A005970 Partial sums of squares of Lucas numbers. (Formerly M4689)

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

1,2

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).

Formula

G.f.: [1+7x-4x^2]/[(1-x)(1+x)(1-3x+x^2)]. - Ralf Stephan, Apr 23 2004

From Amiram Eldar, Jan 13 2022: (Start)

a(n) = Sum_{k=1..n} L(k), 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. (End)

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 A. Sellers, May 29 2000
A005970:=(-1-7*z+4*z**2)/(z-1)/(z+1)/(z**2-3*z+1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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

Crossrefs

Cf. A000032.

Sequence in context: A144255 A259290 A072379 * A192254 A368502 A220155

Adjacent sequences: A005967 A005968 A005969 * A005971 A005972 A005973

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 29 2000

Definition clarified by Harvey P. Dale, Dec 06 2019

Status

approved