A306672 Partial sums of the even Lucas numbers (A014448).

2, 6, 24, 100, 422, 1786, 7564, 32040, 135722, 574926, 2435424, 10316620, 43701902, 185124226, 784198804, 3321919440, 14071876562, 59609425686, 252509579304, 1069647742900, 4531100550902, 19194049946506, 81307300336924, 344423251294200, 1459000305513722, 6180424473349086

Offset

0,1

Links

Robert Israel, Table of n, a(n) for n = 0..1593

Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Formula

a(n) = L(0) + L(3) + L(6) + L(9) + ... + L(3n), L(n) = Lucas numbers A000032.

a(n) = Sum_{i=0..n} L(3i).

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

G.f.: -2*(2*x-1)/((x-1)*(x^2+4*x-1)). - Alois P. Heinz, Mar 04 2019

Example

L(0) + L(3) = 6;
L(0) + L(3) + L(6) = 24;
L(0) + L(3) + L(6) + L(9) = 100.

Maple

f:= gfun:-rectoproc({a(n + 3) - 5*a(n + 2) + 3*a(n + 1) + a(n), a(0) = 2, a(1) = 6, a(2) = 24}, a(n), remember):
map(f, [$0..60]); # Robert Israel, Mar 05 2019

Mathematica

Table[(Lucas[3n+2]-1)/2+1, {n, 0, 25}]
Accumulate[Select[LucasL[Range[0, 100]], EvenQ]] (* or *) LinearRecurrence[ {5, -3, -1}, {2, 6, 24}, 30] (* Harvey P. Dale, Jan 18 2021 *)

Prog

(PARI) L(n) = fibonacci(n+1)+fibonacci(n-1);
a(n) = sum(k=0, n, L(3*k)); \\ Michel Marcus, Mar 05 2019
(Perl) use ntheory ":all"; sub a { vecsum(map{lucasv(1, -1, 3*$_)}0..$_[0]) } # Dana Jacobsen, Mar 05 2019

Crossrefs

Cf. A000032, A014448, A099919.

Sequence in context: A150299 A094012 A141253 * A324063 A078486 A352364

Adjacent sequences: A306669 A306670 A306671 * A306673 A306674 A306675

Keyword

nonn

Author

Rigoberto Florez, Mar 04 2019

Status

approved