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A154404
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Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number.
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8
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0, 0, 0, 0, 1, 2, 3, 3, 5, 5, 5, 4, 6, 5, 6, 5, 7, 6, 6, 9, 9, 8, 8, 6, 8, 10, 9, 6, 9, 7, 5, 8, 10, 8, 8, 7, 6, 9, 9, 8, 8, 7, 6, 9, 9, 13, 10, 9, 8, 12, 10, 10, 10, 9, 9, 11, 9, 11, 9, 10, 8, 11, 13, 11, 10, 12, 11, 11, 10, 10, 7, 8, 10, 14, 10, 16, 11, 9, 11, 11, 10, 12, 10, 7, 9, 16, 10, 12
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OFFSET
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1,6
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COMMENTS
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Motivated by Zhi-Wei Sun's conjecture that each integer n>4 can be expressed as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number (cf. A154257), during their visit to Nanjing Univ. Qing-Hu Hou (Nankai Univ.) and Jiang Zeng (Univ. of Lyon-I) conjectured on Jan 09 2009 that a(n)>0 for every n=5,6,.... and verified this up to 5*10^8. D. S. McNeil has verified the conjecture up to 5*10^13 and Hou and Zeng have offered prizes for settling their conjecture (see Sun 2009).
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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+F_s+C_t=n with p an odd prime and s>1}|.
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EXAMPLE
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For n=7 the a(7)=3 solutions are 3+2+2, 3+3+1, 5+1+1.
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MAPLE
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Cata:=proc(n) binomial(2*n, n)/(n+1); end proc: Fibo:=proc(n) if n=1 then return(1); elif n=2 then return(2); else return(Fibo(n-1) + Fibo(n-2)); fi; end proc: for n from 1 to 10^3 do rep_num:=0; for i from 1 while Fibo(i) < n do for j from 1 while Fibo(i)+Cata(j) < n do p:=n-Fibo(i)-Cata(j); if (p>2) and isprime(p) then rep_num:=rep_num+1; fi; od; od; printf("%d %d\n", n, rep_num); od:
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MATHEMATICA
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a[n_] := (pp = {}; p = 2; While[ Prime[p] < n, AppendTo[pp, Prime[p++]] ]; ff = {}; f = 2; While[ Fibonacci[f] < n, AppendTo[ff, Fibonacci[f++]]]; cc = {}; c = 1; While[ CatalanNumber[c] < n, AppendTo[cc, CatalanNumber[c++]]]; Count[Outer[Plus, pp, ff, cc], n, 3]); Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Nov 22 2011 *)
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PROG
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(PARI) a(n)=my(i=1, j, f, c, t, s); while((f=fibonacci(i++))<n, t=n-f; j=0; while((c=binomial(2*j++, j)/(j+1))<t-2, s+=isprime(t-c))); s \\ Charles R Greathouse IV, Nov 22 2011
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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Qing-Hu Hou (hou(AT)nankai.edu.cn), Jan 09 2009, Jan 18 2009
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EXTENSIONS
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Added the new verification record and Hou and Zeng's prize for settling the conjecture. Edited by Zhi-Wei Sun, Feb 01 2009
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STATUS
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approved
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