This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A205492 #25 Oct 24 2024 23:52:10
%S A205492 1,7,31,109,334,937,2475,6267,15393,36976,87369,203915,471546,1082849,
%T A205492 2473535,5627684,12765052,28887838,65260270,147233926,331842395,
%U A205492 747355066,1682185342,3784718431,8512408455,19141037360,43032743620
%N A205492 Expansion of (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)).
%C A205492 See array A205497 regarding association of this sequence with generating functions for the rows of the array form of A050446.
%H A205492 G. C. Greubel, <a href="/A205492/b205492.txt">Table of n, a(n) for n = 0..1000</a>
%H A205492 L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H A205492 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,12,15,-26,3,13,-5,-2,1).
%F A205492 a(n) = 7*a(n-1) - 17*a(n-2) + 12*a(n-3) + 15*a(n-4) - 26*a(n-5) + 3*a(n-6) + 13*a(n-7) - 5*a(n-8) - 2*a(n-9) + a(n-10), n>9, {a(m)} = {1, 7, 31, 109, 334, 937, 2475, 6267, 15393, 36976}, m=0,...,9.
%F A205492 CONJECTURE 1. a(n) = M_{n,2} = M_{2,n}, where M = A205497.
%F A205492 CONJECTURE 2. lim_{n->oo} a(n+1)/a(n) = (2*cos(Pi/7))^2-1 = A116425-1 = spectral radius of the 3 X 3 unit-primitive matrix (see [Jeffery]) A_{7,2} = [0,0,1; 0,1,1; 1,1,1].
%p A205492 seq(coeff(series((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Jan 04 2020
%t A205492 LinearRecurrence[{7,-17,12,15,-26,3,13,-5,-2,1},{1,7,31,109,334,937,2475,6267, 15393,36976},30] (* _Harvey P. Dale_, Mar 26 2013 *)
%t A205492 CoefficientList[Series[(1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)), {x,0,30}], x] (* _G. C. Greubel_, Jan 04 2020 *)
%o A205492 (PARI) my(x='x+O('x^30)); Vec((1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3))) \\ _G. C. Greubel_, Jan 04 2020
%o A205492 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+ x^3)) )); // _G. C. Greubel_, Jan 04 2020
%o A205492 (Sage)
%o A205492 def A205492_list(prec):
%o A205492 P.<x> = PowerSeriesRing(ZZ, prec)
%o A205492 return P( (1-x^2-x^3-x^4+x^5)/((1-x)^3*(1-x-x^2)^2*(1-2*x-x^2+x^3)) ).list()
%o A205492 A205492_list(30) # _G. C. Greubel_, Jan 04 2020
%Y A205492 Cf. A050446, A050447, A205497.
%K A205492 nonn
%O A205492 0,2
%A A205492 _L. Edson Jeffery_, Jan 28 2012