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%I #59 Sep 08 2022 08:45:56
%S 2,5,12,30,77,200,522,1365,3572,9350,24477,64080,167762,439205,
%T 1149852,3010350,7881197,20633240,54018522,141422325,370248452,
%U 969323030,2537720637,6643838880,17393796002,45537549125,119218851372,312119004990,817138163597,2139295485800
%N Partial sums of A005248.
%C Different from A024851.
%C Luo proves that these integers cannot be uniquely decomposed as the sum of distinct and nonconsecutive terms of the Lucas number sequence. - _Michel Marcus_, Apr 20 2020
%H Robert Israel, <a href="/A188378/b188378.txt">Table of n, a(n) for n = 0..1000</a>
%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Kimberling/kimber12.html">Lucas Representations of Positive Integers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
%H David C. Luo, <a href="https://arxiv.org/abs/2004.08316">Nonuniqueness Properties of Zeckendorf Related Decompositions</a>, arXiv:2004.08316 [math.NT], 2020.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F a(n) = A000032(2n+1)+1 = A002878(n)+1 = 2*A027941(n+1)-3*A027941(n).
%F G.f.: ( -2+3*x ) / ( (x-1)*(x^2-3*x+1) ). - _R. J. Mathar_, Mar 30 2011
%F a(n) = 5*A001654(n) + 1 + (-1)^n, n>=0. [_Wolfdieter Lang_, Jul 23 2012]
%F (a(n)^3 + (a(n)-2)^3) / 2 = A000032(A016945(n)) = Lucas(6*n+3) = A267797(n), for n>0. - _Altug Alkan_, Jan 31 2016
%F a(n) = 2^(-1-n)*(2^(1+n)-(3-sqrt(5))^n*(-1+sqrt(5))+(1+sqrt(5))*(3+sqrt(5))^n). - _Colin Barker_, Nov 02 2016
%p f:= gfun:-rectoproc({a(n+3)-4*a(n+2)+4*a(n+1)-a(n), a(0) = 2, a(1) = 5, a(2) = 12}, a(n), remember):
%p map(f, [$0..60]); # _Robert Israel_, Feb 02 2016
%t LinearRecurrence[{4,-4,1},{2,5,12},30] (* _Harvey P. Dale_, Oct 05 2015 *)
%t Accumulate@ LucasL@ Range[0, 58, 2] (* _Michael De Vlieger_, Jan 24 2016 *)
%o (PARI) a(n) = 5*fibonacci(n)*fibonacci(n+1) + 1 + (-1)^n; \\ _Michel Marcus_, Aug 26 2013
%o (PARI) Vec((-2+3*x)/((x-1)*(x^2-3*x+1)) + O(x^100)) \\ _Altug Alkan_, Jan 24 2016
%o (Magma) [5*Fibonacci(n)*Fibonacci(n+1)+1+(-1)^n: n in [0..40]]; // _Vincenzo Librandi_, Jan 24 2016
%Y Cf. A000032, A002878, A027941, A001654.
%Y Cf. A016945, A267797.
%K nonn,easy
%O 0,1
%A _Gabriele Fici_, Mar 29 2011