%I #35 Sep 08 2022 08:46:05

%S 4,16,96,512,2560,24576,229376,2097152,17956864,142606336,1107296256,

%T 8589934592,67612180480,541165879296,4363686772736,35184372088832,

%U 282583078273024,2260595906707456,18049582881570816,144115188075855872,1151793405676748800

%N Number of solutions to Sum_{i=1...n} x_i^2 == 1 (mod 8) with x_i in 0..7.

%H Colin Barker, <a href="/A229138/b229138.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (16,-96,256,-256,4096,-24576,65536).

%F G.f.: 4*x*(1 -12*x +56*x^2 -128*x^3 +128*x^4 -1024*x^5 +2048*x^6)/((1-8*x)*(1+256*x^4)*(1-8*x+32*x^2)). - _Colin Barker_, Nov 10 2014

%p seq(coeff(series(4*x*(1 -12*x +56*x^2 -128*x^3 +128*x^4 -1024*x^5 +2048*x^6)/( (1-8*x)*(1+256*x^4)*(1-8*x+32*x^2)), x, n+1), x, n), n = 1..30); # _G. C. Greubel_, Dec 21 2019

%t a[n_]:= a[n]= 16a[n-1] -96a[n-2] +256a[n-3] -256a[n-4] +4096a[n-5] -24576 a[n-6] +65536 a[n-7]; Do[a[i]={4, 16, 96, 512, 2560, 24576, 229376}[[i]], {i,7}]; Array[a, 33]

%o (PARI) Vec(4*x*(1-12*x+56*x^2-128*x^3+128*x^4-1024*x^5+2048*x^6)/((1-8*x)*(1+256*x^4)*(1-8*x+32*x^2)) + O(x^30)) \\ _Colin Barker_, Nov 10 2014

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 4*x*(1 -12*x +56*x^2 -128*x^3 +128*x^4 -1024*x^5 +2048*x^6)/((1-8*x)*(1+256*x^4)*(1-8*x+32*x^2)) )); // _G. C. Greubel_, Dec 21 2019

%o (Sage)

%o def A229138_list(prec):

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

%o return P( 4*x*(1 -12*x +56*x^2 -128*x^3 +128*x^4 -1024*x^5 +2048*x^6)/( (1-8*x)*(1+256*x^4)*(1-8*x+32*x^2)) ).list()

%o a=A229138_list(30); a[1:] # _G. C. Greubel_, Dec 21 2019

%Y Cf. A101990, A228920, A228921, A229136, A318609, A318610, A330607, A330619.

%K nonn,easy

%O 1,1

%A _José María Grau Ribas_, Sep 15 2013