A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8).
0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, 915814, 3504600, 13419898, 51371996, 196683278, 752970528, 2882724002, 11036241700, 42251551414, 161756794728, 619274449354, 2370846461804, 9076614069086, 34749153370800
Offset: 0
Examples
Array ((1+y)^n - (1-y)^n)/y with y = sqrt(k). [k\n] [1] 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ... [2] 0, 2, 4, 10, 24, 58, 140, 338, 816, 1970, 4756, ... [3] 0, 2, 4, 12, 32, 88, 240, 656, 1792, 4896, 13376, ... [4] 0, 2, 4, 14, 40, 122, 364, 1094, 3280, 9842, 29524, ... [5] 0, 2, 4, 16, 48, 160, 512, 1664, 5376, 17408, 56320, ... [6] 0, 2, 4, 18, 56, 202, 684, 2378, 8176, 28242, 97364, ... [7] 0, 2, 4, 20, 64, 248, 880, 3248, 11776, 43040, 156736, ... [8] 0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996, ... [9] 0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, ...
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,7).
Crossrefs
Let f(n, y) = ((1 + y)^n - (1 - y)^n)/y.
f(n, 1 ) = A000079(n);
f(n, sqrt(2)) = A163271(n+1);
f(n, sqrt(3)) = A028860(n+2);
f(n, 2 ) = A152011(n) for n>0;
f(n, sqrt(5)) = A103435(n);
f(n, sqrt(6)) = A083694(n);
f(n, sqrt(7)) = A274520(n);
f(n, sqrt(8)) = a(n);
f(n, 3 ) = A192382(n+1);
Cf. A305491.
Equals 2 * A015519.
Programs
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Maple
egf := (n,x) -> 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n): ser := series(egf(8,x), x, 26): seq(n!*coeff(ser,x, n), n=0..24);
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Mathematica
Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}] -
PARI
concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ Colin Barker, Jun 05 2018
Formula
E.g.f.: 2*exp(x)*sinh(sqrt(n)*x)/sqrt(n) for n = 8.
From Colin Barker, Jun 02 2018: (Start)
G.f.: 2*x / (1 - 2*x - 7*x^2).
a(n) = 2*a(n-1) + 7*a(n-2) for n>1.
(End)
Comments