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

1,1

Comments

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)

Links

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

Formula

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

Example

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;
  ...

Mathematica

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

Crossrefs

Cf. A094616, A094618, A138247.

Cf. A034472, A052919, A199108, A199111.

Sequence in context: A118463 A214652 A137929 * A254318 A047502 A356311

Adjacent sequences: A094614 A094615 A094616 * A094618 A094619 A094620

Keyword

nonn,tabl

Author

Clark Kimberling, May 14 2004

Status

approved