%I #18 Jun 07 2019 02:57:18
%S 1,2,3,4,4,4,4,4,5,7,5,8,6,5,6,4,6,8,7,10,8,6,4,8,8,8,10,10,6,9,6,4,
%T 10,9,11,14,8,8,9,10,8,11,8,9,13,6,5,8,9,10,11,13,7,14,8,10,13,9,11,
%U 16,7,7,13,4,12,12,10,12,10,13,5,14,13,9,17,12,7,12,6,10
%N Number of ways to write n as a^2 + 2*b^2 + 2^c*3^d, where a,b,c,d are nonnegative integers.
%C As 3*(a^2 + 2*b^2 + 2^c*3^d) = (a+2*b)^2 + 2*(a-b)^2 + 2^c*3^(d+1), we have a(3*n) > 0 if a(n) > 0.
%C The first positive integer n with a(n) = 0 is 139571911. We also have a(142991573) = 0.
%H Zhi-Wei Sun, <a href="/A308547/b308547.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 1 with 1 = 0^2 + 2*0^2 + 2^0*3^0.
%e a(2) = 2 with 2 = 0^2 + 2*0^2 + 2^1*3^0 = 1^2 + 2*0^2 + 2^0*3^0.
%e a(1117) = 2 with 1117 = 10^2 + 2*12^2 + 2^0*3^6 = 19^2 + 2*18^2 + 2^2*3^3.
%e a(78373) = 1 with 78373 = 271^2 + 2*48^2 + 2^2*3^4.
%e a(448159) = 1 with 448159 = 610^2 + 2*195^2 + 2^0*3^2.
%e a(82816213) = 2 with 82816213 = 4353^2 + 2*5651^2 + 2^1*3^0 = 3681^2 + 2*5885^2 + 2^1*3^0.
%e a(90685253) = 2 with 90685253 = 7007^2 + 2*4560^2 + 2^2*3^0 = 607^2 + 2*6720^2 + 2^2*3^0.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-3^k*2^m-2x^2],r=r+1],{k,0,Log[3,n]},{m,0,Log[2,n/3^k]},{x,0,Sqrt[(n-3^k*2^m)/2]}];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y Cf. A000079, A000244, A000290, A002479, A302984, A303637, A303656, A308411.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Jun 06 2019