A094616 -id:A094616 - cp's OEIS frontend

A094617 Triangular array T of numbers generated by these rules: 2 is in T; and if x is in T, then 2x-1 and 3x-2 are in T.

2, 3, 4, 5, 7, 10, 9, 13, 19, 28, 17, 25, 37, 55, 82, 33, 49, 73, 109, 163, 244, 65, 97, 145, 217, 325, 487, 730, 129, 193, 289, 433, 649, 973, 1459, 2188, 257, 385, 577, 865, 1297, 1945, 2917, 4375, 6562, 513, 769, 1153, 1729, 2593, 3889, 5833, 8749, 13123, 19684

Offset: 1

Clark Kimberling, May 14 2004

To obtain row n from row n-1, apply 2x-1 to each x in row n-1 and then put 1+3^n at the end. Or, instead, apply 3x-2 to each x in row n-1 and then put 1+2^n at the beginning.
From Lamine Ngom, Feb 10 2021: (Start)
Triangle read by diagonals provides all the sequences of the form 1+2^(k-1)*3^n, where k is the k-th diagonal.
For instance, the terms of the first diagonal form the sequence 2, 4, 10, 28, ..., i.e., 1+3^n (A034472).
The 2nd diagonal leads to the sequence 3, 7, 19, 55, ..., i.e., 1+2*3^n (A052919).
The 3rd diagonal is the sequence 5, 13, 37, 109, ..., i.e., 1+4*3^n (A199108).
And for k = 4, we obtain the sequence 9, 25, 73, 217, ..., i.e., 1+8*3^n (A199111). (End)

			Rows of this triangle begin:
    2;
    3,   4;
    5,   7,   10;
    9,  13,   19,   28;
   17,  25,   37,   55,   82;
   33,  49,   73,  109,  163,  244;
   65,  97,  145,  217,  325,  487,  730;
  129, 193,  289,  433,  649,  973, 1459, 2188;
  257, 385,  577,  865, 1297, 1945, 2917, 4375,  6562;
  513, 769, 1153, 1729, 2593, 3889, 5833, 8749, 13123, 19684;
  ...

Ivan Neretin, Table of n, a(n) for n = 1..5151

Cf. A094616, A094618, A138247.

Cf. A034472, A052919, A199108, A199111.

FoldList[Append[2 #1 - 1, 1 + 3^#2] &, {2}, Range[9]] // Flatten (* Ivan Neretin, Mar 30 2016 *)

When offset is zero, then the first term is T(0,0) = 2, and
T(n,0) = 1 + 2^n = A000051(n),
T(n,n) = 1 + 3^n = A048473(n),
T(2n,n) = 1 + 6^n = A062394(n).
Row sums = A094618.
a(n) = A036561(n-1) + 1. - Filip Zaludek, Nov 19 2016

A094615 Triangular array T of numbers generated by these rules: 1 is in T; and if x is in T, then 2x+1 and 3x+2 are in T.

Original entry on oeis.org

1, 3, 5, 7, 11, 17, 15, 23, 35, 53, 31, 47, 71, 107, 161, 63, 95, 143, 215, 323, 485, 127, 191, 287, 431, 647, 971, 1457, 255, 383, 575, 863, 1295, 1943, 2915, 4373, 511, 767, 1151, 1727, 2591, 3887, 5831, 8747, 13121, 1023, 1535, 2303, 3455, 5183, 7775, 11663, 17495, 26243, 39365

Offset: 0

Clark Kimberling, May 14 2004

To obtain row n from row n-1, apply 2x+1 to each x in row n-1 and then put -1+2*3^n at the end. Or, instead, apply 3x+2 to each x in row n-1 and then put -1+2^(n+1) at the beginning.

Subtriangle of the triangle in A230445. - Philippe Deléham, Oct 31 2013

			Triangle begins:
  n\k|   1    2    3    4    5    6     7
  ---+-----------------------------------
  0  |   1;
  1  |   3,   5;
  2  |   7,  11,  17;
  3  |  15,  23,  35,  53;
  4  |  31,  47,  71, 107, 161;
  5  |  63,  95, 143, 215, 323, 485;
  6  | 127, 191, 287, 431, 647, 971, 1457;

Michel Marcus, Rows n=0..99 of triangle, flattened

Cf. A094616 (row sums), A094617, A230445.

Cf. A048473, A171498, A198644

tabl(nn) = {my(row = [1], nrow); for (n=1, nn, print (row); nrow = vector(n+1, k, if (k<=n, (2*row[k]+1), -1+2*3^n)); row = nrow;);} \\ Michel Marcus, Nov 14 2020

T(n,0) = -1+2^(n+1) = A000225(n+1).
T(n,n) = -1+2*3^n = A048473(n).
T(2n,n) = -1+2*6^n.
T(n,k) = -1 + 2^(n+1-k)*3^k. - Lamine Ngom, Feb 10 2021

Offset 0 and more terms from Michel Marcus, Nov 14 2020

cp's OEIS Frontend

A094617 Triangular array T of numbers generated by these rules: 2 is in T; and if x is in T, then 2x-1 and 3x-2 are in T.

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