A147614 a(n) = number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2.

0, 3, 7, 13, 19, 27, 39, 53, 63, 71, 83, 99, 119, 147, 183, 217, 235, 243, 255, 271, 291, 319, 355, 391, 419, 447, 487, 539, 607, 699, 803, 885, 919, 927, 939, 955, 975, 1003, 1039, 1075, 1103, 1131, 1171, 1223, 1291, 1383, 1487, 1571, 1615

Offset

0,2

Comments

a(n) is also the number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250. [From Omar E. Pol, May 18 2009]

It appears that the number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2*k, is equal to (2*k-2) * A139250(n) + a(n), k>0. See formulas in A160420 and A160422. [From Omar E. Pol, Nov 15 2010]

More generally, it appears that a(n) is also the number of grid points that are covered by the "special points" of the toothpicks of A139250, after n-th stage, assuming the toothpicks have length 2*k, k>0 and that each toothpick has three special points: the midpoint and two endpoints.

Note that if k>1 then also there are 2*k-2 grid points that are covered by each toothpick, but these points are not considered for this sequence. [From Omar E. Pol, Nov 15 2010]

Contribution from Omar E. Pol, Sep 16 2012 (Start):

It appears that a(n)/A139250(n) converge to 4/3.

It appears that a(n)/A160124(n) converge to 2.

It appears that a(n)/A139252(n) converge to 4.

(End)

Links

David Applegate, Table of n, a(n) for n = 0..10135

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

Since A160124(n) = 1+2*A139250(n)-A147614(n), n>0 (see A160124), and we have recurrences for A160125 (hence A160124) and A139250, we have a recurrence for this sequence. - N. J. A. Sloane, Feb 02 2010

a(n) = A187220(n+1)-A160124(n), n>0. - Omar E. Pol, Feb 15 2013

Crossrefs

Cf. A139250, A139251, A139252, A160124, A160420, A160422, A170884.

Sequence in context: A100458 A193765 A000960 * A171747 A031215 A099957

Adjacent sequences: A147611 A147612 A147613 * A147615 A147616 A147617

Keyword

nonn,look

Author

David Applegate, Apr 29 2009

Status

approved