A192422 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
0, 1, 1, 5, 7, 20, 35, 83, 161, 355, 720, 1541, 3185, 6733, 14027, 29500, 61663, 129403, 270865, 567911, 1189440, 2492905, 5222449, 10943813, 22928815, 48044900, 100665083, 210927155, 441948689, 926020171, 1940274000, 4065458669, 8518311809
Offset: 0
Keywords
Examples
The first five polynomials p(n,x) and their reductions are as follows: p(0,x) = 2 -> 2 p(1,x) = x -> x p(2,x) = 2 + x^2 -> 3 + x p(3,x) = 3*x + x^3 -> 1 + 5*x p(4,x) = 2 + 4*x^2 + x^4 -> 8 + 7*x. From these, read A192421 = (2, 0, 3, 1, 8, ...) and a(n) = (0, 1, 1, 5, 7, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
- Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x^2)/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 11 2023 -
Mathematica
(See A192421.) LinearRecurrence[{1,3,-1,-1}, {0,1,1,5}, 40] (* G. C. Greubel, Jul 11 2023 *)
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Maxima
a(n):=n*sum((binomial(n-i-1,i))/(n-2*i)*fib(n-2*i),i,0,(n-1)/2); /* Vladimir Kruchinin, Mar 20 2016 */
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SageMath
@CachedFunction def a(n): # a = A192422 if (n<4): return (0,1,1,5)[n] else: return a(n-1) +3*a(n-2) -a(n-3) -a(n-4) [a(n) for n in range(41)] # G. C. Greubel, Jul 11 2023
Formula
From Colin Barker, May 12 2014: (Start)
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
G.f.: x*(1 + x^2)/(1 - x - 3*x^2 + x^3 + x^4). (End)
From Vladimir Kruchinin, Mar 20 2016: (Start)
G.f.: ((1+x^2)/(1-x^2)) * F(x/(1-x^2)), where F(x) is g.f. of Fibonacci numbers (A000045).
a(n) = n*Sum_{i=0..floor((n-1)/2)} (binomial(n-i-1,i)/(n-2*i))*Fibonacci(n-2*i). (End)
a(n) = Sum_{j=0..n} T(n, j)*Fibonacci(j), where T(n, k) = [x^k] ((x + sqrt(x^2+4))^n + (x - sqrt(x^2+4))^n)/2^n. - G. C. Greubel, Jul 11 2023
Comments