A160704 - OEIS

A160704

Jacobsthal sequence A001045 convolved with A139251 (first differences of toothpick numbers).

1, 3, 9, 19, 41, 87, 181, 363, 729, 1463, 2933, 5871, 11753, 23523, 47061, 94123, 188249, 376503, 753013, 1506031, 3012073

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OFFSET

1,2

LINKS

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.]

FORMULA

a(n) = 2*a(n-1) + A160552, where A160552 begins with offset 1: (1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 7, 5,...).

EXAMPLE

a(4) = 19 = (1, 1, 3, 5) dot (4, 4, 2, 1) = (4 + 4 + 6 + 5).

a(4) = 19 = 2*9 + 1, where "1" = A160552(4), A160552 = (1, 1, 3, 1, 3, 5, 7,...)

Using the latter method, we create a heading = row 1:

1,...1,...3,...1,....3,...5,....7,....1,....3,.....5, = A160552 with offset 1.

1...3,....9,...19,..41,..87,..181,..363,..729,..1463,...

...such that a(n) = 2*a(n-1) = term above a(n), or a(n) = 2*a(n-1) + A160552(n). Example: a(6) = 87 = 2*41 + 5, where 5 = A160552(6).

CROSSREFS