cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

Original entry on oeis.org

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

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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)

Crossrefs

Programs

  • 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 *)
  • 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

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

Extensions

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