

A130695


Number of ways to write n as (a+1)(b+1)(c+1)  abc with a, b, c nonnegative integers.


4



1, 3, 3, 6, 3, 9, 4, 9, 6, 12, 3, 15, 6, 12, 9, 15, 3, 21, 7, 15, 9, 18, 6, 24, 9, 15, 9, 24, 6, 30, 6, 15, 15, 24, 9, 30, 7, 21, 12, 30, 3, 33, 15, 21, 15, 24, 6, 39, 12, 27, 12, 27, 9, 42, 12, 21, 15, 36, 6, 45, 13, 18, 21, 36, 12, 39, 6, 33, 15, 45, 9, 42, 12, 24, 24, 30, 9, 57, 18, 30
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OFFSET

1,2


LINKS



FORMULA



EXAMPLE

a(7) = 4 because 7 = 7*1*16*0*0 = 1*7*10*6*0 = 1*1*70*0*6 = 2*2*21*1*1.
G.f. = x + 3*x^2 + 3*x^3 + 6*x^4 + 3*x^5 + 9*x^6 + 4*x^7 + 9*x^8 + 6*x^9 + ...


MATHEMATICA

f[{a_, b_, c_}]:=(a+1)(b+1)(c+1)a*b*c; nn=80; Take[Transpose[Sort[Tally[f/@ Tuples[Range[0, nn], 3]], #1[[1]]<#2[[1]]&]] [[2]], nn] (* Harvey P. Dale, Mar 05 2012 *)
a[ n_] := Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 04 2015 *)
a[ n_] := (2 + (1)^n) Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2  (x^2  x + z^2  z) / 2 == n}, {x, y, z}, Integers, 10^9]; (* Michael Somos, Jul 04 2015 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



