A163069 a(n) = ((4+sqrt(5))*(1+sqrt(5))^n + (4-sqrt(5))*(1-sqrt(5))^n)/2.

4, 9, 34, 104, 344, 1104, 3584, 11584, 37504, 121344, 392704, 1270784, 4112384, 13307904, 43065344, 139362304, 450985984, 1459421184, 4722786304, 15283257344, 49457659904, 160048349184, 517927337984, 1676048072704

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Keywords

• nonn

Comments

Binomial transform A163141. Inverse binomial transform A163070.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,4).

Crossrefs

Cf. A163141, A163070.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(1+r)^n+(4-r)*(1-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

Mathematica

LinearRecurrence[{2, 4}, {4, 9}, 30] (* G. C. Greubel, Jan 08 2018 *)

PARI

x='x+O('x^30); Vec((4+x)/(1-2*x-4*x^2)) \\ G. C. Greubel, Jan 08 2018

Formula

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 4, a(1) = 9.

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

a(n) = 2^(n+1) * A000032(n) + 5 * 2^(n-1) * A000045(n). - Diego Rattaggi, Jun 27 2020

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009