cp's OEIS Frontend

A192429 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

%I A192429 #20 Jul 14 2023 09:04:36
%S A192429 0,1,4,21,76,329,1256,5157,20216,81505,322924,1293189,5144644,
%T A192429 20550089,81881168,326756661,1302722672,5196774145,20723304532,
%U A192429 82657204533,329642305468,1314745861769,5243461810232,20912613564549,83404589311592
%N A192429 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C A192429 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
%C A192429 The polynomial p(n,x) is defined by ((x+d)^n + (x-d)^n)/2 + ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x^2+4), as at A163762. For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.
%H A192429 G. C. Greubel, <a href="/A192429/b192429.txt">Table of n, a(n) for n = 0..1000</a>
%H A192429 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,10,-6,-9).
%F A192429 From _Colin Barker_, May 12 2014: (Start)
%F A192429 a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).
%F A192429 G.f.: x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)
%F A192429 a(n) = Sum_{k=0..n} T(n,k)*Fibonacci(k), where T(n, k) = [x^k] ( ((x + sqrt(x+4))^n + (x - sqrt(x+4))^n)/2 + ((x + sqrt(x+4))^n - (x - sqrt(x+4))^n)/(2*sqrt(x+4)) ). - _G. C. Greubel_, Jul 13 2023
%e A192429 The first five polynomials p(n,x) and their reductions are as follows:
%e A192429   p(0,x) = 1 -> 1
%e A192429   p(1,x) = 1 + x -> 1 + x
%e A192429   p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x
%e A192429   p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x
%e A192429   p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x.
%e A192429 From these, read A192428 = (1, 1, 5, 11, 57, 185, ...) and a(n) = (0, 1, 4, 21, 76, 329, ...).
%t A192429 (See A192428.)
%t A192429 LinearRecurrence[{2,10,-6,-9}, {0,1,4,21}, 40] (* _G. C. Greubel_, Jul 13 2023 *)
%o A192429 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4) )); // _G. C. Greubel_, Jul 13 2023
%o A192429 (SageMath)
%o A192429 @CachedFunction
%o A192429 def a(n): # a = A192429
%o A192429     if (n<4): return (0,1,4,21)[n]
%o A192429     else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4)
%o A192429 [a(n) for n in range(41)] # _G. C. Greubel_, Jul 13 2023
%Y A192429 Cf. A000045, A163762, A192232, A192428.
%K A192429 nonn
%O A192429 0,3
%A A192429 _Clark Kimberling_, Jun 30 2011