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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.

%I A123329 #51 Sep 23 2021 11:18:00
%S A123329 0,1,3,8,14,26,39,59,83,115,148,197,247,307,376,460,545,651,758,887,
%T A123329 1027,1181,1336,1527,1724,1937,2163,2417,2672,2969,3267,3596,3940,
%U A123329 4304,4681,5113,5546,6001,6473,6995,7518,8095,8673,9291,9942,10619,11297,12051
%N 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.
%C A123329 From _Omar E. Pol_, Jan 20 2021: (Start)
%C A123329 Conjectures:
%C A123329 1. Convolution of A001065 and A000027.
%C A123329 2. Partial sums of A153485.
%C A123329 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).
%C A123329 In the three conjectures assuming that here the offset is 1.
%C A123329 For more information about the first pyramid see A237593. (End)
%H A123329 Ray Chandler, <a href="/A123329/b123329.txt">Table of n, a(n) for n = 0..10000</a> (corrected original b-file from Michael S. Branicky missing two terms at request of Christian Krause)
%F A123329 a(n) = binomial(n+2,3) - A072481(n+1). - _Robert Israel_, Aug 13 2015
%F A123329 a(n) = A175254(n+1) - A000292(n+1), conjectured by _Omar E. Pol_, Jan 20 2021
%F A123329 a(n) = Sum_{i=2..(n+2)} Sum_{j=1..i-1} (M(i,j)-M(j,i)). - _Michael S. Branicky_, Jan 20 2021
%p A123329 b:= proc(n) option remember; `if`(n=0, [0$2], (p-> p
%p A123329       +[numtheory[sigma](n)-n$2]+[0, p[1]])(b(n-1)))
%p A123329     end:
%p A123329 a:= n-> b(n+1)[2]:
%p A123329 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 21 2021
%t A123329 b[n_] := b[n] = If[n == 0, {0, 0}, With[{p = b[n-1]}, p +
%t A123329      DivisorSigma[1, n] - n + {0, p[[1]]}]];
%t A123329 a[n_] := b[n+1][[2]];
%t A123329 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 10 2021, after _Alois P. Heinz_ *)
%o A123329 (Python)
%o A123329 def M(n,k): return 1 + (k-1)%n
%o A123329 def a(n):
%o A123329   return sum(M(i,j)-M(j,i) for i in range(2, n+3) for j in range(1, i))
%o A123329 print([a(n) for n in range(48)]) # _Michael S. Branicky_, Jan 20 2021
%Y A123329 Cf. A072481, A111490.
%Y A123329 Cf. also A000027, A000292, A001065, A153485, A175254, A237593, A245092.
%K A123329 nonn
%O A123329 0,3
%A A123329 _Paolo P. Lava_ and _Giorgio Balzarotti_, Oct 06 2006
%E A123329 a(14) and beyond from _Michael S. Branicky_, Jan 20 2021