A084128 a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

1, 2, 12, 56, 272, 1312, 6336, 30592, 147712, 713216, 3443712, 16627712, 80285696, 387653632, 1871757312, 9037643776, 43637604352, 210700992512, 1017354387456, 4912221519872, 23718303629312, 114522100596736, 552961616904192, 2669934870003712

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• nonn
• easy

Comments

Original name was: Generalized Fibonacci sequence.

Binomial transform of A084058.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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.
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (4,4).

Crossrefs

Cf. A001333 A001541, A002203, A057087, A084058, A084128, A201730.

Appears in A086346, A086347 and A086348. - Johannes W. Meijer, Aug 01 2010

Programs

Magma

[2^(n-1)*Evaluate(DicksonFirst(n,-1), 2): n in [0..40]]; // G. C. Greubel, Oct 13 2022

Maple

a:=proc(n) option remember; if n=0 then 1 elif n=1 then 2 else
4*a(n-1)+4*a(n-2); fi; end: seq(a(n), n=0..40); # Wesley Ivan Hurt, Jan 31 2017
a := n -> (2*I)^n*ChebyshevT(n, -I):
seq(simplify(a(n)), n = 0..23); # Peter Luschny, Dec 03 2023

Mathematica

CoefficientList[Series[(2 z - 1)/(4 z^2 + 4 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 01 2011 *)
Table[2^(n-1) LucasL[n, 2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)
LinearRecurrence[{4,4},{1,2},30] (* Harvey P. Dale, Mar 01 2018 *)

PARI

a(n)=if(n<0,0,polsym(4+4*x-x^2,n)[n+1]/2)

Sage

[lucas_number2(n,4,-4)/2 for n in range(0, 23)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = 2^n * A001333(n).

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

a(n) = 4*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

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

E.g.f.: exp(2*x)*cosh(2*x*sqrt(2)).

From Johannes W. Meijer, Aug 01 2010: (Start)

Lim_{k->infinity} a(n+k)/a(k) = A084128(n) + 2*A057087(n-1)*sqrt(2).

Lim_{n->infinity} A084128(n)/A057087(n-1) = sqrt(2). (End)

a(n) = Sum_{k=0..n} A201730(n,k)*7^k. - Philippe Deléham, Dec 06 2011

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(4*k-2)/(x*(4*k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 27 2013

a(n) = 2^(n-1)*A002203(n). - Vladimir Reshetnikov, Oct 07 2016