A005970 Partial sums of squares of Lucas numbers.

1, 10, 26, 75, 196, 520, 1361, 3570, 9346, 24475, 64076, 167760, 439201, 1149850, 3010346, 7881195, 20633236, 54018520, 141422321, 370248450, 969323026, 2537720635, 6643838876, 17393796000, 45537549121, 119218851370

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 20.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Crossrefs

Cf. A000032, A215602.

Programs

Maple

lucas := proc(n) option remember: if n=1 then RETURN(1) fi: if n=2 then RETURN(3) fi: lucas(n-1)+lucas(n-2) end: l[0] := 0: for i from 1 to 50 do l[i] := l[i-1]+lucas(i)^2; printf(`%d,`,l[i]) od: # James Sellers, May 29 2000

Mathematica

Accumulate[LucasL[Range[30]]^2] (* Harvey P. Dale, Dec 06 2019 *)

Formula

a(n) - a(n-1) = A001254(n).

G.f.: (1+7*x-4*x^2)/((1-x)*(1+x)*(1-3*x+x^2)). - Simon Plouffe in his 1992 dissertation

From Amiram Eldar, Jan 13 2022: (Start)

a(n) = Sum_{k=1..n} L(k)^2, where L(k) is the k-th Lucas number (A000032).

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4), for n > 4.

a(n) = L(n)*L(n+1) - 2 = A215602(n) - 2. (End)

Extensions

More terms from James Sellers, May 29 2000

Definition clarified by Harvey P. Dale, Dec 06 2019