A163070 a(n) = ((4+sqrt(5))*(2+sqrt(5))^n + (4-sqrt(5))*(2-sqrt(5))^n)/2.

4, 13, 56, 237, 1004, 4253, 18016, 76317, 323284, 1369453, 5801096, 24573837, 104096444, 440959613, 1867934896, 7912699197, 33518731684, 141987625933, 601469235416, 2547864567597, 10792927505804, 45719574590813

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A163069. Second binomial transform of A163141. Inverse binomial transform of A163071.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,1).

Crossrefs

Cf. A000032, A000045, A001076, A014448, A163069, A163071, A163141.

Programs

Magma

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

Mathematica

LinearRecurrence[{4,1},{4,13},30] (* Harvey P. Dale, Sep 19 2011 *)

PARI

x='x+O('x^30); Vec((4-3*x)/(1-4*x-x^2)) \\ G. C. Greubel, Jan 08 2018

Formula

a(n) = 4*a(n-1) + a(n-2) for n > 1; a(0) = 4, a(1) = 13.

G.f.: (4-3*x)/(1-4*x-x^2).

a(n) = 2*A000032(3*n) + 5*A000045(3*n)/2 = 2*A014448(n) + 5*A001076(n). - Diego Rattaggi, Aug 09 2020

Extensions

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