A164593 a(n) = ((5 + sqrt(18))*(2 + sqrt(8))^n + (5 - sqrt(18))*(2 - sqrt(8))^n)/2.

5, 22, 108, 520, 2512, 12128, 58560, 282752, 1365248, 6592000, 31828992, 153683968, 742051840, 3582943232, 17299980288, 83531694080, 403326697472, 1947433566208, 9403041054720, 45401898483712, 219219758153728, 1058486626549760, 5110825538813952, 24677248661454848

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A096980 without initial 1. Second binomial transform of A164737. Inverse binomial transform of A101386.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..100 from Vincenzo Librandi)
• Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Index entries for linear recurrences with constant coefficients, signature (4,4).

Crossrefs

Cf. A000129, A096980, A101386, A164737.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((5+3*r)*(2+2*r)^n+(5-3*r)*(2-2*r)^n)/2: n in [0..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

Maple

seq(coeff(series( (5+2*x)/(1-4*x-4*x^2) , x, n+1), x, n), n = 0..25); # G. C. Greubel, Apr 16 2020

Mathematica

LinearRecurrence[{4, 4}, {5, 22}, 25] (* G. C. Greubel, Aug 12 2017 *)
Table[2^n*(Fibonacci[n+2, 2] + 3*Fibonacci[n+1, 2]), {n,0,25}] (* G. C. Greubel, Apr 16 2020 *)

PARI

my(x='x+O('x^25)); Vec((5+2*x)/(1-4*x-4*x^2)) \\ G. C. Greubel, Aug 12 2017

Sage

[2^n*(lucas_number1(n+2, 2, -1) + 3*lucas_number1(n+1, 2, -1)) for n in range(25)] # G. C. Greubel, Apr 16 2020

Formula

a(n) = 4*a(n-1) + 4*a(n-2) for n > 1; a(0) = 5, a(1) = 22.

G.f.: (5 + 2*x)/(1-4*x-4*x^2).

E.g.f.: exp(2*x)*(5*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

a(n) = 2^n*(Pell(n+2) + 3*Pell(n+1)), where Pell(n) = A000129(n). - G. C. Greubel, Apr 16 2020