A307268 Partial sums of the Lucas numbers of the form L(3n+2) (A163063).

3, 14, 61, 260, 1103, 4674, 19801, 83880, 355323, 1505174, 6376021, 27009260, 114413063, 484661514, 2053059121, 8696898000, 36840651123, 156059502494, 661078661101, 2800374146900, 11862575248703, 50250675141714, 212865275815561, 901711778403960, 3819712389431403

Offset: 0

Author

Rigoberto Florez, Apr 01 2019

Keywords

• nonn
• easy

Examples

			L(2) + L(5) = 14;
L(2) + L(5) + L(8) = 61;
L(2) + L(5) + L(8) + L(11) = 260.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Crossrefs

Cf. A001076, A014448, A049651, A099919, A163063.

Programs

Mathematica

Table[(LucasL[3*n + 4] - 1)/2, {n, 0, 20}]
LinearRecurrence[{5,-3,-1},{3,14,61},30] (* Harvey P. Dale, Aug 10 2022 *)

PARI

L(n) = fibonacci(n+1)+fibonacci(n-1);
a(n) = (L(3*n+4)-1)/2; \\ Michel Marcus, Apr 01 2019

PARI

Vec((3 - x) / ((1 - x)*(1 - 4*x - x^2)) + O(x^25)) \\ Colin Barker, Apr 02 2019

Formula

a(n) = A001076(n+1) + A099919(n+1).

a(n) = Sum_{i=0..n} L(3i+2), L(i) = A000032(i).

a(n) = (L(3*n+4)-1)/2.

From Colin Barker, Apr 02 2019: (Start)

G.f.: (3 - x) / ((1 - x)*(1 - 4*x - x^2)).

a(n) = (-2 + (7-3*sqrt(5))*(2-sqrt(5))^n + (2+sqrt(5))^n*(7+3*sqrt(5))) / 4.

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3) for n > 2.

(End)