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

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A305492 a(n) = ((1 + y)^n - (1 - y)^n)/y with y = sqrt(8). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
KEYWORD
nonn
AUTHOR
Peter Luschny, Jun 02 2018
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.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 19 15:11 EDT 2024. Contains 371794 sequences. (Running on oeis4.)