A008768 - OEIS

%I #24 Sep 08 2022 08:44:36

%S 1,1,2,3,5,6,9,12,16,20,26,32,40,48,58,69,82,95,111,128,147,167,190,

%T 214,241,269,300,333,369,406,447,490,536,584,636,690,748,808,872,939,

%U 1010,1083,1161,1242,1327,1415,1508,1604,1705,1809,1918,2031,2149,2270

%N Expansion of (1+x^7)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

%H G. C. Greubel, <a href="/A008768/b008768.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,-1,1,-1,-1,1,-1,2,-1).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) - a(n-5) + a(n-6) - a(n-7) + 2*a(n-8) - a(n-9); a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=6, a(6)=9, a(7)=12, a(8)=16. - _Harvey P. Dale_, Jul 06 2011

%F a(n) = floor((2*n^3 + 9*n^2 + 72*n + 160)/144). - _Tani Akinari_, May 13 2014

%p seq(coeff(series((1+x^7)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)), x, n+1), x, n), n = 0 .. 60); # _G. C. Greubel_, Sep 10 2019

%t CoefficientList[Series[(1+x^7)/(1-x)/(1-x^2)/(1-x^3)/(1-x^4), {x,0,50}], x] (* or *) LinearRecurrence[{2,-1,1,-1,-1,1,-1,2,-1}, {1,1,2,3,5,6,9, 12,16}, 50] (* _Harvey P. Dale_, Jul 06 2011 *)

%o (PARI) my(x='x+O('x^60)); Vec((1+x^7)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))) \\ _G. C. Greubel_, Sep 10 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1+x^7)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) )); // _G. C. Greubel_, Sep 10 2019

%o (Sage)

%o def A008768_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P((1+x^7)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4))).list()

%o A008768_list(60) # _G. C. Greubel_, Sep 10 2019

%o (GAP) a:=[1,1,2,3,5,6,9,12,16];; for n in [10..60] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-a[n-4]-a[n-5]+a[n-6]-a[n-7]+2*a[n-8]-a[n-9]; od; a; # _G. C. Greubel_, Sep 10 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_