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 A192386 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments. 3
 1, 0, 8, 8, 96, 224, 1408, 4608, 22784, 86016, 386048, 1548288, 6676480, 27467776, 116490240, 484409344, 2040135680, 8521777152, 35786063872, 149761818624, 628140015616, 2630784909312, 11028578435072, 46205266558976, 193656954814464 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+5). 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 = 1..1000 Index entries for linear recurrences with constant coefficients, signature (2,12,-8,-16). FORMULA From Colin Barker, May 11 2014: (Start) a(n) = 2*a(n-1) + 12*a(n-2) - 8*a(n-3) - 16*a(n-4). G.f.: x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4). (End) From G. C. Greubel, Jul 10 2023: (Start) T(n, k) = [x^k] ((x+sqrt(x+5))^n - (x-sqrt(x+5))^n)/(2*sqrt(x+5)). a(n) = Sum_{k=0..n-1} T(n, k)*Fibonacci(k-1). (End) EXAMPLE The first five polynomials p(n,x) and their reductions are as follows: p(0, x) = 1 -> 1 p(1, x) = 2*x -> 2*x p(2, x) = 3 + x + 3*x^2 -> 8 + 4*x p(3, x) = 12*x + 4*x^2 + 4*x^3 -> 8 + 32*x p(4, x) = 9 + 6*x + 31*x^2 + 10*x^3 + 5*x^4 -> 96 + 96*x. From these, read A192386 = (1, 0, 8, 8, 96, ...) and A192387 = (0, 2, 4, 32, 96, ...). MATHEMATICA q[x_]:= x+1; d= Sqrt[x+5]; p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2*d); (* suggested by A162517 *) Table[Expand[p[n, x]], {n, 6}] reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[ FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]; Table[Coefficient[Part[t, n], x, 0], {n, 30}] (* A192386 *) Table[Coefficient[Part[t, n], x, 1], {n, 30}] (* A192387 *) Table[Coefficient[Part[t, n]/2, x, 1], {n, 30}] (* A192388 *) LinearRecurrence[{2, 12, -8, -16}, {1, 0, 8, 8}, 40] (* G. C. Greubel, Jul 10 2023 *) PROG (Magma) R:=PowerSeriesRing(Integers(), 41); Coefficients(R!( x*(1-2*x-4*x^2)/(1-2*x-12*x^2+8*x^3+16*x^4) )); // G. C. Greubel, Jul 10 2023 (SageMath) @CachedFunction def a(n): # a = A192386 if (n<5): return (0, 1, 0, 8, 8)[n] else: return 2*a(n-1) +12*a(n-2) -8*a(n-3) -16*a(n-4) [a(n) for n in range(1, 41)] # G. C. Greubel, Jul 10 2023 CROSSREFS Cf. A083087, A192232, A192387, A192388. Sequence in context: A082798 A286068 A228071 * A119932 A270118 A270150 Adjacent sequences: A192383 A192384 A192385 * A192387 A192388 A192389 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 30 2011 STATUS approved

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Last modified September 13 03:33 EDT 2024. Contains 375857 sequences. (Running on oeis4.)