A224524 -id:A224524 - cp's OEIS frontend

A224525 Number of idempotent 3 X 3 0..n matrices of rank 1.

27, 69, 123, 195, 273, 375, 477, 603, 735, 885, 1035, 1221, 1395, 1593, 1803, 2031, 2253, 2511, 2757, 3039, 3321, 3615, 3909, 4251, 4575, 4917, 5271, 5649, 6015, 6429, 6819, 7245, 7671, 8109, 8559, 9051, 9513, 9999, 10497, 11031, 11541, 12099, 12633

Offset: 1

R. H. Hardin, Apr 09 2013

nonn

Row 3 of A224524.

			Some solutions for n=3:
  0 1 0   1 3 2   0 1 0   1 3 1   0 0 0   0 0 0   1 0 0
  0 1 0   0 0 0   0 1 0   0 0 0   0 1 2   3 0 3   1 0 0
  0 3 0   0 0 0   0 2 0   0 0 0   0 0 0   1 0 1   2 0 0

Robert Israel, Table of n, a(n) for n = 1..10000 (first 80 terms from R. H. Hardin)

Cf. A002541, A224524.

f:= k -> 6*k^2 + 18*k + 3 + 6 * add(floor(k/m),m=2..k):
map(f, [$1..50]); # Robert Israel, Dec 15 2019

a(n) = 6*n^2 + 18*n + 3 + 6 * A002541(n). - Robert Israel, Dec 15 2019

A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

Original entry on oeis.org

108, 404, 892, 1716, 2732, 4324, 6060, 8516, 11308, 14820, 18572, 23668, 28716, 34916, 41836, 49860, 58076, 68164, 78252, 90356, 102988, 116868, 131276, 148564, 165660, 184532, 204604, 226788, 249116, 274900, 300252, 328628, 357868, 389028, 421580, 457924, 493500

Offset: 1

R. H. Hardin, Apr 09 2013

nonn

Row 4 of A224524

			Some solutions for n=3:
  0 0 0 0     0 0 0 0     1 0 0 0     0 0 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     1 1 0 0     0 0 0 0
  2 0 1 0     2 0 0 1     0 0 0 0     3 3 0 0     0 0 0 0
  0 0 0 0     2 0 0 1     0 0 0 0     2 2 0 0     1 0 1 1

Robert Israel, Table of n, a(n) for n = 1..4000

Cf. A002541, A024917, A224524.

F4 := k -> 8*k^3 + 36*k^2 + 24*add(m*floor(k/m), m = 2 .. k) + 12*add(floor(k/m), m = 2 .. k) + 12*add(floor(k/m)^2, m = 2 .. k) + 60*k + 4:
map(F4, [$1..100]); # Robert Israel, Dec 15 2019

Table[8*n^3+36*n^2+60*n+4+24*Sum[k*Floor[n/k],{k, 2, n}]+12*Sum[Floor[(n-k)/k],{k, n-1}]+12*Sum[Floor[(n/k)]^2,{k,2,n}],{n,1,100}] (* Metin Sariyar, Dec 15 2019 *)

a(n) = 8*n^3 + 36*n^2 + 60*n + 4 + 24*A024917(n) + 12*A002541(n) + 12*Sum_{m=2..n} floor(n/m)^2. - Robert Israel, Dec 15 2019

More terms from Metin Sariyar, Dec 15 2019

A224527 Number of idempotent 5X5 0..n matrices of rank 1.

Original entry on oeis.org

405, 2155, 5845, 13525, 24575, 44545, 69195, 107925, 156505, 223355, 301605, 413755, 535845, 694695, 882245

Offset: 1

R. H. Hardin Apr 09 2013

nonn

Row 5 of A224524

			Some solutions for n=3
..0..0..0..0..0....0..3..0..2..1....0..0..0..0..0....0..0..0..0..0
..0..0..0..0..0....0..0..0..0..0....3..1..1..1..3....1..0..0..0..1
..2..3..0..1..3....0..3..0..2..1....0..0..0..0..0....2..0..0..0..2
..2..3..0..1..3....0..0..0..0..0....0..0..0..0..0....3..0..0..0..3
..0..0..0..0..0....0..3..0..2..1....0..0..0..0..0....1..0..0..0..1

A224528 Number of idempotent 6X6 0..n matrices of rank 1.

Original entry on oeis.org

1458, 10830, 36042, 99774, 208146, 430278, 745650, 1289622, 2048514

Offset: 1

R. H. Hardin Apr 09 2013

nonn

Row 6 of A224524

			Some solutions for n=2
..0..0..1..0..0..0....0..0..0..0..0..0....0..0..0..0..0..0....0..0..1..0..0..0
..0..0..0..0..0..0....0..0..0..0..2..2....0..0..0..0..0..0....0..0..2..0..0..0
..0..0..1..0..0..0....0..0..0..0..1..1....1..0..0..0..1..2....0..0..1..0..0..0
..0..0..2..0..0..0....0..0..0..0..2..2....0..0..0..0..0..0....0..0..0..0..0..0
..0..0..1..0..0..0....0..0..0..0..0..0....1..0..0..0..1..2....0..0..2..0..0..0
..0..0..1..0..0..0....0..0..0..0..1..1....0..0..0..0..0..0....0..0..1..0..0..0

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A224525 Number of idempotent 3 X 3 0..n matrices of rank 1.

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A224526 Number of idempotent 4 X 4 0..n matrices of rank 1.

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A224527 Number of idempotent 5X5 0..n matrices of rank 1.

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A224528 Number of idempotent 6X6 0..n matrices of rank 1.

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