A162956 a(0) = 0, a(1) = 1; a(2^i + j) = 3a(j) + a(j + 1) for 0 <= j < 2^i.

0, 1, 1, 4, 1, 4, 7, 13, 1, 4, 7, 13, 7, 19, 34, 40, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121, 7, 19, 34, 46, 40, 91, 142, 127, 40, 91, 148, 178, 211, 415, 547, 364, 1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46

Offset

0,4

Comments

2^n term triangle by rows, analogous to A160552 but multiplier is "3" instead of "2"

Row sums = powers of 5: (1, 5, 25, 125, 625,...).

Rows tend to A162957, obtained by taking (1, 3, 0, 0, 0,...) * A162956.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..16383

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

Follows the same analogous procedure as A160552 but multiplier is 3 instead of 2. (n+1)-th row brings down n-th row and appends to the right and equal number of terms following the rules: from left to right,let a = last term, b = current term, c = next term. Then c = 3*a + b except for the rightmost term = 3*a + 1.

Example

The triangle begins:
0;
1;
1, 4;
1, 4, 7, 13;
1, 4, 7, 13, 7, 19, 34, 40;
1, 4, 7, 13, 7, 19, 34, 40, 7, 19, 34, 46, 40, 91, 142, 121;
...
Row 4 = (1, 4, 7, 13, 7, 19, 34, 40): brings down (1, 4, 7, 13) then 7 = 3*1 + 4, 19 = 3*4 + 7, 34 = 3*7 + 13, 40 = 3*13 + 1.

Maple

a:= proc(n) option remember; `if`(n<2, n,
      (j-> 3*a(j)+a(j+1))(n-2^ilog2(n)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 28 2017

Mathematica

row[0] = {0}; row[1] = {1}; row[n_] := row[n] = Join[row[n-1], 3 row[n-1] + Append[Rest[row[n-1]], 1]]; Table[row[n], {n, 0, 7}] // Flatten (* Jean-François Alcover, Mar 13 2017 *)

Crossrefs

Cf. A162957, A162958, A160552.

Cf. A170838-A170852, A170854-A170872.

Sequence in context: A051006 A072812 A244097 * A131112 A141225 A344947

Adjacent sequences: A162953 A162954 A162955 * A162957 A162958 A162959

Keyword

nonn

Author

Gary W. Adamson, Jul 18 2009

Extensions

Edited with more terms by N. J. A. Sloane, Jan 02 2010

Status

approved