A341318 -id:A341318 - cp's OEIS frontend

A341706 Row 2 of semigroup multiplication table shown in A341317 and A341318.

0, 2, 7, 8, 16, 17, 18, 29, 30, 31, 32, 46, 47, 48, 49, 50, 67, 68, 69, 70, 71, 72, 92, 93, 94, 95, 96, 97, 98, 121, 122, 123, 124, 125, 126, 127, 128, 154, 155, 156, 157, 158, 159, 160, 161, 162, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 232, 233, 234

Offset: 0

N. J. A. Sloane, Feb 17 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A341317, A341318.

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)][])(f(n)):
a:= n-> (h-> t(h[1]*h[3])+h[2]*h[4])(map(g, [n, 2])):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 17 2021

More terms from Alois P. Heinz, Feb 17 2021

A341317 Array read by antidiagonals of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 7, 3, 0, 0, 4, 8, 8, 4, 0, 0, 5, 16, 10, 16, 5, 0, 0, 6, 17, 17, 17, 17, 6, 0, 0, 7, 18, 19, 37, 19, 18, 7, 0, 0, 8, 29, 21, 38, 38, 21, 29, 8, 0, 0, 9, 30, 30, 39, 40, 39, 30, 30, 9, 0, 0, 10, 31, 32, 67, 42, 42, 67, 32, 31, 10, 0

Offset: 0

N. J. A. Sloane, Feb 17 2021

Consider the semigroup S consisting of the pairs (0,0) and {(i,j): i >= j >= 1}, with componentwise products. Label the elements 0 = (0,0), 1 = (1,1), 2 = (2,1), 3 = (2,2), 4 = (3,1), 5 = (3,2), 6 = (3,3), 7 = (4,1), ... Form the array A(n,k) = label of product of n-th and k-th elements, for n>=0, k>=0, and read it by antidiagonals.

			The third and fourth elements of S are (2,2) and (3,1), and their product is (6,2), which is the 17th element.
The first few rows of the multiplication table A are:
  0, [0, 0,  0,  0,  0,  0,  0,   0,   0, ...]
  1, [0, 1,  2,  3,  4,  5,  6,   7,   8, ...]
  2, [0, 2,  7,  8, 16, 17, 18,  29,  30, ...]
  3, [0, 3,  8, 10, 17, 19, 21,  30,  32, ...]
  4, [0, 4, 16, 17, 37, 38, 39,  67,  68, ...]
  5, [0, 5, 17, 19, 38, 40, 42,  68,  70, ...]
  6, [0, 6, 18, 21, 39, 42, 45,  69,  72, ...]
  7, [0, 7, 29, 30, 67, 68, 69, 121, 122, ...]
  8, [0, 8, 30, 32, 68, 70, 72, 122, 124, ...]
  ...
The first few antidiagonals are:
   0, [0]
   1, [0, 0]
   2, [0, 1,  0]
   3, [0, 2,  2,  0]
   4, [0, 3,  7,  3,  0]
   5, [0, 4,  8,  8,  4,  0]
   6, [0, 5, 16, 10, 16,  5,  0]
   7, [0, 6, 17, 17, 17, 17,  6,  0]
   8, [0, 7, 18, 19, 37, 19, 18,  7,  0]
   9, [0, 8, 29, 21, 38, 38, 21, 29,  8, 0]
  10, [0, 9, 30, 30, 39, 40, 39, 30, 30, 9, 0]
  ...

J. M. Howie, An Introduction to Semigroup Theory, Academic Press (1976). [Background information.]

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Cf. A341318. See A341706 for row 2.

Main diagonal gives A341736.

# Build table of elements
M:=100; ct:=0; id[0,0]:=0; x[0]:=0; y[0]:=0;
for m from 1 to M do for n from 1 to m do
ct:=ct+1; x[ct]:=m; y[ct]:=n; id[m,n]:=ct;
od: od:
# Build multiplication table:
for m from 0 to 10 do
ro:=[];
for n from 0 to m do
a1:=x[m-n]; a2:=y[m-n]; b1:=x[n]; b2:=y[n];
c1:=a1*b1; c2:=a2*b2; d:=id[c1,c2];
ro:=[op(ro),d];
od:
lprint(m,ro);
od:

A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

Original entry on oeis.org

0, 1, 7, 10, 37, 40, 45, 121, 124, 129, 136, 301, 304, 309, 316, 325, 631, 634, 639, 646, 655, 666, 1177, 1180, 1185, 1192, 1201, 1212, 1225, 2017, 2020, 2025, 2032, 2041, 2052, 2065, 2080, 3241, 3244, 3249, 3256, 3265, 3276, 3289, 3304, 3321, 4951, 4954, 4959

Offset: 0

Alois P. Heinz, Feb 17 2021

nonn

The product in S is computed componentwise.

For the labeling of the elements in S and further information see A341317.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Main diagonal of A341317 and of A341318.

Cf. A000217, A000290, A037270, A123578, A161680.

t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)])(f(n)):
a:= n-> (h-> t(h[1]^2)+h[2]^2)(g(n)):
seq(a(n), n=0..60);

t[n_] := n*(n - 1)/2;
f[n_] := Ceiling[(Sqrt[1 + 8*n] - 1)/2];
g[n_] := Function[x, {x, n - t[x]}][f[n]];
a[n_] := Function[h, t[h[[1]]^2] + h[[2]]^2][g[n]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

a(n) = A341317(n,n) = A341318(n,n).

a(A000217(n)) = A037270(n) = A000217(A000290(n)).

cp's OEIS Frontend

A341706 Row 2 of semigroup multiplication table shown in A341317 and A341318.

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A341317 Array read by antidiagonals of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition).

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A341736 a(n) is the label of the square of the n-th element in the semigroup S = {(0,0), (i,j): i >= j >= 1}.

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