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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.
1
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(14) and beyond from Michael S. Branicky, Jan 20 2021
STATUS
approved