A087935 Perrin sequence of order 5.
5, 0, 0, 0, 4, 5, 0, 0, 4, 9, 5, 0, 4, 13, 14, 5, 4, 17, 27, 19, 9, 21, 44, 46, 28, 30, 65, 90, 74, 58, 95, 155, 164, 132, 153, 250, 319, 296, 285, 403, 569, 615, 581, 688, 972, 1184, 1196, 1269, 1660, 2156, 2380, 2465, 2929, 3816, 4536, 4845, 5394, 6745, 8352, 9381
Offset: 0
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).
Programs
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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
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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
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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
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Mathematica
LinearRecurrence[{0,0,0,1,1},{5,0,0,0,4},60] (* Harvey P. Dale, Oct 03 2016 *) -
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
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Sage
((5-x^4)/(1-x^4-x^5)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019
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)
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