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%I #22 Sep 10 2018 14:02:15
%S 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,
%T 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,
%U 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
%N Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and a Catalan number.
%C 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).
%D R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%D R. P. Stanley, Enumerative Combinatorics, Vol. II, Cambridge Univ. Press, 1999, Chapter 6.
%H Jon E. Schoenfield, <a href="/A154404/b154404.txt">Table of n, a(n) for n = 1..100000</a>
%H D. S. McNeil, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bda1acb4.0812">Sun's strong conjecture</a>
%H 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 Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a9e706e.0812">A summary concerning my conjecture n=p+F_s+F_t (II)</a>
%H Z.-W. Sun and R. Tauraso, <a href="http://arxiv.org/abs/0709.1665">Congruences involving Catalan numbers</a>, arXiv:0709.1665v5.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/0901.3075">Mixed sums of primes and other terms</a>, preprint, 2009
%F a(n) = |{<p,s,t>: p+F_s+C_t=n with p an odd prime and s>1}|.
%e For n=7 the a(7)=3 solutions are 3+2+2, 3+3+1, 5+1+1.
%p 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:
%t 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 *)
%o (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
%Y Cf. A000040, A000045, A000108, A154257, A154290, A154285, A154952, A156695.
%K nice,nonn
%O 1,6
%A Qing-Hu Hou (hou(AT)nankai.edu.cn), Jan 09 2009, Jan 18 2009
%E More terms from _Jon E. Schoenfield_, Jan 17 2009
%E Added the new verification record and Hou and Zeng's prize for settling the conjecture. Edited by _Zhi-Wei Sun_, Feb 01 2009
%E Comment edited by _Charles R Greathouse IV_, Oct 28 2009
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