A341706
Row 2 of semigroup multiplication table shown in A341317 and A341318.
Original entry on oeis.org
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
-
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
A341318
Lower triangular table of products in the semigroup S = {(0,0), (i,j): i >= j >= 1} (see Comments for precise definition), read by rows.
Original entry on oeis.org
0, 0, 1, 0, 2, 7, 0, 3, 8, 10, 0, 4, 16, 17, 37, 0, 5, 17, 19, 38, 40, 0, 6, 18, 21, 39, 42, 45, 0, 7, 29, 30, 67, 68, 69, 121, 0, 8, 30, 32, 68, 70, 72, 122, 124, 0, 9, 31, 34, 69, 72, 75, 123, 126, 129, 0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136, 0, 11, 46, 47, 106, 107, 108, 191, 192, 193, 194, 301
Offset: 0
Triangle begins:
0, [0]
1, [0, 1]
2, [0, 2, 7]
3, [0, 3, 8, 10]
4, [0, 4, 16, 17, 37]
5, [0, 5, 17, 19, 38, 40]
6, [0, 6, 18, 21, 39, 42, 45]
7, [0, 7, 29, 30, 67, 68, 69, 121]
8, [0, 8, 30, 32, 68, 70, 72, 122, 124]
9, [0, 9, 31, 34, 69, 72, 75, 123, 126, 129]
10, [0, 10, 32, 36, 70, 74, 78, 124, 128, 132, 136]
...
-
t:= n-> n*(n-1)/2:
f:= n-> ceil((sqrt(1+8*n)-1)/2):
g:= n-> (x-> [x, n-t(x)][])(f(n)):
T:= (n, k)-> (h-> t(h[1]*h[3])+h[2]*h[4])(map(g, [n, k])):
seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Feb 17 2021
-
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]];
T[n_, k_] := (Function[h, t[h[[1]]*h[[3]]] + h[[2]]*h[[4]]])[Flatten @ Map[g, {n, k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)
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
-
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 *)
Showing 1-3 of 3 results.
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