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%I #7 Mar 28 2021 00:19:16
%S 1,1,2,1,2,1,2,2,2,1,2,1,2,1,3,1,2,2,2,1,2,1,2,2,2,1,2,1,2,1,2,2,2,1,
%T 3,1,2,1,2,1,2,1,3,1,2,1,2,2,2,2,2,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1,2,1,
%U 2,1,2,2,2,1,2,1,2,1,2,2,2,1,2,2,2,1,2
%N Number of ways n appears as a cross-polytope number (A142978).
%C Every entry in the first column (of A142978) is 1, so this sequence starts at a(2).
%C a(n) is always positive, as the first row lists the positive integers.
%C a(n) >= 3 infinitely often. This happens, in particular, at every even square > 4. (The second row contains the squares, and the second column the positive even numbers.)
%C For n <= 10000, the only instance of a(n) > 3 is a(1156) = 4. This occurs because 1156 is even, square, and octahedral (third row of A142978).
%o (Sage) def a(n) : return len([K for K in [2..n] if n == next(A142978(N, K) for N in (1..) if A142978(N, K) >= n)])
%Y Cf. A142978.
%K nonn
%O 2,3
%A _Eric M. Schmidt_, Mar 22 2021
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