A161941 a(n) = ((4+sqrt(2))*(2+sqrt(2))^n + (4-sqrt(2))*(2-sqrt(2))^n)/4.

2, 5, 16, 54, 184, 628, 2144, 7320, 24992, 85328, 291328, 994656, 3395968, 11594560, 39586304, 135156096, 461451776, 1575494912, 5379076096, 18365314560, 62703106048, 214081795072, 730920968192, 2495520282624, 8520239194112

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Jun 22 2009

Keywords

• nonn
• easy

Comments

Second binomial transform of A135530.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), #12.7.8.
• Index entries for linear recurrences with constant coefficients, signature (4, -2).

Crossrefs

Cf. A135530, A161944 (third binomial transform of A135530).

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((4+r)*(2+r)^n+(4-r)*(2-r)^n)/4: n in [0..24] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 01 2009

Mathematica

LinearRecurrence[{4,-2},{2,5},30] (* Harvey P. Dale, May 26 2012 *)

PARI

x='x+O('x^30); Vec((2-3*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jan 27 2018

Formula

a(n) = 4*a(n-1) - 2*a(n-2) for n>1; a(0) = 2; a(1) = 5.

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

a(n) = 2*A007070(n) - 3*A007070(n-1). - R. J. Mathar, Oct 20 2017

Extensions

Edited and extended beyond a(4) by Klaus Brockhaus, Jul 01 2009