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Partial sums of squares of Lucas numbers.
(Formerly M4689)
3

%I M4689 #45 Jul 09 2024 09:07:17

%S 1,10,26,75,196,520,1361,3570,9346,24475,64076,167760,439201,1149850,

%T 3010346,7881195,20633236,54018520,141422321,370248450,969323026,

%U 2537720635,6643838876,17393796000,45537549121,119218851370

%N Partial sums of squares of Lucas numbers.

%D Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 20.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Harvey P. Dale, <a href="/A005970/b005970.txt">Table of n, a(n) for n = 1..1000</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-3,1).

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

%F G.f.: (1+7*x-4*x^2)/((1-x)*(1+x)*(1-3*x+x^2)). - _Simon Plouffe_ in his 1992 dissertation

%F From _Amiram Eldar_, Jan 13 2022: (Start)

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

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

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

%p 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

%t Accumulate[LucasL[Range[30]]^2] (* _Harvey P. Dale_, Dec 06 2019 *)

%Y Cf. A000032, A215602.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, May 29 2000

%E Definition clarified by _Harvey P. Dale_, Dec 06 2019