A008814 Expansion of (1+x^7)/((1-x)^2*(1-x^7)).

1, 2, 3, 4, 5, 6, 7, 10, 13, 16, 19, 22, 25, 28, 33, 38, 43, 48, 53, 58, 63, 70, 77, 84, 91, 98, 105, 112, 121, 130, 139, 148, 157, 166, 175, 186, 197, 208, 219, 230, 241, 252, 265, 278, 291, 304, 317, 330, 343, 358, 373, 388, 403, 418, 433, 448, 465, 482, 499

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of 0..n arrays of 8 elements with zero second differences. - R. H. Hardin, Nov 16 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,1,-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), this sequence (m=7), A008815 (m=8), A008816 (m=9), A008817 (m=10).

Programs

GAP

a:=[1,2,3,4,5,6,7,10,13];; for n in [10..70] do a[n]:=2*a[n-1]-a[n-2]+a[n-7]-2*a[n-8]+a[n-9]; od; a; # G. C. Greubel, Sep 12 2019

Magma

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^7)/((1-x)^2*(1-x^7)) )); // G. C. Greubel, Sep 12 2019

Maple

seq(coeff(series((1+x^7)/((1-x)^2*(1-x^7)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 12 2019

Mathematica

CoefficientList[Series[(1+x^7)/(1-x)^2/(1-x^7), {x,0,70}], x] (* or *)
LinearRecurrence[{2,-1,0,0,0,0,1,-2,1}, {1,2,3,4,5,6,7,10,13}, 70] (* Harvey P. Dale, Dec 18 2012 *)

PARI

a(n)=(n*(n+2)+[7,11,13,13,11,7,1][n%7+1])/7 \\ Charles R Greathouse IV, Nov 16 2011

PARI

a(n)=(n*(n+2)+13-6*(n%7==6))\7  \\ Tani Akinari, Jul 25 2013

Sage

def A008814_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x^7)/((1-x)^2*(1-x^7))).list()
A008814_list(70) # G. C. Greubel, Sep 12 2019

Formula

G.f.: (1+x^7)/((1-x)^2*(1-x^7)).

a(n) = 2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9). - R. H. Hardin, Nov 16 2011

Extensions

More terms added by G. C. Greubel, Sep 12 2019