|
%I #25 Feb 24 2021 02:48:18
%S 0,0,0,2,4,4,8,18,24,24,28,36,40,44,64,94,108,108,112,120,124,128,148,
%T 176,188,192,208,228,240,268,340,418,448,448,452,460,464,468,488,516,
%U 528,532,548,568,580,608,680,756,784,788,804,824,836,864,932,1000,1028
%N Total number of squares and rectangles after n stages in the toothpick structure of A139250.
%C From _Omar E. Pol_, Sep 16 2012: (Start)
%C It appears that A147614(n)/a(n) converge to 2.
%C It appears that A139250(n)/a(n) converge to 3/2.
%C It appears that a(n)/A139252(n) converge to 2.
%C (End)
%C Also 0 together with the rows sums of A211008. - _Omar E. Pol_, Sep 24 2012
%H David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H Brian Hayes, <a href="http://bit-player.org/2013/joshua-trees-and-toothpicks">Joshua Trees and Toothpicks</a>
%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F See A160125 for a recurrence. - _N. J. A. Sloane_, Feb 03 2010
%F a(n) = 1+2*A139250(n)-A147614(n), n>0 (Euler's formula). [From _R. J. Mathar_, Jan 22 2010]
%F a(n) = A187220(n+1) - A147614(n), n>0. - _Omar E. Pol_, Feb 15 2013
%t 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]]];
%t 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]]];
%t Join[{0}, Array[r, 100]] // Accumulate (* _Jean-François Alcover_, Apr 15 2020, after Maple code in A160125 *)
%Y Cf. A139250, A139252, A147614, A159786, A159787, A159788, A159789, A160125, A160126, A160127.
%K nonn
%O 0,4
%A _Omar E. Pol_, May 03 2009
%E More terms from _R. J. Mathar_, Jan 21 2010
|
