A163064 a(n) = ((3+sqrt(5))*(4+sqrt(5))^n + (3-sqrt(5))*(4-sqrt(5))^n)/2.

3, 17, 103, 637, 3963, 24697, 153983, 960197, 5987763, 37339937, 232854103, 1452093517, 9055353003, 56469795337, 352149479663, 2196028088597, 13694580432483, 85400334485297, 532562291125063, 3321094649662237

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A098648 without initial 1. Fourth binomial transform of A163114. Inverse binomial transform of A163065.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-11).

Crossrefs

Cf. A098648, A163114, A163065.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(4+r)^n+(3-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

Magma

I:=[3,17]; [n le 2 select I[n] else 8*Self(n-1) - 11*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 22 2017

Mathematica

CoefficientList[Series[(3-7*x)/(1-8*x+11*x^2), {x,0,50}], x] (* or *) LinearRecurrence[{8,-11}, {3,17}, 30] (* G. C. Greubel, Dec 22 2017 *)

PARI

x='x+O('x^30); Vec((3-7*x)/(1-8*x+11*x^2)) \\ G. C. Greubel, Dec 22 2017

Formula

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 3, a(1) = 17.

G.f.: (3-7*x)/(1-8*x+11*x^2).

Extensions

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