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A130695 Number of ways to write n as (a+1)(b+1)(c+1) - abc with a, b, c nonnegative integers.

%I A130695 #13 Feb 04 2019 14:14:45
%S A130695 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,
%T A130695 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,
%U A130695 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
%N A130695 Number of ways to write n as (a+1)(b+1)(c+1) - abc with a, b, c nonnegative integers.
%H A130695 _Jan Kristian Haugland_, Jul 10 2007, <a href="/A130695/b130695.txt">Table of n, a(n) for n = 1..200</a>
%F A130695 a(2*n) = A238872(2*n) / 3 if n>0. a(2*n + 1) = A238872(2*n + 1). - _Michael Somos_, Jul 04 2015
%e A130695 a(7) = 4 because 7 = 7*1*1-6*0*0 = 1*7*1-0*6*0 = 1*1*7-0*0*6 = 2*2*2-1*1*1.
%e A130695 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 + ...
%t A130695 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 *)
%t A130695 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 *)
%t A130695 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 *)
%Y A130695 Cf. A238872.
%K A130695 nonn
%O A130695 1,2
%A A130695 _Jan Kristian Haugland_, Jul 10 2007