A192429 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
0, 1, 4, 21, 76, 329, 1256, 5157, 20216, 81505, 322924, 1293189, 5144644, 20550089, 81881168, 326756661, 1302722672, 5196774145, 20723304532, 82657204533, 329642305468, 1314745861769, 5243461810232, 20912613564549, 83404589311592
Offset: 0
Keywords
Examples
The first five polynomials p(n,x) and their reductions are as follows: p(0,x) = 1 -> 1 p(1,x) = 1 + x -> 1 + x p(2,x) = 4 + 3*x + x^2 -> 5 + 4*x p(3,x) = 4 + 13*x + 6*x^2 + x^3 -> 11 + 21*x p(4,x) = 16 + 24*x + 29*x^2 + 10*x^3 + x^4 -> 57 + 76*x. From these, read A192428 = (1, 1, 5, 11, 57, 185, ...) and a(n) = (0, 1, 4, 21, 76, 329, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,10,-6,-9).
Programs
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Magma
R
:=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 -
Mathematica
(See A192428.) LinearRecurrence[{2,10,-6,-9}, {0,1,4,21}, 40] (* G. C. Greubel, Jul 13 2023 *)
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SageMath
@CachedFunction def a(n): # a = A192429 if (n<4): return (0,1,4,21)[n] else: return 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4) [a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023
Formula
From Colin Barker, May 12 2014: (Start)
a(n) = 2*a(n-1) + 10*a(n-2) - 6*a(n-3) - 9*a(n-4).
G.f.: x*(1+2*x+3*x^2)/(1-2*x-10*x^2+6*x^3+9*x^4). (End)
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
Comments