A188378 Partial sums of A005248.

2, 5, 12, 30, 77, 200, 522, 1365, 3572, 9350, 24477, 64080, 167762, 439205, 1149852, 3010350, 7881197, 20633240, 54018522, 141422325, 370248452, 969323030, 2537720637, 6643838880, 17393796002, 45537549125, 119218851372, 312119004990, 817138163597, 2139295485800

Offset: 0

Author

Gabriele Fici, Mar 29 2011

Keywords

• nonn
• easy

Comments

Different from A024851.

Luo proves that these integers cannot be uniquely decomposed as the sum of distinct and nonconsecutive terms of the Lucas number sequence. - Michel Marcus, Apr 20 2020

Links

• Robert Israel, Table of n, a(n) for n = 0..1000
• Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
• David C. Luo, Nonuniqueness Properties of Zeckendorf Related Decompositions, arXiv:2004.08316 [math.NT], 2020.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A000032, A002878, A027941, A001654.

Cf. A016945, A267797.

Programs

Magma

[5*Fibonacci(n)*Fibonacci(n+1)+1+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Jan 24 2016

Maple

f:= gfun:-rectoproc({a(n+3)-4*a(n+2)+4*a(n+1)-a(n), a(0) = 2, a(1) = 5, a(2) = 12}, a(n), remember):
map(f, [$0..60]); # Robert Israel, Feb 02 2016

Mathematica

LinearRecurrence[{4,-4,1},{2,5,12},30] (* Harvey P. Dale, Oct 05 2015 *)
Accumulate@ LucasL@ Range[0, 58, 2] (* Michael De Vlieger, Jan 24 2016 *)

PARI

a(n) = 5*fibonacci(n)*fibonacci(n+1) + 1 + (-1)^n; \\ Michel Marcus, Aug 26 2013

PARI

Vec((-2+3*x)/((x-1)*(x^2-3*x+1)) + O(x^100)) \\ Altug Alkan, Jan 24 2016

Formula

a(n) = A000032(2n+1)+1 = A002878(n)+1 = 2*A027941(n+1)-3*A027941(n).

G.f.: ( -2+3*x ) / ( (x-1)*(x^2-3*x+1) ). - R. J. Mathar, Mar 30 2011

a(n) = 5*A001654(n) + 1 + (-1)^n, n>=0. [Wolfdieter Lang, Jul 23 2012]

(a(n)^3 + (a(n)-2)^3) / 2 = A000032(A016945(n)) = Lucas(6*n+3) = A267797(n), for n>0. - Altug Alkan, Jan 31 2016

a(n) = 2^(-1-n)*(2^(1+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 02 2016