A160570 Triangle read by rows, A160552 convolved with (1, 2, 2, 2, ...); row sums = A139250, the Toothpick sequence.

1, 1, 2, 3, 2, 2, 1, 6, 2, 2, 3, 2, 6, 2, 2, 5, 6, 2, 6, 2, 2, 7, 10, 6, 2, 6, 2, 2, 1, 14, 10, 6, 2, 6, 2, 2, 3, 2, 14, 10, 6, 2, 6, 2, 2, 5, 6, 2, 14, 10, 6, 2, 6, 2, 2, 7, 10, 6, 2, 14, 10, 6, 2, 6, 2, 2, 5, 14, 10, 6, 2, 14, 10, 6, 2, 6, 2, 2, 11, 10, 14, 10, 6, 2, 14, 10, 6, 2, 6

Offset: 1

Author

Gary W. Adamson, May 19 2009

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle:
  1;
  1,  2;
  3,  2,  2;
  1,  6,  2,  2;
  3,  2,  6,  2,  2;
  5,  6,  2,  6,  2,  2;
  7, 10,  6,  2,  6,  2,  2;
  1, 14, 10,  6,  2,  6,  2,  2;
  3,  2, 14, 10,  6,  2,  6,  2,  2;
  5,  6,  2, 14, 10,  6,  2,  6,  2,  2;
  ...
Example: Row 4 = (1, 6, 2, 2) = (1, 3, 1, 1) dot (1, 2, 2, 2); where (1 + 6 + 2 + 2) = A139250(4), i.e., 4th term in the Toothpick sequence.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• 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

Crossrefs

Cf. A160552, A139250.

Programs

Maple

T:=proc(n,k)if(k=1)then return A160552(n):else return 2*A160552(n-k+1):fi:end:
for n from 1 to 8 do for k from 1 to n do print(T(n,k));od:od: # Nathaniel Johnston, Apr 13 2011

Formula

Construct triangle M = an infinite lower triangular Toeplitz matrix with A160552: (1, 1, 3, 1, 3, 5, 7, ...) in every column. Let Q = an infinite lower triangular matrix with (1, 2, 2, 2, 2, ...) as the main diagonal and the rest zeros. A160570 = M * Q.