login
A307268
Partial sums of the Lucas numbers of the form L(3n+2) (A163063).
1
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
0,1
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)
EXAMPLE
L(2) + L(5) = 14;
L(2) + L(5) + L(8) = 61;
L(2) + L(5) + L(8) + L(11) = 260.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Rigoberto Florez, Apr 01 2019
STATUS
approved