A154940 Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Jan 17 2009

Keywords

• nonn

Comments

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).

Examples

			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.
		

References

• R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
• R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.

Links

• Zhi-Wei Sun, Table of n, a(n), n=1..100000
• D. S. McNeil, Sun's strong conjecture
• Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
• Zhi-Wei Sun, Offer prizes for solutions to my main conjectures involving primes
• Z.-W. Sun and R. Tauraso, Congruences involving Catalan numbers, arXiv:0709.1665.

Crossrefs

A000040, A000032, A000108, A154257, A154285, A154290, A154404, A154536, A156695.

Programs

Mathematica

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}]

Formula

a(n) = |{: p+L_s+C_t=n with p an odd prime, s>=0 and t>0}|.

Extensions

More terms (from b-file) added by N. J. A. Sloane, Aug 31 2009