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%I #16 Sep 11 2018 08:48:05
%S 0,0,0,0,0,1,1,3,2,5,2,5,2,4,3,4,4,5,2,6,2,7,5,7,3,9,3,9,4,7,3,6,4,9,
%T 3,10,3,8,4,6,5,8,6,8,3,9,4,8,6,8
%N Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.
%C On Jan 09 2009, _Zhi-Wei Sun_ conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, _D. S. McNeil_ found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.
%D R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
%H Zhi-Wei SUN, <a href="/A154421/b154421.txt">Table of n, a(n), n=1..50000.</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 summary concerning my 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 Terence Tao, <a href="http://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%H K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp. 78 (2009) 1853, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02212-1">[DOI]</a>, <a href="http://arxiv.org/abs/math/0702382">arXiv:math.NT/0702382</a>
%F a(n) = |{<p,s,t>: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.
%e For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.
%t PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[3x+1]+Fibonacci[3x]-Fibonacci[y]],1,0], {x,0,Log[2,n]},{y,2,2*Log[2,Max[2,n-2*Fibonacci[3x+1]+Fibonacci[3x]]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]
%Y Cf. A000040, A000045, A000032, A154257, A154285, A154290, A154364, A154417, A156695.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Jan 09 2009
%E McNeil's counterexample added by _Zhi-Wei Sun_, Jan 20 2009
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