A008815 Expansion of (1+x^8)/((1-x)^2*(1-x^8)).

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 37, 42, 47, 52, 57, 62, 67, 72, 79, 86, 93, 100, 107, 114, 121, 128, 137, 146, 155, 164, 173, 182, 191, 200, 211, 222, 233, 244, 255, 266, 277, 288, 301, 314, 327, 340, 353, 366, 379, 392, 407, 422

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Crossrefs

Cf. Expansions of the form (1+x^m)/((1-x)^2*(1-x^m)): A000290 (m=1), A000982 (m=2), A008810 (m=3), A008811 (m=4), A008812 (m=5), A008813 (m=6), A008814 (m=7), this sequence (m=8), A008816 (m=9), A008817 (m=10).

Programs

GAP

a:=[1,2,3,4,5,6,7,8,11,14];; for n in [11..50] do a[n]:=2*a[n-1] -a[n-2]+a[n-8]-2*a[n-9]+a[n-10]; od; a; # G. C. Greubel, Sep 12 2019

Magma

I:=[1,2,3,4,5,6,7,8,11,14]; [n le 10 select I[n] else 2*Self(n-1) -Self(n-2)+Self(n-8)-2*Self(n-9)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, May 14 2019

Maple

seq(coeff(series((1+x^8)/((1-x)^2*(1-x^8)), x, n+1), x, n), n = 0..50); # G. C. Greubel, Sep 12 2019

Mathematica

CoefficientList[Series[(1+x^8)/(1-x)^2/(1-x^8), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,8,11,14}, 50] (* Harvey P. Dale, Dec 17 2016 *)

PARI

a(n)=(n*(n+2)+14+4*(n%4-1)*(-1)^(n\4))\8  \\ Tani Akinari, Jul 25 2013

Sage

def A008815_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^8)/((1-x)^2*(1-x^8))).list()
A008815_list(50) # G. C. Greubel, Sep 12 2019

Formula

G.f.: (1 + x^8)/((1 - x)^3*(1 + x)*(1 + x^2)*(1 + x^4)).

a(n) = floor( (n*(n+2) + 14 + 4*((n mod 4) - 1)*(-1)^floor(n/4))/8 ). - Tani Akinari, Jul 25 2013

a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - Vincenzo Librandi, May 14 2019