A193592 -id:A193592 - cp's OEIS frontend

A193593 Augmentation of the triangle A193592. See Comments.

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 31, 40, 23, 1, 15, 75, 166, 187, 105, 1, 21, 155, 530, 958, 993, 549, 1, 28, 287, 1415, 3786, 5988, 5865, 3207, 1, 36, 490, 3311, 12441, 28056, 40380, 37947, 20577, 1, 45, 786, 7000, 35469, 109451, 217720, 292092

Offset: 0

Clark Kimberling, Jul 31 2011

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Regarding A193592, (column 1)=A014616, (column 2)=A090809, (right edge)=A113227.

			First 5 rows:
1
1...1
1...3...2
1...6...10...6
1...10..31...40...23
Rows reversed as in Callan's n-edge increasing ordered trees with outdegree k:
 1
0      1
0      1      1
0      2      3      1
0      6     10      6      1
0     23     40     31     10      1
0    105    187    166     75     15      1
0    549    993    958    530    155     21     1
0   3207   5865   5988   3786   1415    287    28    1
0  20577  37947  40380  28056  12441   3311   490   36   1
0 143239 265901 292092 217720 109451  35469  7000  786  45 1

D. Callan, A bijection to count (1-23-4)-avoiding permutations, arXiv:1008.2375 (rows reversed)

Cf. A193091, A193592, A113227 (row sums and diagonal), A090809 (3rd col).

p[n_, 0] := 1; p[n_, k_] := n + 1 - k /; k > 0;
Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A193592 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 12}]]  (* A193593 *)
Flatten[Table[v[n], {n, 0, 10}]]

A334228 a(n) is the n-th completed odd-numbered level 2k-1 containing k ON-cells in a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^j >= 2k-1 generations of the automaton in A151723 have been computed).

Original entry on oeis.org

1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 55, 59, 63, 67, 83, 91, 95, 99, 107, 111, 115, 119, 123, 127, 131, 163, 179, 187, 191, 195, 211, 219, 223, 227, 235, 239, 243, 247, 251, 255, 259, 323, 355, 371, 379, 383, 387, 419, 435, 443, 447, 451, 467, 475, 479, 483, 491, 495, 499, 503, 507

Offset: 1

Hartmut F. W. Hoft, Apr 19 2020

nonn

			The sequence appears to be the triangle L(m, j) below read by rows where each row m >= 2 contains the level numbers in the sequence between 2^m - 1 and 2^(m+1) - 5:
m/j  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
1:   1
2:   3
3:   7  11
4:  15  19  23  27
5:  31  35  43  47  51  55  59
6:  63  67  83  91  95  99 107 111 115 119 123
7: 127 131 163 179 187 191 195 211 219 223 227 235 239 243 247 251
...
If T(i) = (i-2)*(i-1)/2 then row m >= 2 contains T(m) + 1 values.
The difference structure of this triangle L(m, j) is given by the triangle P(m, 1) = 4 and P(m, j) = 2^(m+2-j), for 2 <= j <= m, of powers of 2 as follows:
m/j  1  2  3  4  5
1:   4
2:   4  4
3:   4  8  4
4:   4 16  8  4
5:   4 32 16  8  4
...
Applying function log_2(k) - 1 to an entry k in this triangle gives the corresponding entry in the triangle of A193592.
Going backwards in triangle P(m, j) from the row labeled m - 2, left to right up to its vertex, starting with 2^m - 1 and computing the cumulative differences using the entries in triangle P(m, j) produces the numbers in row m of triangle L(m, j).

Hartmut F. W. Hoft, Plot of 3 rows of triangle L

Cf. A002024, A002260, A004736, A151723, A193592.

row[1] = 1; row[2] = 3; row[n_] := (2^n - 1) + Prepend[Accumulate[Flatten[Table[If[i==0||==j, 4, 2^(2+j-i)], {j, n-3, 0, -1}, {i, 0, j}]]], 0]/;n>=3
a334228[n_] := Flatten[Map[row,Range[n]]] (* first n rows in triangle L *)
a334228[8] (* sequence data *)

L(m, 0) = 2^(m-1); L(m, j) = 2^(m-1) + Sum_{i=k+1-j..k} P( t_1(i), t_2(i) ), for m >= 3 and for j=1..k, where k = (m-2)*(m-1)/2. Functions t_1(n) = floor(1/2 + sqrt(2*n)), A002024, and t_2(n) = binomial(floor(3/2 + sqrt(2*n)), 2) - n + 1, A004736, by Michael Somos, Jul 12 2003, are listed in triangle #7 in his link in A002260.

The formula for the count of ON-cells was verified through level 16384.

cp's OEIS Frontend

A193593 Augmentation of the triangle A193592. See Comments.

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