A267652 a(n) = 4*a(n - 1) + 4*a(n - 2) for n>1, a(0)=2, a(1)=3.

2, 3, 20, 92, 448, 2160, 10432, 50368, 243200, 1174272, 5669888, 27376640, 132186112, 638251008, 3081748480, 14879997952, 71846985728, 346907934720, 1675019681792, 8087710466048, 39050920591360, 188554524229632, 910421779283968, 4395905214054400, 21225307973353472, 102484852749631488

Offset: 0

Author

Ilya Gutkovskiy, Jan 19 2016

Keywords

• nonn
• easy

Comments

Generalized Fibonacci sequence.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,4).

Crossrefs

Cf. A057087, A084128.

Programs

Mathematica

Table[2^(n - 5/2) ((1 + 4 Sqrt[2]) (1 - Sqrt[2])^n - (1 - 4 Sqrt[2]) (1 + Sqrt[2])^n), {n, 0, 25}]
RecurrenceTable[{a[0] == 2, a[1] == 3, a[n] == 4 a[n - 1] + 4 a[n - 2]}, a, {n, 25}] (* Bruno Berselli, Jan 19 2016 *)
LinearRecurrence[{4, 4}, {2, 3}, 20] (* Vincenzo Librandi, Jan 19 2016 *)

PARI

Vec((2-5*x)/(1-4*x-4*x^2) + O(x^100)) \\ Altug Alkan, Jan 19 2016

Formula

G.f.: (2 - 5*x)/(1 - 4*x - 4*x^2).

a(n) = 2^(n-5/2)*((1+4*sqrt(2))*(1-sqrt(2))^n - (1-4*sqrt(2))*(1+sqrt(2))^n).

Lim_{n -> infinity} a(n)/a(n - 1) = 2 + 2*sqrt(2) = 2*A014176 = 4.82842712...

a(n) = 2*A057087(n)-5*A057087(n-1). - R. J. Mathar, Jun 07 2016