The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A192427 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments. 2
 0, 1, 1, 8, 11, 45, 80, 251, 517, 1432, 3195, 8317, 19360, 48827, 116213, 288360, 694331, 1708397, 4138480, 10138363, 24636645, 60217912, 146570491, 357833309, 871703360, 2126857275, 5183425493, 12642971912, 30819571387, 75160150861 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The polynomial p(n,x) is defined by ((x+d)/2)^n + ((x-d)/2)^n, where d = sqrt(x^2+8). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (1,5,-2,-4). FORMULA From Colin Barker, May 12 2014: (Start) a(n) = a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4). G.f.: x*(1+2*x^2)/(1-x-5*x^2+2*x^3+4*x^4). (End) a(n) = Sum_{k=0..n} T(n, k)*Fibonacci(k), where T(n, k) = [x^k] ((x + sqrt(x^2+8))^n + (x - sqrt(x^2+8))^n)/2^n. - G. C. Greubel, Jul 13 2023 EXAMPLE 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) = 4 + x^2 -> 5 + x p(3,x) = 6*x + x^3 -> 1 + 8*x p(4,x) = 8 + 8*x^2 + x^4 -> 18 + 11*x. From these, read A192426 = (2, 0, 5, 1, 18, ...) and a(n) = (0, 1, 1, 8, 11, ...). MATHEMATICA (See A192426.) LinearRecurrence[{1, 5, -2, -4}, {0, 1, 1, 8}, 40] (* G. C. Greubel, Jul 13 2023 *) PROG (Magma) R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+2*x^2)/(1-x-5*x^2+2*x^3+4*x^4) )); // G. C. Greubel, Jul 13 2023 (SageMath) @CachedFunction def a(n): # a = A192427 if (n<4): return (0, 1, 1, 8)[n] else: return a(n-1) + 5*a(n-2) - 2*a(n-3) - 4*a(n-4) [a(n) for n in range(41)] # G. C. Greubel, Jul 13 2023 CROSSREFS Cf. A000045, A192232, A192426. Sequence in context: A056873 A303883 A304889 * A258285 A166673 A107788 Adjacent sequences: A192424 A192425 A192426 * A192428 A192429 A192430 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 30 2011 EXTENSIONS Typo in name corrected by G. C. Greubel, Jul 13 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 6 18:55 EDT 2024. Contains 374981 sequences. (Running on oeis4.)