A162770 a(n) = ((2+sqrt(5))*(1+sqrt(5))^n + (2-sqrt(5))*(1-sqrt(5))^n)/2.

2, 7, 22, 72, 232, 752, 2432, 7872, 25472, 82432, 266752, 863232, 2793472, 9039872, 29253632, 94666752, 306348032, 991363072, 3208118272, 10381688832, 33595850752, 108718456832, 351820316672, 1138514460672, 3684310188032

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Keywords

• nonn

Comments

Binomial transform of A162963. Inverse binomial transform of A001077 without initial 1.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1960
• Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 35.
• Index entries for linear recurrences with constant coefficients, signature (2,4).

Crossrefs

Cf. A000032, A001077, A162963.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((2+r)*(1+r)^n+(2-r)*(1-r)^n)/2: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

Mathematica

LinearRecurrence[{2,4},{2,7},30] (* Harvey P. Dale, Jan 13 2015 *)

Formula

a(n) = 2*a(n-1) + 4*a(n-2) for n > 1; a(0) = 2, a(1) = 7.

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

a(n) = 2^(n-1) * A000032(n+3). - Diego Rattaggi, Jun 24 2020

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009