cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

Omar E. Pol, May 03 2009

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A147614 a(n) = number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 3, 7, 13, 19, 27, 39, 53, 63, 71, 83, 99, 119, 147, 183, 217, 235, 243, 255, 271, 291, 319, 355, 391, 419, 447, 487, 539, 607, 699, 803, 885, 919, 927, 939, 955, 975, 1003, 1039, 1075, 1103, 1131, 1171, 1223, 1291, 1383, 1487, 1571, 1615
Offset: 0

Views

Author

David Applegate, Apr 29 2009

Keywords

Comments

a(n) is also the 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. [From Omar E. Pol, May 18 2009]
It appears that the number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2*k, is equal to (2*k-2) * A139250(n) + a(n), k>0. See formulas in A160420 and A160422. [From Omar E. Pol, Nov 15 2010]
More generally, it appears that a(n) is also the number of grid points that are covered by the "special points" of the toothpicks of A139250, after n-th stage, assuming the toothpicks have length 2*k, k>0 and that each toothpick has three special points: the midpoint and two endpoints.
Note that if k>1 then also there are 2*k-2 grid points that are covered by each toothpick, but these points are not considered for this sequence. [From Omar E. Pol, Nov 15 2010]
Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that a(n)/A139250(n) converge to 4/3.
It appears that a(n)/A160124(n) converge to 2.
It appears that a(n)/A139252(n) converge to 4.
(End)

Links

Crossrefs

Cf. A139250, A139251, A139252, A160124, A160420, A160422, A170884.

Formula

Since A160124(n) = 1+2*A139250(n)-A147614(n), n>0 (see A160124), and we have recurrences for A160125 (hence A160124) and A139250, we have a recurrence for this sequence. - N. J. A. Sloane, Feb 02 2010
a(n) = A187220(n+1)-A160124(n), n>0. - Omar E. Pol, Feb 15 2013

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

Views

Author

Omar E. Pol, Sep 18 2012

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

Zero together with the row sums gives A160124.
Cf. A139250, A159786, A160125, A168131, A187210, A188346, A211016, A211017, A211019.

Formula

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

Views

Author

Omar E. Pol, May 03 2009

Keywords

Links

Crossrefs

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

Formula

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

Extensions

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

A168131 Number of squares and rectangles that are created at the n-th stage in the corner toothpick structure (see A152980, A153006).

Original entry on oeis.org

0, 0, 1, 2, 1, 1, 5, 7, 3, 1, 4, 5, 3, 7, 18, 19, 7, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 56, 47, 15, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6, 13, 13, 13, 31, 51, 41, 20, 25, 39, 39, 58, 120, 160, 111, 31, 1, 4, 5, 3, 7, 17, 17, 7, 6, 13, 13, 13, 32, 55, 45, 15, 6
Offset: 0

Views

Author

Omar E. Pol, Jan 18 2010

Keywords

Comments

Essentially the first differences of A170926. - Omar E. Pol, Feb 16 2013

Examples

			If written as a triangle:
0,
0,
1,2,
1,1,5,7,
3,1,4,5,3,7,18,19,
7,1,4,5,3,7,17,17,7,6,13,13,13,32,56,47,
15,1,4,5,3,7,17,17,7,6,13,13,13,32,55,45,15,6,13,13,13,31,51,41,20,...
The rows (omitting the first term) converge to A170929.

Links

Crossrefs

Cf. A152980, A153006, A170926, A160124, A160125, A139250.

Programs

  • Maple
    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;
[seq(w(n),n=0..256)];
  • Mathematica
    a[n_] := a[n] = Module[{k, i}, Which[n==0, 0, n <= 3, n-1, True, k = Floor[Log2[n]]; i = n-2^k; Which[i==0, 2^(k-1)-1, i < 2^k-2, 2*a[i]+a[i+1], i==2^k-2, 2*a[i]+a[i+1]+1, True, 2*a[i]+a[i+1]+2]]];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, Sep 25 2022, after Maple code *)

Formula

See Maple program for recurrence.

Extensions

Edited and extended by N. J. A. Sloane, Feb 01 2010

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

Views

Author

Omar E. Pol, May 03 2009

Keywords

Links

Crossrefs

Toothpick sequence: A139250.
Cf. A159786, A159787, A159788, A159789, A160124, A160125, A160126.

Extensions

More terms from Colin Barker, Apr 19 2015

A160018 A175099 with a(3) changed from 0 to 1.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 2, 2, 0, 0, 4, 7, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 15, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 14, 0, 0, 4, 4, 0, 0, 8, 12, 0, 0, 8, 8, 0, 0, 16, 31, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 14, 0, 0, 4, 4, 0, 0, 8, 12, 0, 0, 8, 8, 0, 0, 16, 30, 0, 0, 4, 4, 0
Offset: 0

Views

Author

N. J. A. Sloane, Feb 07 2010

Keywords

Links

Crossrefs

Cf. A160125, A175098, A175099.

Formula

a(n) = a(1) = 0; for k >= 1, a(2^k) = 2^(k-1)-1, a(2^k+i) = 2*a(i) for 1 <= i < 2^k.

A175099 Number of closed rectangles added at the n-th stage of the leftist toothpicks A151566.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 3, 0, 0, 2, 2, 0, 0, 4, 7, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 15, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 14, 0, 0, 4, 4, 0, 0, 8, 12, 0, 0, 8, 8, 0, 0, 16, 31, 0, 0, 2, 2, 0, 0, 4, 6, 0, 0, 4, 4, 0, 0, 8, 14, 0, 0, 4, 4, 0, 0, 8, 12, 0, 0, 8, 8, 0, 0, 16, 30, 0, 0, 4, 4, 0
Offset: 0

Views

Author

R. J. Mathar, Feb 04 2010

Keywords

Links

Crossrefs

Cf. A160125, A175098

Formula

For a recurrence see A160018. - N. J. A. Sloane, Feb 07 2010
Showing 1-8 of 8 results.

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