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Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.
9

%I #7 Jun 11 2019 11:22:38

%S 1,2,2,3,3,2,3,5,2,4,4,3,3,5,3,3,6,4,4,5,2,6,5,4,4,5,2,4,6,3,4,8,5,3,

%T 5,4,5,8,5,5,4,3,5,7,4,5,8,4,2,8,2,6,7,4,3,4,6,5,8,4,4,6,5,5,5,5,6,8,

%U 4,6,7,4,6,10,4,4,7,5,2,10,4,7,7,4,8,4,4,7,8,2,4,9,5,5,9,5,5,7,5,6

%N Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.

%C Conjecture: a(n) > 0 for all n > 0. Equivalently, any positive integer n can be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*10^(2d) with a,b,c,d nonnegative integers.

%C This was motivated by A308566, and we verified a(n) > 0 for all n = 1..2*10^8. Then, on the author's request, _Giovanni Resta_ verified the above conjecture for n up to 10^10. G. Resta also noted that 729546026 cannot be written as a*(a+1)/2 + b*(b+1)/2 + 2^c*3^d with a,b,c,d nonnegative integers.

%C See also A308566, A308594 and A308621 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A308623/b308623.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 1 with 1 = 0*1/2 + 0*1/2 + 2^0*10^(2*0).

%e a(10107) = 1 with 10107 = 82*83/2 + 96*97/2 + 2^11*10^(2*0).

%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];

%t tab={};Do[r=0;Do[If[TQ[n-10^(2k)*2^m-x(x+1)/2],r=r+1],{k,0,Log[10,n]/2},{m,0,Log[2,n/10^(2k)]},{x,0,(Sqrt[4(n-10^(2k)*2^m)+1]-1)/2}];tab=Append[tab,r],{n,1,100}];Print[tab]

%Y Cf. A000079, A000217, A011557, A308566, A308584, A308621.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jun 11 2019