A160124 Total number of squares and rectangles after n stages in the toothpick structure of A139250.

0, 0, 0, 2, 4, 4, 8, 18, 24, 24, 28, 36, 40, 44, 64, 94, 108, 108, 112, 120, 124, 128, 148, 176, 188, 192, 208, 228, 240, 268, 340, 418, 448, 448, 452, 460, 464, 468, 488, 516, 528, 532, 548, 568, 580, 608, 680, 756, 784, 788, 804, 824, 836, 864, 932, 1000, 1028

Offset

0,4

Comments

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

It appears that A147614(n)/a(n) converge to 2.

It appears that A139250(n)/a(n) converge to 3/2.

It appears that a(n)/A139252(n) converge to 2.

(End)

Also 0 together with the rows sums of A211008. - Omar E. Pol, Sep 24 2012

Links

Table of n, a(n) for n=0..56.

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.]

Brian Hayes, Joshua Trees and Toothpicks

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

Formula

See A160125 for a recurrence. - N. J. A. Sloane, Feb 03 2010

a(n) = 1+2*A139250(n)-A147614(n), n>0 (Euler's formula). [From R. J. Mathar, Jan 22 2010]

a(n) = A187220(n+1) - A147614(n), n>0. - Omar E. Pol, Feb 15 2013

Mathematica

w [n_] := w[n] = Module[{k, i}, Which[n == 0, 0, n <= 3, n - 1, True, k = Floor[Log[2, n]]; i = n - 2^k; Which[i == 0, 2^(k - 1) - 1, i < 2^k - 2, 2 w[i] + w[i + 1], i == 2^k - 2, 2 w[i] + w[i + 1] + 1, True, 2 w[i] + w[i + 1] + 2]]];
r[n_] := r[n] = Module[{k, i}, Which[n <= 2, 0, n <= 4, 2, True, k = Floor[Log[2, n]]; i = n - 2^k; Which[i == 0, 2^k - 2, i <= 2^k - 2, 4 w[i], True, 4 w[i] + 2]]];
Join[{0}, Array[r, 100]] // Accumulate (* Jean-François Alcover, Apr 15 2020, after Maple code in A160125 *)

Crossrefs

Cf. A139250, A139252, A147614, A159786, A159787, A159788, A159789, A160125, A160126, A160127.

Sequence in context: A089887 A161816 A240347 * A295293 A080007 A239649

Adjacent sequences: A160121 A160122 A160123 * A160125 A160126 A160127

Keyword

nonn

Author

Omar E. Pol, May 03 2009

Extensions

More terms from R. J. Mathar, Jan 21 2010

Status

approved