \\ G. C. Greubel, Mar 06 2019
(PARI) polsym(x^5-x-1, 66) \\ Joerg Arndt, Mar 10 2019
(Magma) I:=[5, 0, 0, 0, 4]; [n le 5 select I[n] else Self(n-4) +Self(n-5): n in [1..60]]; // G. C. Greubel, Mar 06 2019
(SageMath) ((5-x^4)/(1-x^4-x^5)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Oct 27 2003
STATUS
approved
