A139250 -id:A139250 - cp's OEIS frontend

A211016 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

0, 0, 4, 8, 12, 4, 40, 52, 12, 4, 168, 212, 52, 12, 4, 680, 852, 212, 52, 12, 4, 2728, 3412, 852, 212, 52, 12, 4, 10920, 13652, 3412, 852, 212, 52, 12, 4, 43688, 54612, 13652, 3412, 852, 212, 52, 12, 4, 174760, 218452, 54612, 13652, 3412, 852, 212, 52, 12, 4

Offset: 1

Omar E. Pol, Sep 18 2012

All internal regions in the toothpick structure are squares and rectangles.

			For n = 5 in the toothpick structure after 2^5 stages we have that:
T(5,1) = 168 is the number of squares of size 1 X 1.
T(5,2) = 212 is the number of rectangles of size 1 X 2.
T(5,3) = 52 is the total number of squares of size 2 X 2 and of rectangles of size 1 X 4.
T(5,4) = 12 is the number of rectangles of size 2 X 4.
T(5,5) = 4 is the number of rectangles of size 2 X 8.
Triangle begins:
       0;
       0,      4;
       8,     12,     4;
      40,     52,    12,     4;
     168,    212,    52,    12,    4;
     680,    852,   212,    52,   12,   4;
    2728,   3412,   852,   212,   52,  12,   4;
   10920,  13652,  3412,   852,  212,  52,  12,  4;
   43688,  54612, 13652,  3412,  852, 212,  52, 12,  4;
  174760, 218452, 54612, 13652, 3412, 852, 212, 52, 12, 4;

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

Row sums give 0 together with A145655.

Cf. A020988, A072197, A080675, A139250, A160124, A211008, A211017, A211018, A211019.

T(n,k) = A211008(2^n,k) = 4*A211019(n,k).

T(n,1) = 4*A020988(n-2), n>=2.

A160128 a(n) = number of grid points that are covered after (2^n)th stage of A139250.

Original entry on oeis.org

3, 7, 19, 63, 235, 919, 3651, 14575, 58267, 233031, 932083, 3728287, 14913099, 59652343, 238609315, 954437199, 3817748731, 15270994855, 61083979347, 244335917311, 977343669163, 3909374676567, 15637498706179

Offset: 0

Omar E. Pol, May 09 2009

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 linear recurrences with constant coefficients, signature (6,-9,4).

Cf. A000079, A007583, A139250, A139251, A139252, A139560, A147614.

Cf. Same recurrence: A073724, A210985, A014825.

Vec((3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)) + O(x^40)) \\ Colin Barker, May 13 2020

a(n) = A147614(A000079(n)).
a(n) = (1/9)*(2^(2*n+3) + 12*n + 19). [Nathaniel Johnston, Mar 29 2011]
It appears that a(n) = A139252(2^(n+1)). - Omar E. Pol, Sep 11 2012
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3). - Paul Curtz, May 07 2020
G.f.: (3 - 11*x + 4*x^2) / ((1 - x)^2*(1 - 4*x)). - Colin Barker, May 13 2020

Terms after a(10) from Nathaniel Johnston, Mar 29 2011

A211017 T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2. Triangle read by rows.

Original entry on oeis.org

0, 0, 8, 8, 24, 16, 40, 104, 48, 32, 168, 424, 208, 96, 64, 680, 1704, 848, 416, 192, 128, 2728, 6824, 3408, 1696, 832, 384, 256, 10920, 27304, 13648, 6816, 3392, 1664, 768, 512, 43688, 109924, 54608, 27296, 13632, 6784, 3328, 1536, 1024

Offset: 1

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.

			For n = 5 in the toothpick structure after 2^5 stages we have that:
T(5,1) = 168 is the total area of all squares of size 1 X 1.
T(5,2) = 424 is the total area of all rectangles of size 1 X 2.
T(5,3) = 208 is the total area of all squares of size 2 X 2 and of all rectangles of size 1 X 4.
T(5,4) = 96 is the total area of all rectangles of size 2 X 4.
T(5,5) = 64 is the total area of all rectangles of size 2 X 8.
Triangle begins:
      0;
      0,     8;
      8,    24,    16;
     40,   104,    48,   32;
    168,   424,   208,   96,   64;
    680,  1704,   848,  416,  192,  128;
   2728,  6824,  3408, 1696,  832,  384, 256;
  10920, 27304, 13648, 6816, 3392, 1664, 768, 512;

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

Row sums give A211012.

Cf. A020988, A139250, A160124, A211016, A211018, A211019.

T(n,k) = A211016(n,k)*2^(k-1).

T(n,1) = 4*A020988(n-2), n>=2.

A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 1, 2, 3, 1, 10, 13, 3, 1, 42, 53, 13, 3, 1, 170, 213, 53, 13, 3, 1, 682, 853, 213, 53, 13, 3, 1, 2730, 3413, 853, 213, 53, 13, 3, 1, 10922, 13653, 3413, 853, 213, 53, 13, 3, 1, 43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1, 174762, 218453

Offset: 1

Omar E. Pol, Sep 24 2012

All internal regions in the toothpick structure are squares and rectangles.

			Triangle begins:
0;
0,         1;
2,         3,     1;
10,       13,     3,    1;
42,       53,    13,    3,   1;
170,     213,    53,   13,   3,   1;
682,     853,   213,   53,  13,   3,  1;
2730,   3413,   853,  213,  53,  13,  3,  1;
10922, 13653,  3413,  853, 213,  53, 13,  3, 1;
43690, 54613, 13653, 3413, 853, 213, 53, 13, 3, 1;

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

Row sums give 0 together with A014825.

Cf. A002450, A020988, A072197, A086462, A139250, A160124, A211016, A211017, A211018.

T(n,k) = A211016(n,k)/4.

T(n,1) = A020988(n-2), n>=2.

A151885 Similar to the original toothpick sequence A139250, except that the rule is now: a toothpick changes state if its midpoint is adjacent to exactly one ON toothpick.

Original entry on oeis.org

0, 1, 3, 5, 11, 5, 15, 17, 39, 5, 15, 25, 55, 17, 51, 61, 139, 5, 15, 25, 55, 25, 75, 85, 195, 17, 51, 85, 187, 61, 183, 217, 495, 5, 15, 25, 55, 25, 75, 85, 195, 25, 75, 125, 275, 85, 255, 305, 695, 17, 51, 85, 187, 85, 255, 289, 663

Offset: 0

N. J. A. Sloane, Jul 23 2009

nonn

In the original toothpick sequence A139250, a toothpick simply turned ON (and stayed ON) if its midpoint was adjacent to exactly one ON toothpick.

Related to A139250 in the same way that A079315 is related to A147562.

Nathaniel Johnston, Table of n, a(n) for n = 0..925
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.]
Nathaniel Johnston, C program for computing terms of A151885-A151888
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Illustration of first 8 generations

Cf. A151886-A151888, A139250, A139251, A147562, A079314, A079315.

a(n) = a(n-1) + A151888(n).

Terms after a(8) from Nathaniel Johnston, Apr 02 2011

A160126 Total number of squares and rectangles in the toothpick structure after n stages, divided by 2. (See A139250).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 4, 9, 12, 12, 14, 18, 20, 22, 32, 47, 54, 54, 56, 60, 62, 64, 74, 88, 94, 96, 104, 114, 120, 134, 170, 209, 224, 224, 226, 230, 232, 234, 244, 258, 264, 266, 274, 284, 290, 304, 340, 378, 392, 394, 402, 412, 418, 432, 466, 500, 514

Offset: 0

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

Cf. A139250, A159786, A159787, A159788, A159789, A160124, A160125, A160127.

a(n) = A160124(n)/2. - Nathaniel Johnston, Apr 12 2011

Terms beyond a(10) from Nathaniel Johnston, Apr 12 2011

A195853 Decimal expansion of the lower limit of A139250(i)/i^2.

Original entry on oeis.org

4, 5, 1, 3, 0, 5, 8, 2, 8, 4, 5, 3, 1, 2, 2, 1, 3, 5, 8, 9, 6, 4, 0, 1, 3, 4, 2, 2, 8, 0, 4, 0, 0, 1, 2, 3, 5, 1, 0, 2, 0, 4, 1, 3, 8, 3, 2, 9

Offset: 0

Omar E. Pol, Sep 27 2011

Constant mentioned in the Applegate-Pol-Sloane article, Section 5, the fractal-like structure. It is also mentioned in A139250 and A170927.
Note that at least there are 162 constants starting with 0.451305 in the Plouffe's Inverter.
Also evidently half the lower limit of A147562(n)/n^2; see A260239. - Steven Finch, Jul 21 2015

			0.451305...

D. 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

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.], which is also available at arXiv:1004.3036v2, [math.CO], 2010.
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.

a(6)-a(14) from Robert Price, Aug 18 2012
a(15)-a(47) from Robert Price, Aug 16 2015
Removed first fifteen terms as they were duplicated in the last edit by Robert Price, Dec 17 2018

A211018 Triangle read by rows: T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 8, n>=1, k>=1, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 13, 6, 4, 21, 53, 26, 12, 8, 85, 213, 106, 52, 24, 16, 341, 853, 426, 212, 104, 48, 32, 1365, 3413, 1706, 852, 424, 208, 96, 64, 5461, 13653, 6826, 3412, 1704, 848, 416, 192, 128, 21845, 54613, 27306, 13652, 6824, 3408, 1696, 832, 384, 256

Offset: 1

Omar E. Pol, Sep 24 2012

All internal regions in the toothpick structure are squares and rectangles. The area of every internal region is a power of 2.

			0;
0,        1;
1,        3,    2;
5,       13,    6,    4;
21,      53,   26,   12,    8;
85,     213,  106,   52,   24,  16;
341,    853,  426,  212,  104,  48,  32;
1365,  3413, 1706,  852,  424, 208,  96,  64;
5461, 13653, 6826, 3412, 1704, 848, 416, 192, 128;

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

Rows sums give A006516. Right border gives A131577.

Cf. A002450, A072197, A139250, A160124, A211012, A211016, A211017, A211019

T(n,k) = A211017(n,k)/8.

T(n,1) = A002450(n-2), n>=2.

A159790 Toothpick number A139250(n) minus triangular number A000217(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 7, 7, 2, 0, 1, 1, 4, 18, 35, 35, 22, 12, 5, -3, -8, -2, 7, 3, -6, -4, 5, 17, 48, 106, 155, 155, 126, 100, 77, 53, 32, 22, 15, -5, -30, -44, -51, -55, -40, 2, 35, 23, -10, -32, -47, -59, -52, -18, 11, 11, 14, 48, 101, 181, 328, 522, 651, 651, 590, 532

Offset: 0

Omar E. Pol, May 03 2009

sign

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
OEIS Plot 2, Triangular numbers and toothpick numbers vs n

More terms from R. J. Mathar, May 04 2009

A160570 Triangle read by rows, A160552 convolved with (1, 2, 2, 2, ...); row sums = A139250, the Toothpick sequence.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 1, 6, 2, 2, 3, 2, 6, 2, 2, 5, 6, 2, 6, 2, 2, 7, 10, 6, 2, 6, 2, 2, 1, 14, 10, 6, 2, 6, 2, 2, 3, 2, 14, 10, 6, 2, 6, 2, 2, 5, 6, 2, 14, 10, 6, 2, 6, 2, 2, 7, 10, 6, 2, 14, 10, 6, 2, 6, 2, 2, 5, 14, 10, 6, 2, 14, 10, 6, 2, 6, 2, 2, 11, 10, 14, 10, 6, 2, 14, 10, 6, 2, 6

Offset: 1

Gary W. Adamson, May 19 2009

			First few rows of the triangle:
  1;
  1,  2;
  3,  2,  2;
  1,  6,  2,  2;
  3,  2,  6,  2,  2;
  5,  6,  2,  6,  2,  2;
  7, 10,  6,  2,  6,  2,  2;
  1, 14, 10,  6,  2,  6,  2,  2;
  3,  2, 14, 10,  6,  2,  6,  2,  2;
  5,  6,  2, 14, 10,  6,  2,  6,  2,  2;
  ...
Example: Row 4 = (1, 6, 2, 2) = (1, 3, 1, 1) dot (1, 2, 2, 2); where (1 + 6 + 2 + 2) = A139250(4), i.e., 4th term in the Toothpick sequence.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
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. A160552, A139250.

T:=proc(n,k)if(k=1)then return A160552(n):else return 2*A160552(n-k+1):fi:end:
for n from 1 to 8 do for k from 1 to n do print(T(n,k));od:od: # Nathaniel Johnston, Apr 13 2011

Construct triangle M = an infinite lower triangular Toeplitz matrix with A160552: (1, 1, 3, 1, 3, 5, 7, ...) in every column. Let Q = an infinite lower triangular matrix with (1, 2, 2, 2, 2, ...) as the main diagonal and the rest zeros. A160570 = M * Q.

cp's OEIS Frontend

A211016 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2.

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A160128 a(n) = number of grid points that are covered after (2^n)th stage of A139250.

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A211017 T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, n>=1, k>=1, assuming the toothpicks have length 2. Triangle read by rows.

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A211019 Triangle read by rows: T(n,k) = number of squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 4, n>=1, k>=1, assuming the toothpicks have length 2.

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A151885 Similar to the original toothpick sequence A139250, except that the rule is now: a toothpick changes state if its midpoint is adjacent to exactly one ON toothpick.

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A160126 Total number of squares and rectangles in the toothpick structure after n stages, divided by 2. (See A139250).

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A195853 Decimal expansion of the lower limit of A139250(i)/i^2.

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A211018 Triangle read by rows: T(n,k) = total area of all squares and rectangles of area 2^(k-1) after 2^n stages in the toothpick structure of A139250, divided by 8, n>=1, k>=1, assuming the toothpicks have length 2.

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A159790 Toothpick number A139250(n) minus triangular number A000217(n).

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