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%I #30 Sep 08 2022 08:44:36
%S 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,
%T 98,105,112,121,130,139,148,157,166,175,186,197,208,219,230,241,252,
%U 265,278,291,304,317,330,343,358,373,388,403,418,433,448,465,482,499
%N Expansion of (1+x^7)/((1-x)^2*(1-x^7)).
%C Number of 0..n arrays of 8 elements with zero second differences. - _R. H. Hardin_, Nov 16 2011
%H G. C. Greubel, <a href="/A008814/b008814.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,1,-2,1).
%F G.f.: (1+x^7)/((1-x)^2*(1-x^7)).
%F 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
%p 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
%t CoefficientList[Series[(1+x^7)/(1-x)^2/(1-x^7), {x,0,70}], x] (* or *)
%t 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 *)
%o (PARI) a(n)=(n*(n+2)+[7,11,13,13,11,7,1][n%7+1])/7 \\ _Charles R Greathouse IV_, Nov 16 2011
%o (PARI) a(n)=(n*(n+2)+13-6*(n%7==6))\7 \\ _Tani Akinari_, Jul 25 2013
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x^7)/((1-x)^2*(1-x^7)) )); // _G. C. Greubel_, Sep 12 2019
%o (Sage)
%o def A008814_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+x^7)/((1-x)^2*(1-x^7))).list()
%o A008814_list(70) # _G. C. Greubel_, Sep 12 2019
%o (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
%Y 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).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms added by _G. C. Greubel_, Sep 12 2019
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