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

Author

Omar E. Pol, May 03 2009

Keywords

• nonn

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

• 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

Crossrefs

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

Programs

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 *)

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

Extensions

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