A212151 Number of 2 X 2 matrices M of positive integers such that permanent(M) < n.

0, 0, 0, 1, 5, 13, 27, 47, 75, 112, 156, 214, 278, 358, 444, 552, 660, 796, 930, 1099, 1259, 1457, 1649, 1885, 2101, 2377, 2623, 2933, 3221, 3569, 3879, 4279, 4623, 5056, 5452, 5926, 6334, 6878, 7328, 7892, 8404, 9018, 9540, 10228, 10788, 11504, 12142, 12898

Offset: 0

Author

Clark Kimberling, May 07 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000005, A211795, A212240.

Cf. A055507.

Programs

Maple

N:= 100: # to get a(0)..a(N)
g:= z*(1-z)^(-1)*add(z^i/(1-z^i),i=1..N-2)^2:
S:=series(g,z,N+1):
seq(coeff(S,z,n),n=0..N); # Robert Israel, Nov 16 2017

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x + y*z < n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212151 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Python

from sympy import divisor_count
def A212151(n): return  sum((sum(divisor_count(i+1)*divisor_count(j-i) for i in range(j>>1))<<1)+(divisor_count(j+1>>1)**2 if j&1 else 0) for j in range(1,n-1)) # Chai Wah Wu, Jul 26 2024

Formula

a(n) + A212240(n) = n^4.

a(n) = Sum_{k=1..n-1} Sum_{i=1..n-1} d(k) * floor((n-k-1)/i), where d(k) is the number of divisors of k (A000005). - Wesley Ivan Hurt, Nov 16 2017

G.f.: (x/(1-x))*(Sum_{i>=1} x^i/(1-x^i))^2. - Robert Israel, Nov 16 2017

from Ridouane Oudra, Oct 10 2023: (Start)

a(n) = Sum_{i=1..n-1} Sum_{j=1..n-1} tau(i*j)*floor((n-1)/(i+j)) ;

a(n) = Sum_{i=1..n-1} Sum_{j=1..i-1} tau(j)*tau(i-j) ;

a(n+2) = Sum_{i=1..n} A055507(i). (End)