A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.
0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, 5591040, 22372352, 89473024, 357924864, 1431633920, 5726666752, 22906404864, 91626143744, 366503526400, 1466016202752, 5864060616704, 23456250855424, 93824986644480
Offset: 1
Keywords
Examples
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) = 2 + x + 3*x^2 -> 8 + 4*x. p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x. p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x. From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,8).
Programs
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Magma
[(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // G. C. Greubel, Feb 19 2023
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Maple
seq(4^n*(1-(-1/2)^n)/3, n=0..24); # Peter Luschny, Oct 02 2019
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Mathematica
q[x_]:= x+2; d= Sqrt[x+2]; 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,30}]; Table[Coefficient[Part[t,n], x, 0], {n,30}] (* abs value of A083086 *) Table[Coefficient[Part[t,n], x, 1], {n,30}] (* 2*A003683 *) Table[Coefficient[Part[t,n]/2, x, 1], {n,30}] (* A003683 *) LinearRecurrence[{2,8}, {0,2}, 40] (* G. C. Greubel, Feb 19 2023 *)
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SageMath
[(4^(n-1) - (-2)^(n-1))/3 for n in range(1,41)] # G. C. Greubel, Feb 19 2023
Formula
Conjectures from Colin Barker, May 12 2014: (Start)
a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).
a(n) = 2*a(n-1) + 8*a(n-2).
G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).
a(n) = 4^n*(1 - (-1/2)^n)/3. - Peter Luschny, Oct 02 2019
E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - G. C. Greubel, Feb 19 2023
Comments