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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.
4
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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 14 2004
STATUS
approved