This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A154940 #18 Sep 10 2018 14:16:12
%S A154940 0,0,0,0,1,2,3,4,5,5,6,5,5,5,7,7,6,5,9,8,8,9,10,7,9,10,7,9,7,6,7,9,7,
%T A154940 9,11,9,9,8,8,7,7,7,8,8,9,11,10,10,13,12,10,10,10,10,10,14,9,7,11,11,
%U A154940 9,14,12,10,12,13,9,11,8,7,10,12,10,12,12,12,12,11,11,12,8,11,11,14,10,13,10
%N A154940 Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number.
%C A154940 On Jan 16 2009, _Zhi-Wei Sun_ conjectured that a(n)>0 for n=5,6,... and verified this up to 5*10^6. (Sun also thought that lim inf_n a(n)/log(n) is a positive constant.) _D. S. McNeil_ continued the verification up to 10^13 and found no counterexamples. The conjecture is similar to a conjecture of Qing-Hu Hou and Jiang Zeng related to the sequence A154404; both conjectures were motivated by Sun's recent conjecture on sums of primes and Fibonacci numbers (cf. A154257).
%D A154940 R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%D A154940 R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.
%H A154940 Zhi-Wei Sun, <a href="/A154940/b154940.txt">Table of n, a(n), n=1..100000</a>
%H A154940 D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.081">Sun's strong conjecture</a>
%H A154940 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6434d742.0812">A promising conjecture: n=p+F_s+F_t</a>
%H A154940 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;65ce2465.0901">Offer prizes for solutions to my main conjectures involving primes</a>
%H A154940 Z.-W. Sun and R. Tauraso, <a href="http://arxiv.org/abs/0709.1665">Congruences involving Catalan numbers</a>, arXiv:0709.1665.
%F A154940 a(n) = |{<p,s,t>: p+L_s+C_t=n with p an odd prime, s>=0 and t>0}|.
%e A154940 For n=10 the a(10)=5 solutions are 3 + L_0 + C_3, 5 + L_2 + C_2, 5 + L_3 + C_1, 7 + L_0 + C_1, 7 + L_1 + C_2.
%t A154940 PQ[m_]:=m>2&&PrimeQ[m] L[x_]:=2*Fibonacci[x+1]-Fibonacci[x] RN[n_]:=Sum[If[PQ[n-L[x]-CatalanNumber[y]], 1, 0], {x,0,2*Log[2,n]},{y,1,2*Log[2,Max[2,n-L[x]+1]]}] Do[Print[n, " ",RN[n]]; Continue, {n, 1, 100000}]
%Y A154940 A000040, A000032, A000108, A154257, A154285, A154290, A154404, A154536, A156695.
%K A154940 nonn
%O A154940 1,6
%A A154940 _Zhi-Wei Sun_, Jan 17 2009
%E A154940 More terms (from b-file) added by _N. J. A. Sloane_, Aug 31 2009