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A154940
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Number of ways to express n as the sum of an odd prime, a Lucas number and a Catalan number.
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6
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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
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OFFSET
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1,6
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COMMENTS
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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).
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = |{<p,s,t>: p+L_s+C_t=n with p an odd prime, s>=0 and t>0}|.
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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