A123329 Let M be the matrix defined in A111490. Sequence gives M(2,1)-M(1,2), M(2,1)+M(3,1)+M(3,2)-M(1,2)-M(1,3)-M(2,3), etc.

0, 1, 3, 8, 14, 26, 39, 59, 83, 115, 148, 197, 247, 307, 376, 460, 545, 651, 758, 887, 1027, 1181, 1336, 1527, 1724, 1937, 2163, 2417, 2672, 2969, 3267, 3596, 3940, 4304, 4681, 5113, 5546, 6001, 6473, 6995, 7518, 8095, 8673, 9291, 9942, 10619, 11297, 12051

Offset

0,3

Comments

From Omar E. Pol, Jan 20 2021: (Start)

Conjectures:

1. Convolution of A001065 and A000027.

2. Partial sums of A153485.

3. a(n) is also the difference of volume (the difference of number of cells) between two polycubes: the stepped pyramid described in A245092 which has volume A175254(n) and the stepped pyramid that represents the n-th tetrahedral number which has volume A000292(n).

In the three conjectures assuming that here the offset is 1.

For more information about the first pyramid see A237593. (End)

Links

Ray Chandler, Table of n, a(n) for n = 0..10000 (corrected original b-file from Michael S. Branicky missing two terms at request of Christian Krause)

Formula

a(n) = binomial(n+2,3) - A072481(n+1). - Robert Israel, Aug 13 2015

a(n) = A175254(n+1) - A000292(n+1), conjectured by Omar E. Pol, Jan 20 2021

a(n) = Sum_{i=2..(n+2)} Sum_{j=1..i-1} (M(i,j)-M(j,i)). - Michael S. Branicky, Jan 20 2021

Maple

b:= proc(n) option remember; `if`(n=0, [0$2], (p-> p
      +[numtheory[sigma](n)-n$2]+[0, p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2021

Mathematica

b[n_] := b[n] = If[n == 0, {0, 0}, With[{p = b[n-1]}, p +
     DivisorSigma[1, n] - n + {0, p[[1]]}]];
a[n_] := b[n+1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 10 2021, after Alois P. Heinz *)

Prog

(Python)
def M(n, k): return 1 + (k-1)%n
def a(n):
  return sum(M(i, j)-M(j, i) for i in range(2, n+3) for j in range(1, i))
print([a(n) for n in range(48)]) # Michael S. Branicky, Jan 20 2021

Crossrefs

Cf. A072481, A111490.

Cf. also A000027, A000292, A001065, A153485, A175254, A237593, A245092.

Sequence in context: A055335 A295200 A321047 * A268191 A169929 A129067

Adjacent sequences: A123326 A123327 A123328 * A123330 A123331 A123332

Keyword

nonn

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 06 2006

Extensions

a(14) and beyond from Michael S. Branicky, Jan 20 2021

Status

approved