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

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

Offset

1,2

Links

Jan Kristian Haugland, Jul 10 2007, Table of n, a(n) for n = 1..200

Formula

a(2*n) = A238872(2*n) / 3 if n>0. a(2*n + 1) = A238872(2*n + 1). - Michael Somos, Jul 04 2015

Example

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.
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

Cf. A238872.

Sequence in context: A007425 A260152 A358223 * A308083 A181788 A112163

Adjacent sequences: A130692 A130693 A130694 * A130696 A130697 A130698

Keyword

nonn

Author

Jan Kristian Haugland, Jul 10 2007

Status

approved