A083337 a(n) = 2*a(n-1) + 2*a(n-2); a(0)=0, a(1)=3.

0, 3, 6, 18, 48, 132, 360, 984, 2688, 7344, 20064, 54816, 149760, 409152, 1117824, 3053952, 8343552, 22795008, 62277120, 170144256, 464842752, 1269974016, 3469633536, 9479215104, 25897697280, 70753824768, 193303044096, 528113737728, 1442833563648, 3941894602752, 10769456332800

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 29 2003

Keywords

• easy
• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Martin Burtscher, Igor Szczyrba, 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 (2,2).

Crossrefs

Equals 3 * A002605.

Cf. A026150.

Cf. A028859, A028860, A030195, A080040, A106435, A108898, A125145.

Programs

Haskell

a083337 n = a083337_list !! n
a083337_list =
   0 : 3 : map (* 2) (zipWith (+) a083337_list (tail a083337_list))
-- Reinhard Zumkeller, Oct 15 2011

Mathematica

CoefficientList[Series[3x/(1-2x-2x^2), {x, 0, 25}], x]
s = Sqrt[3]; a[n_] := Simplify[s*((1 + s)^n - (1 - s)^n)/2]; Array[a, 30, 0] (* or *)
LinearRecurrence[{2, 2}, {0, 3}, 31] (* Robert G. Wilson v, Aug 07 2018 *)

PARI

apply( a(n)=([1,1;3,1]^n)[2,1], [0..30]) \\ or: ([2,2;1,0]^n)[2,1]*3. - M. F. Hasler, Aug 06 2018

Formula

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

a(n) = a(n-1) + 3*A026150(n-1). a(n)/A026150(n) converges to sqrt(3).

a(n) = lower left term of [1,1; 3,1]^n. - Gary W. Adamson, Mar 12 2008

Extensions

Edited and definition completed by M. F. Hasler, Aug 06 2018