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%I #32 Oct 01 2022 01:16:57
%S 1,2,3,4,4,5,5,6,6,7,6,8,7,8,8,9,8,10,8,10,10,10,9,12,10,11,11,12,10,
%T 13,11,13,12,12,12,15,12,13,13,15,12,15,13,15,15,14,13,17,14,16,15,16,
%U 14,17,15,17,16,16,15,20,15,16,17,18,17,19,16,18,17,19,16,21,17,18,19,19
%N a(n) is the number of equations in the set {x+2y=n, 2x+3y=n, ..., k*x+(k+1)*y=n, ..., n*x+(n+1)*y=n} which admit at least one nonnegative integer solution.
%C a(n) = ceiling(2*(sqrt(n)-1)) + ceiling(A000005(n)/2).
%e a(5) = 4. Consider the equations: x+2y=5, 2x+3y=5, 3x+4y=5, 4x+5y=5, 5x+6y=5. Only four of them admit at least one nonnegative integer solution, since 3x+4y=5 has no nonnegative integer solution.
%t b[m_] := m;
%t f[n_] := Table[Dimensions[Solve[b[k]*x + b[k + 1]*y == n, {x, y}, NonNegativeIntegers]][[1]], {k, 1, n}];
%t Flatten[Table[Dimensions[DeleteCases[f[k], 0]], {k, 1, 100}]]
%Y Cf. A000005.
%K nonn
%O 1,2
%A _Luca Onnis_, Aug 27 2022
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