A159786 -id:A159786 - cp's OEIS frontend

A159789 a(n) = A159786(n+1)/4.

0, 0, 1, 2, 2, 3, 8, 12, 12, 13, 16, 18, 19, 26, 44, 56, 56, 57, 60, 62, 63, 70, 84, 92, 93, 98, 106, 111, 120, 152, 216, 240, 240, 241, 244, 246, 247, 254, 268, 276, 277, 282, 290, 295, 304, 336, 384, 408, 409, 414

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 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

Toothpick sequence: A139250.

Cf. A139251, A153000, A159786, A159787, A159788, A159799.

More terms from Colin Barker, Apr 19 2015

A159787 a(n) = A159786(n+1)*3/4.

Original entry on oeis.org

0, 0, 3, 6, 6, 9, 24, 36, 36, 39, 48, 54, 57, 78, 132, 168, 168, 171, 180, 186, 189, 210, 252, 276, 279, 294, 318, 333, 360, 456, 648, 720, 720, 723, 732, 738, 741, 762, 804, 828, 831, 846, 870, 885, 912, 1008, 1152, 1224, 1227, 1242

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 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

Toothpick sequence: A139250.

Cf. A139251, A153003, A159786, A159788, A159789, A159797.

a(11)-a(49) from Robert Price, May 10 2019

A159788 a(n) = A159786(n+1)/2.

Original entry on oeis.org

0, 0, 2, 4, 4, 6, 16, 24, 24, 26, 32, 36, 38, 52, 88, 112, 112, 114, 120, 124, 126, 140, 168, 184, 186, 196, 212, 222, 240, 304, 432, 480, 480, 482, 488, 492, 494, 508, 536, 552, 554, 564, 580, 590, 608, 672, 768, 816, 818, 828

Offset: 0

Omar E. Pol, Apr 28 2009, May 02 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

Toothpick sequence: A139250.

Cf. A139251, A152998, A159786, A159787, A159789, A159798.

a(11)-a(49) from Robert Price, May 10 2019

A159796 First differences of A159786.

Original entry on oeis.org

0, 0, 4, 4, 0, 4, 20, 16, 0, 4, 12, 8, 4, 28, 72, 48, 0, 4, 12, 8, 4, 28, 56, 32, 4, 20, 32, 20, 36, 128, 256, 96, 0, 4, 12, 8, 4, 28, 56, 32, 4, 20, 32, 20, 36, 128, 192, 96, 4, 20

Offset: 1

Omar E. Pol, Jun 13 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

Cf. A139250, A159786, A159789, A159799.

More terms from Colin Barker, Apr 19 2015

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

Original entry on oeis.org

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

Omar E. Pol, May 03 2009

nonn

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

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

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

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 *)

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

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

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

Original entry on oeis.org

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

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

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).

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

A160127 First differences of A160126.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 5, 3, 0, 2, 4, 2, 2, 10, 15, 7, 0, 2, 4, 2, 2, 10, 14, 6, 2, 8, 10, 6, 14, 36, 39, 15, 0, 2, 4, 2, 2, 10, 14, 6, 2, 8, 10, 6, 14, 36, 38, 14, 2, 8, 10, 6, 14, 34, 34, 14

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

Toothpick sequence: A139250.

Cf. A159786, A159787, A159788, A159789, A160124, A160125, A160126.

More terms from Colin Barker, Apr 19 2015

cp's OEIS Frontend

A159789 a(n) = A159786(n+1)/4.

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A159787 a(n) = A159786(n+1)*3/4.

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A159788 a(n) = A159786(n+1)/2.

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A159796 First differences of A159786.

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

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

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Maple

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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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PARI

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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.

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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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Author

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Links

Crossrefs

Formula

Extensions

A160127 First differences of A160126.