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A308566 Number of ways to write n as w^2 + x*(x+1) + 4^y*5^z with w,x,y,z nonnegative integers.

%I A308566 #21 Jun 10 2019 10:21:14
%S A308566 1,1,1,2,3,2,4,3,1,3,4,2,2,3,2,4,5,2,3,5,4,6,4,2,6,8,4,4,6,3,6,8,3,4,
%T A308566 6,6,5,5,2,6,8,3,6,4,3,6,9,2,4,7,4,6,4,4,4,8,3,4,6,4,7,8,3,4,6,5,7,5,
%U A308566 3,7,11,3,6,6,4,8,8,2,2,10,7,9,5,5,9,10,3,6,7,3,6,11,5,5,10,7,7,8,4,6
%N A308566 Number of ways to write n as w^2 + x*(x+1) + 4^y*5^z with w,x,y,z nonnegative integers.
%C A308566 Recall an observation of Euler: {w^2 + x*(x+1): w,x = 0,1,2,...} = {a*(a+1)/2 + b*(b+1)/2: a,b = 0,1,...}.
%C A308566 Conjecture: a(n) > 0 for all n > 0. Equivalently, each n = 1,2,3,... can be written as a*(a+1)/2 + b*(b+1)/2 + 4^c*5^d with a,b,c,d nonnegative integers.
%C A308566 See also A308584 for a similar conjecture.
%C A308566 We have verified a(n) > 0 for all n = 1..5*10^8.
%C A308566 a(n) > 0 for 0 < n < 10^10. - _Giovanni Resta_, Jun 08 2019
%H A308566 Zhi-Wei Sun, <a href="/A308566/b308566.txt">Table of n, a(n) for n = 1..10000</a>
%H A308566 Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.
%e A308566 a(1) = 1 with 1 = 0^2 + 0*1 + 4^0*5^0.
%e A308566 a(2) = 1 with 2 = 1^2 + 0*1 + 4^0*5^0.
%e A308566 a(3) = 1 with 3 = 0^2 + 1*2 + 4^0*5^0.
%e A308566 a(9) = 1 with 9 = 2^2 + 0*1 + 4^0*5^1.
%e A308566 a(303) = 1 with 303 = 16^2 + 6*7 + 4^0*5^1.
%e A308566 a(585) = 1 with 585 = 5^2 + 15*16 + 4^3*5^1.
%e A308566 a(37863) = 2 with 37863 = 166^2 + 101*102 + 4^0*5^1 = 179^2 + 26*27 + 4^5*5^1.
%t A308566 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A308566 tab={};Do[r=0;Do[If[SQ[n-4^k*5^m-x(x+1)],r=r+1],{k,0,Log[4,n]},{m,0,Log[5,n/4^k]},{x,0,(Sqrt[4(n-4^k*5^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A308566 Cf. A000217, A000290, A000302, A000351, A002378, A303656, A303637, A308411, A308547, A308584.
%K A308566 nonn
%O A308566 1,4
%A A308566 _Zhi-Wei Sun_, Jun 07 2019