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

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, 7).

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)

Example

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,    2048, ...
[2]   0, 2, 4, 10, 24, 58,  140,  338,    816,  1970,   4756,   11482, ...
[3]   0, 2, 4, 12, 32, 88,  240,  656,   1792,  4896,  13376,   36544, ...
[4]   0, 2, 4, 14, 40, 122, 364,  1094,  3280,  9842,  29524,   88574, ...
[5]   0, 2, 4, 16, 48, 160, 512,  1664,  5376, 17408,  56320,  182272, ...
[6]   0, 2, 4, 18, 56, 202, 684,  2378,  8176, 28242,  97364,  335938, ...
[7]   0, 2, 4, 20, 64, 248, 880,  3248, 11776, 43040, 156736,  571712, ...
[8]   0, 2, 4, 22, 72, 298, 1100, 4286, 16272, 62546, 238996,  915814, ...
[9]   0, 2, 4, 24, 80, 352, 1344, 5504, 21760, 87552, 349184, 1398784, ...

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);

Mathematica

Table[Simplify[((1 + Sqrt[8])^n - (1 - Sqrt[8])^n)/ Sqrt[8]], {n, 0, 24}]

Prog

(PARI) concat(0, Vec(2*x / (1 - 2*x - 7*x^2) + O(x^40))) \\ Colin Barker, Jun 05 2018

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.

Sequence in context: A072355 A134246 A254867 * A220456 A071298 A152104

Adjacent sequences: A305489 A305490 A305491 * A305493 A305494 A305495

Keyword

nonn

Author

Peter Luschny, Jun 02 2018

Status

approved