A211017 -id:A211017 - cp's OEIS frontend

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

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).

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.

Original entry on oeis.org

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.

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.

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.

cp's OEIS Frontend

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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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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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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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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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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