A163606 a(n) = ((3 + 2*sqrt(2))*(3 + sqrt(2))^n + (3 - 2*sqrt(2))*(3 - sqrt(2))^n)/2.

3, 13, 57, 251, 1107, 4885, 21561, 95171, 420099, 1854397, 8185689, 36133355, 159500307, 704068357, 3107907993, 13718969459, 60558460803, 267317978605, 1179998646009, 5208766025819, 22992605632851, 101494271616373

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

• nonn

Comments

Binomial transform of A048580. Inverse binomial transform of A163604.

For n >= 1, a(n-1) is the number of generalized compositions of n when there are 2^(i-1)+1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6, -7).

Crossrefs

Cf. A048580, A163604.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+2*r)*(3+r)^n+(3-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009

Mathematica

LinearRecurrence[{6,-7},{3,13},40] (* Harvey P. Dale, Dec 24 2011 *)

PARI

x='x+O('x^50); Vec((3-5*x)/(1-6*x+7*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 3, a(1) = 13.

G.f.: (3-5*x)/(1-6*x+7*x^2).

E.g.f.: exp(3*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 07 2009