A163071 a(n) = ((4+sqrt(5))*(3+sqrt(5))^n + (4-sqrt(5))*(3-sqrt(5))^n)/2.

4, 17, 86, 448, 2344, 12272, 64256, 336448, 1761664, 9224192, 48298496, 252894208, 1324171264, 6933450752, 36304019456, 190090313728, 995325804544, 5211593572352, 27288258215936, 142883175006208, 748146017173504

Offset: 0

Author

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

Keywords

• nonn

Comments

Binomial transform of A163070. Third binomial transform of A163141. Inverse binomial transform of A108404 without initial 1.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6, -4).

Crossrefs

Cf. A163070, A163141, A108404.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(3+r)^n+(4-r)*(3-r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

Mathematica

LinearRecurrence[{6,-4},{4,17},40] (* Harvey P. Dale, Feb 12 2013 *)

Formula

a(n) = 6*a(n-1) - 4*a(n-2) for n > 1; a(0) = 4, a(1) = 17.

G.f.: (4-7*x)/(1-6*x+4*x^2).

a(n) = 2^(n+1) * A000032(2*n) + 5 * 2^(n-1) * A000045(2*n) = 2^(n+1) * A005248(n) + 5 * 2^(n-1) * A001906(n). - Diego Rattaggi, Aug 02 2020

Extensions

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