A164541 a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 15.

1, 15, 89, 519, 3025, 17631, 102761, 598935, 3490849, 20346159, 118586105, 691170471, 4028436721, 23479449855, 136848262409, 797610124599, 4648812485185, 27095264786511, 157922776233881, 920441392616775

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164540. Third binomial transform of A164675. Inverse binomial transform of A164542.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..155
• Index entries for linear recurrences with constant coefficients, signature (6, -1).

Crossrefs

Cf. A164540, A164675, A164542.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+3*r)*(3+2*r)^n+(1-3*r)*(3-2*r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 20 2009

Mathematica

LinearRecurrence[{6,-1},{1,15},20] (* Harvey P. Dale, Feb 04 2023 *)

Formula

a(n) = 6*a(n-1) - a(n-2) for n > 1; a(0) = 1, a(1) = 15.

G.f: (1+9*x)/(1-6*x+x^2).

a(n) = ((1+3*sqrt(2))*(3+2*sqrt(2))^n + (1-3*sqrt(2))*(3-2*sqrt(2))^n)/2.

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 20 2009