A139250 -id:A139250 - cp's OEIS frontend

A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250).

0, 0, 2, 2, 0, 4, 10, 6, 0, 4, 8, 4, 4, 20, 30, 14, 0, 4, 8, 4, 4, 20, 28, 12, 4, 16, 20, 12, 28, 72, 78, 30, 0, 4, 8, 4, 4, 20, 28, 12, 4, 16, 20, 12, 28, 72, 76, 28, 4, 16, 20, 12, 28, 68, 68, 28, 24, 52, 52, 52, 128, 224, 190, 62, 0, 4, 8, 4, 4, 20, 28, 12, 4, 16, 20, 12, 28, 72

Offset: 1

Omar E. Pol, May 03 2009

nonn

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

First differences of A160124.
Cf. toothpick sequence A139250.
Cf. A159786, A159787, A159788, A159789, A160124, A160126, A160127.

# First construct A168131:
w := proc(n) option remember; local k,i;
if (n=0) then RETURN(0)
elif (n <= 3) then RETURN(n-1)
else
k:=floor(log(n)/log(2)); i:=n-2^k;
if (i=0) then RETURN(2^(k-1)-1)
elif (i<2^k-2) then RETURN(2*w(i)+w(i+1));
elif (i=2^k-2) then RETURN(2*w(i)+w(i+1)+1);
else RETURN(2*w(i)+w(i+1)+2);
fi; fi; end;
# Then construct A160125:
r := proc(n) option remember; local k,i;
if (n<=2) then RETURN(0)
elif (n <= 4) then RETURN(2)
else
k:=floor(log(n)/log(2)); i:=n-2^k;
if (i=0) then RETURN(2^k-2)
elif (i<=2^k-2) then RETURN(4*w(i));
else RETURN(4*w(i)+2);
fi; fi; end;
[seq(r(n),n=0..200)];
# N. J. A. Sloane, Feb 01 2010

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]]];
Array[r, 78] (* Jean-François Alcover, Apr 15 2020, from Maple *)

See Maple program for recurrence.

Terms beyond a(10) from R. J. Mathar, Jan 21 2010

A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 17, 1, 5, 5, 17, 5, 17, 21, 49, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 1, 5, 5, 17, 5, 17, 21, 49, 5, 17, 21, 49, 21, 53, 81, 129, 5, 17, 21, 49, 21, 53, 81, 129

Offset: 1

Omar E. Pol, Jul 14 2009

nonn

It appears that a(2^k) = 1, for k >= 0. [From Omar E. Pol, Feb 22 2010]

			Contribution from _Omar E. Pol_, Feb 22 2010: (Start)
If written as a triangle:
1;
1,5;
1,5,5,17;
1,5,5,17,5,17,21,49;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129;
1,5,5,17,5,17,21,49,5,17,21,49,21,53,81,129,5,17,21...
Rows converge to A173464.
(End)
Contribution from Omar E. Pol, Apr 01 2011 (Start):
It appears that the final terms of rows give A000337.
It appears that row sums give A006516.
(End)

Nathaniel Johnston, Table of n, a(n) for n = 1..94
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

Cf. A139250, A139251, A159791, A159792, A162793, A162794, A162795, A162796.

Cf. A000337, A058922, A173464. [From Omar E. Pol, Feb 22 2010]

a(n) = A162796(n) - A162795(n).

Edited by Omar E. Pol, Jul 18 2009
More terms from Omar E. Pol, Feb 22 2010
More terms (a(51)-a(55)) from Nathaniel Johnston, Mar 30 2011

A159791 Bisection of toothpick sequence A139250.

Original entry on oeis.org

0, 3, 11, 23, 43, 55, 79, 123, 171, 183, 207, 251, 303, 347, 423, 571, 683, 695, 719, 763, 815, 859, 935, 1083, 1199, 1243, 1319, 1467, 1607, 1759, 2011, 2475, 2731, 2743, 2767, 2811, 2863, 2907, 2983, 3131, 3247, 3291, 3367, 3515, 3655, 3807, 4059, 4523, 4783, 4827, 4903, 5051, 5191, 5343, 5595, 6059, 6343, 6495, 6747, 7215

Offset: 1

Omar E. Pol, Apr 28 2009

nonn

Robert Price, Table of n, a(n) for n = 1..32768
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

Cf. A139250.

a(28)-a(60) from Robert Price, May 10 2019

A160422 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 but with toothpicks of length 6.

Original entry on oeis.org

0, 7, 19, 41, 63, 87, 131, 193, 235, 259, 303, 367, 435, 527, 675, 837, 919, 943, 987, 1051, 1119, 1211, 1359, 1523, 1631, 1723, 1875, 2071, 2299, 2631, 3087, 3489, 3651, 3675, 3719, 3783, 3851, 3943, 4091, 4255, 4363, 4455, 4607, 4803, 5031, 5363, 5819, 6223, 6411

Offset: 0

Omar E. Pol, May 20 2009

nonn

a(n) is also the number of grid points that are covered after n-th stage by an polyedge as the toothpick structure of A139250, but with toothpicks of length 6.

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

Cf. A139250, A139251, A147614, A160118, A160120, A160170, A160420, A160430.

a(n) = A147614(n)+4*A139250(n) = A160420(n)+2*A139250(n) since each toothpick covers exactly four more grid points than the corresponding toothpick in A147614.

More terms and formula from Nathaniel Johnston, Nov 13 2010

A162793 Number of toothpicks added to the toothpick structure A139250 at the n-th odd round.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 16, 32, 4, 12, 16, 32, 16, 36, 60, 80, 4, 12, 16, 32, 16, 36, 60, 80, 16, 36, 60, 84, 60, 112, 208, 192, 4, 12, 16, 32, 16, 36, 60, 80, 16, 36, 60, 84, 60, 112, 208, 192, 16, 36, 60, 84, 60, 112, 208, 196, 60, 112, 208, 224, 212, 364, 672, 448, 4, 12, 16, 32, 16

Offset: 1

Omar E. Pol, Jul 14 2009

Bisection of A139251.
Note that these toothpicks are parallel to the initial toothpick in the structure.
First differences of A162795. - Omar E. Pol, Feb 23 2015

			From _Omar E. Pol_, Feb 23 2015: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,12;
4,12,16,32;
4,12,16,32,16,36,60,80;
4,12,16,32,16,36,60,80,16,36,60,84,60,112,208,192;
4,12,16,32,16,36,60,80,16,36,60,84,60,112,208,192,16,36,60,84,60,112,208,196,60,112,208,224,212,364,672,448;
...
It appears that right border gives the positive terms of A001787.
It appears that row sums give A000302.
(End)

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
Index entries for sequences related to cellular automata

Cf. A000302, A001787, A139250, A139251, A147582, A159791, A159792, A162794, A162795, A162796, A162797, A169708.

More terms from N. J. A. Sloane, Dec 28 2009

A211012 Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 8, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224, 17179607040, 68718952448, 274876858368, 1099509530624, 4398042316800, 17592177655808, 70368727400448, 281474943156224

Offset: 0

Omar E. Pol, Sep 21 2012

All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.
Similar to A271061. - Robert Price, Mar 30 2016
For n=3,5,..., also the number of minimum vertex colorings in the n-sunlet graph. - Eric W. Weisstein, Mar 03 2024

			For n = 3 the area of all squares and rectangles in the toothpick structure after 2^3 stages equals the area of a rectangle of size 8X6, so a(3) = 8*6 = 48.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Sunlet Graph.
Index entries for linear recurrences with constant coefficients, signature (6,-8).

Row sums of triangle A211017, n>=1.

Cf. A000079, A006516, A139250, A159786, A160124, A211008, A211016, A211018.

concat(vector(2), Vec(8*x^2/((1-2*x)*(1-4*x)) + O(x^50))) \\ Colin Barker, Mar 30 2016

a(n) = 2^n * (2^n-2) = A000079(n)*(A000079(n) - 2) = A159786(2^n) = 8*A006516(n-1), n>=1.
From Colin Barker, Mar 30 2016: (Start)
G.f.: 8*x^2 / ((1-2*x)*(1-4*x)).
a(n) = 6*a(n-1)-8*a(n-2) for n>2. (End)
E.g.f.: (1 - exp(2*x))^2. - Stefano Spezia, Mar 12 2025

A159792 Bisection of toothpick sequence A139250.

Original entry on oeis.org

1, 7, 15, 35, 47, 67, 95, 155, 175, 195, 223, 283, 319, 383, 483, 651, 687, 707, 735, 795, 831, 895, 995, 1163, 1215, 1279, 1379, 1551, 1667, 1871, 2219, 2667, 2735, 2755, 2783, 2843, 2879, 2943, 3043, 3211, 3263, 3327, 3427, 3599, 3715, 3919, 4267, 4715

Offset: 1

Omar E. Pol, Apr 28 2009

nonn

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

Cf. A139250, A159791.

More terms from R. J. Mathar, May 26 2009

A162794 Number of toothpicks added to the toothpick structure A139250 at the n-th even round.

Original entry on oeis.org

0, 2, 4, 8, 8, 8, 12, 28, 16, 8, 12, 28, 20, 28, 40, 88, 32, 8, 12, 28, 20, 28, 40, 88, 36, 28, 40, 88, 56, 92, 140, 256, 64, 8, 12, 28, 20, 28, 40, 88, 36, 28, 40, 88, 56, 92, 140, 256, 68, 28, 40, 88, 56, 92, 140, 256, 88, 92, 140, 260, 172, 296, 488, 704, 128, 8, 12, 28, 20, 28

Offset: 0

Omar E. Pol, Jul 14 2009

nonn

Note that these toothpicks are orthogonal to the initial toothpick in the sieve.

A bisection of A139251.

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

Cf. A139250, A139251, A159791, A159792, A162793, A162795, A162796, A162797.

Extended by R. J. Mathar, Sep 27 2009

A170927 Consider the 2^n values of A139250(i)/i^2 for 2^n <= i < 2^(n+1); a(n) = value of i where this quantity is minimized.

Original entry on oeis.org

1, 2, 5, 12, 21, 44, 89, 180, 362, 728, 1459, 2921, 5843, 11690, 23384, 46770, 93544, 187094, 374193, 748391, 1496786, 2993576, 5987158, 11974321, 23948647, 47897300, 95794608, 191589222, 383178450, 766356910, 1532713828, 3065427664, 6130855333, 12261710675

Offset: 0

Benoit Jubin, Jan 22 2010, Feb 06 2010

nonn

{log_2 a(n)} converges to about 0.513441 and equivalently 2^{log_2 a(n)}-1 converges to about 1.427451, and the corresponding values T(i)/i^2 converge to about 0.4513058.

For all values listed, a(n) = 2 * a(n-1) + c(n), where c(n) is a small positive integer, except for a(4) where c(4)=-3. - Robert Price, Aug 16 2015

			The values of A139250(i)/i^2 for i = 1 .., 15 are 1.0, 0.7500000000, 0.7777777778, 0.6875000000, 0.6000000000, 0.6388888889, 0.7142857143, 0.6718750000, 0.5802469136, 0.5500000000, 0.5537190083, 0.5486111111, 0.5621301775, 0.6275510204, 0.6888888889, 0.6679687500. The minimal value for 4 <= i <= 7 is 0.6000000000 at i=5.

Robert Price, Table of n, a(n) for n = 0..169
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.]
Steven R. Finch, Toothpicks and Live Cells, July 21, 2015. [Cached copy, with permission of the author]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A170927, A147562, A260239, A261313.

a(26)-a(33) from Robert Price, Aug 18 2012

A211008 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after n-th stage in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 0, 2, 0, 4, 0, 4, 4, 4, 8, 8, 2, 8, 12, 4, 8, 12, 4, 12, 12, 4, 16, 16, 4, 16, 20, 4, 20, 20, 4, 32, 28, 4, 40, 44, 8, 2, 40, 52, 12, 4, 40, 52, 12, 4, 44, 52, 12, 4, 48, 56, 12, 4, 48, 60, 12, 4, 52, 60, 12, 4, 64, 68, 12, 4, 72, 84, 16, 4

Offset: 1

Omar E. Pol, Sep 18 2012

It appears that the number of rectangles of area 2 in the toothpick structure of A139250 equals the number of hearts in the Q-toothpick cellular automaton of A187210. See conjecture in formula section.

			For n = 8 in the toothpick structure after 8 stages we have that:
T(8,1) = 8 is the number of squares of size 1 X 1.
T(8,2) = 12 is the number of rectangles of size 1 X 2.
T(8,3) = 4 is the number of squares of size 2 X 2.
Written as an irregular array the sequence begins:
   0;
   0;
   0,  2;
   0,  4;
   0,  4;
   4,  4;
   8,  8,  2;
   8, 12,  4;
   8, 12,  4;
  12, 12,  4;
  16, 16,  4;
  16, 20,  4;
  20, 20,  4;
  32, 28,  4;
  40, 44,  8,  2;
  40, 52, 12,  4;

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

Zero together with the row sums gives A160124.

Cf. A139250, A159786, A160125, A168131, A187210, A188346, A211016, A211017, A211019.

It appears that T(n,2) = A188346(n+2) (checked by hand up to n = 128 in the toothpick structure of A139250).

cp's OEIS Frontend

A160125 Number of squares and rectangles that are created at the n-th stage in the toothpick structure (see A139250).

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A162797 a(n) = difference between the number of toothpicks of A139250 that are orthogonal to the initial toothpick and the number of toothpicks that are parallel to the initial toothpick, after n even rounds.

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A159791 Bisection of toothpick sequence A139250.

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A160422 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 but with toothpicks of length 6.

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A162793 Number of toothpicks added to the toothpick structure A139250 at the n-th odd round.

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A211012 Total area of all squares and rectangles after 2^n stages in the toothpick structure of A139250, assuming the toothpicks have length 2.

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A159792 Bisection of toothpick sequence A139250.

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A162794 Number of toothpicks added to the toothpick structure A139250 at the n-th even round.

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A170927 Consider the 2^n values of A139250(i)/i^2 for 2^n <= i < 2^(n+1); a(n) = value of i where this quantity is minimized.

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