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A003263 Number of representations of n as a sum of distinct Lucas numbers 1, 3, 4, 7, 11, ... (A000204).

%I A003263 M0045 #36 Jul 02 2025 16:01:54
%S A003263 1,0,1,2,1,0,2,2,0,1,3,2,0,2,3,1,0,3,3,0,2,4,2,0,3,3,0,1,4,3,0,3,5,2,
%T A003263 0,4,4,0,2,5,3,0,3,4,1,0,4,4,0,3,6,3,0,5,5,0,2,6,4,0,4,6,2,0,5,5,0,3,
%U A003263 6,3,0,4,4,0,1,5,4,0,4,7,3,0,6,6,0,3,8,5,0,5,7,2,0,6,6,0,4,8,4,0,6,6,0,2,7
%N A003263 Number of representations of n as a sum of distinct Lucas numbers 1, 3, 4, 7, 11, ... (A000204).
%D A003263 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 58.
%D A003263 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003263 T. D. Noe, <a href="/A003263/b003263.txt">Table of n, a(n) for n = 1..9349</a>
%H A003263 Alfred Brousseau, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/fibonacci-tables.html">Fibonacci and Related Number Theoretic Tables</a>, Fibonacci Association, San Jose, CA, 1972. See p. 58.
%H A003263 Casey Mongoven, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from175to192.pdf">Sonification of multiple Fibonacci-related sequences</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.
%F A003263 G.f.: Product_{n>=1} (1 + x^L(n)) where L(n) = A000204(n). - _Joerg Arndt_, Jul 14 2013
%t A003263 n1 = 10; n2 = LucasL[n1]; Product[1 + x^LucasL[n], {n, 1, n1}] + O[x]^n2 // CoefficientList[#, x]& // Rest (* _Jean-François Alcover_, Feb 17 2017, after _Joerg Arndt_ *)
%o A003263 (PARI)
%o A003263 L(n)=fibonacci(n+1) + fibonacci(n-1);
%o A003263 N = 66;  x = 'x + O('x^N);
%o A003263 gf = prod(n=1, 11, 1 + x^L(n) );
%o A003263 Vec(gf) \\ _Joerg Arndt_, Jul 14 2013
%Y A003263 Cf. A054770, A000204.
%K A003263 nonn,easy
%O A003263 1,4
%A A003263 _N. J. A. Sloane_
%E A003263 More terms from _James Sellers_, May 29 2000