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A008815 Expansion of (1+x^8)/((1-x)^2*(1-x^8)). 11

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

%S 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,

%T 86,93,100,107,114,121,128,137,146,155,164,173,182,191,200,211,222,

%U 233,244,255,266,277,288,301,314,327,340,353,366,379,392,407,422

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

%H Vincenzo Librandi, <a href="/A008815/b008815.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,0,0,1,-2,1).

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

%F a(n) = floor( (n*(n+2) + 14 + 4*((n mod 4) - 1)*(-1)^floor(n/4))/8 ). - _Tani Akinari_, Jul 25 2013

%F a(n) = 2*a(n-1) - a(n-2) + a(n-8) - 2*a(n-9) + a(n-10). - _Vincenzo Librandi_, May 14 2019

%p 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

%t 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 *)

%o (PARI) a(n)=(n*(n+2)+14+4*(n%4-1)*(-1)^(n\4))\8 \\ _Tani Akinari_, Jul 25 2013

%o (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

%o (Sage)

%o def A008815_list(prec):

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

%o return P((1+x^8)/((1-x)^2*(1-x^8))).list()

%o A008815_list(50) # _G. C. Greubel_, Sep 12 2019

%o (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

%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), A008814 (m=7), this sequence (m=8), A008816 (m=9), A008817 (m=10).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)