A209084 a(n) = 2*a(n-1) + 4*a(n-2) with n>1, a(0)=0, a(1)=4.

0, 4, 8, 32, 96, 320, 1024, 3328, 10752, 34816, 112640, 364544, 1179648, 3817472, 12353536, 39976960, 129368064, 418643968, 1354760192, 4384096256, 14187233280, 45910851584, 148570636288, 480784678912, 1555851902976, 5034842521600, 16293092655104

Offset: 0

Author

Seiichi Kirikami, Mar 06 2012

Keywords

• nonn
• easy

Comments

a(n)/A063727(n) are convergents for A134972.

Abs(Sum_{i=0..n} C(n,n-i)*a(i)-(sqrt(5)-1)* A033887(n))->0. - Seiichi Kirikami, Jan 20 2016

References

• E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, Inc., 1966.

Links

• Bruno Berselli, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (2,4).

Crossrefs

Cf. A063727, A033887, A085449, A103435, A134972, A269991.

Cf. A086344 (this sequence with signs).

Programs

Magma

I:=[0,4]; [n le 2 select I[n] else 2*Self(n-1)+4*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 16 2016

Mathematica

RecurrenceTable[{a[n]==2*a[n-1]+4*a[n-2], a[0]==0, a[1]==4}, a, {n, 30}]
LinearRecurrence[{2, 4}, {0, 4}, 40] (* Vincenzo Librandi, Jan 16 2016 *)

PARI

concat(0, Vec(4*x/(1-2*x-4*x^2) + O(x^40))) \\ Michel Marcus, Jan 16 2016

Formula

a(n) = (2/sqrt(5))*((1+sqrt(5))^n-(1-sqrt(5))^n).

G.f.: 4*x/(1-2*x-4*x^2). - Bruno Berselli, Mar 08 2012

a(n) = 4*A085449(n) = 2*A103435(n). - Bruno Berselli, Mar 08 2012

Sum_{n>=1} 1/a(n) = (1/4) * A269991. - Amiram Eldar, Feb 01 2021