A163348 a(n) = 6*a(n-1) - 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

1, 7, 35, 161, 721, 3199, 14147, 62489, 275905, 1218007, 5376707, 23734193, 104768209, 462469903, 2041441955, 9011362409, 39778080769, 175588947751, 775087121123, 3421400092481, 15102790707025, 66666943594783

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 25 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A111566. Third binomial transform of A143095. Inverse binomial transform of A081180 without initial 0.

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. A111566, A143095 (1,4,2,8,4,16,...), A081180.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(3+r)^n+(1-2*r)*(3-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 26 2009

Mathematica

LinearRecurrence[{6, -7}, {1, 7}, 50] (* G. C. Greubel, Dec 19 2016 *)

PARI

Vec((1+x)/(1-6*x+7*x^2) + O(x^50)) \\ G. C. Greubel, Dec 19 2016

Formula

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

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

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

E.g.f.: exp(3*x)*( cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Dec 19 2016

a(n) = A081179(n)+A081179(n+1). - R. J. Mathar, Feb 04 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 26 2009