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A307268
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Partial sums of the Lucas numbers of the form L(3n+2) (A163063).
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1
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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
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = Sum_{i=0..n} L(3i+2), L(i) = A000032(i).
a(n) = (L(3*n+4)-1)/2.
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)
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EXAMPLE
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L(2) + L(5) = 14;
L(2) + L(5) + L(8) = 61;
L(2) + L(5) + L(8) + L(11) = 260.
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MATHEMATICA
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Table[(LucasL[3*n + 4] - 1)/2, {n, 0, 20}]
LinearRecurrence[{5, -3, -1}, {3, 14, 61}, 30] (* Harvey P. Dale, Aug 10 2022 *)
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PROG
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(PARI) L(n) = fibonacci(n+1)+fibonacci(n-1);
(PARI) Vec((3 - x) / ((1 - x)*(1 - 4*x - x^2)) + O(x^25)) \\ Colin Barker, Apr 02 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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