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%I #12 Mar 14 2023 03:49:17
%S 0,1,8,145,505,9728,187561,654841,12626288,243453385,849982465,
%T 16388911448,316002305521,1103276584081,21272794432568,
%U 410170749112225,1432052156154025,27612070784561168,532401316345361881,1858802595411339721,35840446605565962848
%N a(n) = 1298*a(n-3) - a(n-6) - 648, for n>6, with a(0)=0, a(1)=1, a(2)=8, a(3)=145, a(4)=505, a(5)=9728, a(6)=187561.
%C It appears this sequence gives all nonnegative m such that 13*m^2 - 13*m + 1 is a square and that a(n+1) = A104240(n) + 1. (A104240 is nonnegative n such that 13*n^2 + 13*n + 1 is a square.)
%H G. C. Greubel, <a href="/A105046/b105046.txt">Table of n, a(n) for n = 0..950</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1298,-1298,0,-1,1).
%F a(n) = 1298*a(n-3) - a(n-6) - 648 for n > 6.
%F G.f.: x*(1+7*x+137*x^2-938*x^3+137*x^4+7*x^5+x^6) / ((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)). - _R. J. Mathar_, Sep 09 2008
%t CoefficientList[Series[x (1+7x+137x^2-938x^3+137x^4+7x^5+x^6)/((1-x) (1-11x+x^2)(1+11x+120x^2+11x^3+x^4)),{x,0,30}],x] (* or *) LinearRecurrence[{1,0,1298,-1298,0,-1,1},{0,1,8,145,505,9728,187561, 654841},30] (* _Harvey P. Dale_, Jun 12 2012 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+7*x+137*x^2-938*x^3+137*x^4+7*x^5+x^6) / ((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)) )); // _G. C. Greubel_, Mar 14 2023
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A105046
%o if (n<8): return (0,1,8,145,505,9728,187561,654841)[n]
%o else: return a(n-1) +1298*a(n-3) -1298*a(n-4) -a(n-6) +a(n-7)
%o [a(n) for n in range(41)] # _G. C. Greubel_, Mar 14 2023
%Y Cf. A104240.
%K nonn
%O 0,3
%A _Gerald McGarvey_, Apr 03 2005
%E More terms from _Harvey P. Dale_, Jun 12 2012
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