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.

User: Yi Yang

Yi Yang's wiki page.

Yi Yang has authored 11 sequences. Here are the ten most recent ones:

A381979 Decimal expansion of the expected number of steps to termination by self-trapping of a self-avoiding random walk on the square lattice.

Original entry on oeis.org

7, 0, 7, 5, 9
Offset: 2

Views

Author

Yi Yang, Mar 11 2025

Keywords

Comments

The average walk length determined by 1.2*10^12 simulations is 70.75915 +- 0.00010

Examples

			70.759...

References

Links

Crossrefs

Cf. A378903 (The expected walk length on the cubic lattice).
Cf. A077483 (Probability of the occurrence of each walk length).
Cf. A322831.

A357516 Number of snake-like polyominoes in an n X n square that start at the NW corner and end at the SE corner and have the maximum length.

Original entry on oeis.org

1, 2, 6, 20, 2, 64, 44, 512, 28, 4, 64, 520, 480, 6720, 43232, 14400
Offset: 1

Views

Author

Yi Yang, Oct 01 2022

Keywords

Comments

The maximum length is given by A357234(n).
If the lower bounds of A357234(n) are tight, then a(14)-a(19) are 6720, 43232, 14400, 226560, 1646080, 403712.
For n > 1, a(n) is even since for every solution there is also the symmetrical solution reflected in the main diagonal.

Examples

			For n = 5, there are 2 such snakes shown as follows:
  X . X X X         X X X X X
  X . X . X         . . . . X
  X . X . X         X X X X X
  X . X . X         X . . . .
  X X X . X         X X X X X

Links

Crossrefs

Cf. A331986, A357234.

Extensions

a(14)-a(16) from Andrew Howroyd, Feb 28 2023

A357234 a(n) is the maximum length of a snake-like polyomino in an n X n square that starts and ends at opposite corners.

Original entry on oeis.org

1, 3, 5, 7, 17, 23, 31, 39, 51, 63, 75, 89, 105, 121, 139, 159
Offset: 1

Views

Author

Yi Yang, Sep 18 2022

Keywords

Comments

Snake-like polyominoes have all cells with at most two neighbor cells, and have at least one cell that has only one neighbor cell, where neighbors are horizontal or vertical (not diagonal).
Lower bounds for a(10)-a(22) are 63, 75, 89, 105, 121, 139, 159, 179, 201, 225, 249, 275, 303. Is it true that a(n) = round((2*n*n-4*n+28)/3) for n >= 9?

Examples

			Longest snakes for 5 <= n <= 8:
  X X X X X   X X X X X X   X X X . X X X   X . X X X X X X
  . . . . X   . . . . . X   . . X . X . X   X . X . . . . X
  X X X X X   X X X X X X   X X X . X . X   X . X X X X . X
  X . . . .   X . . . . .   X . . X X . X   X X . . . X . X
  X X X X X   X . X X X X   X . . X . X X   . X . X X X . X
              X X X . . X   X . . X . X .   X X . X . . X X
                            X X X X . X X   X . . X . . X .
                                            X X X X . . X X

Links

Crossrefs

Cf. A331968, A357516.

Formula

a(n) ~ 2*n^2/3. - Pontus von Brömssen, Sep 19 2022
a(n) <= A331968(n). - Pontus von Brömssen, Sep 21 2022

Extensions

a(1)-a(9) confirmed by Pontus von Brömssen, Sep 21 2022. - N. J. A. Sloane, Sep 30 2022
a(10)-a(13) confirmed by Elijah Beregovsky, Nov 27 2022
a(14)-a(16) from Andrew Howroyd, Feb 28 2023

A293289 Number of level n squares on a Sierpinski carpet that intersect the edge of a circle with the same center and diameter.

Original entry on oeis.org

1, 8, 28, 76, 204, 580, 1556, 4180, 11204, 29724, 79276, 212076, 565692, 1509332, 4026028, 10740796, 28646804, 76396620, 203728972, 543283204, 1448779164, 3863345612, 10302538780, 27473690092, 73263231116, 195369181668, 520985280228, 1389296277316, 3704793953044
Offset: 0

Views

Author

Yi Yang, Oct 05 2017

Keywords

Comments

There are 8^n level n squares on a Sierpinski carpet.
The terms of this sequence have a common factor 4 except a(0).
Lim_{n->infinity} a(n)/a(n-1) = 8/3.
Lim_{n->infinity} a(n)/(8/3)^n = 4.38167....

Links

Crossrefs

Cf. A094976, A242118, A293288.

A293288 Number of level n squares on a Sierpinski carpet that are enclosed in a circle with the same center and diameter.

Original entry on oeis.org

0, 0, 32, 332, 2908, 23900, 192844, 1547068, 12388068, 99135316, 793162780, 6345516140, 50764701756, 406119132924, 3248957101772, 25991667561644, 207933369171996, 1663467029827132, 13307736442512524, 106461892083564380, 851695138117736668, 6813561108806027412
Offset: 0

Views

Author

Yi Yang, Oct 04 2017

Keywords

Comments

There are 8^n level n squares on a Sierpinski carpet.
The terms of this sequence have a common factor 4.
Lim_{n->infinity} a(n)/a(n-1) = 8.
Lim_{n->infinity} a(n)/8^n = 0.73872777586247800994....

Links

Crossrefs

Cf. A001018, A261849, A293289.

A292621 a(n) = a(n-1) + a(floor(log(n))) with a(1) = 1, a(2) = 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 139, 143, 147, 151, 155, 159, 163, 167
Offset: 1

Views

Author

Yi Yang, Sep 20 2017

Keywords

Comments

a(n) > c*n*log(n)*log(log(n))*log(log(log(n)))*...*log(log...(log(n))...) (k layers) for any sufficient large n, any constant c and any positive integer k.
The sum of 1/a(i) for i = 1, 2, 3, ... diverges extremely slowly.

Links

Crossrefs

Cf. A000195, A031876, A292620.

Programs

  • Maple
    f:= proc(n) option remember;
procname(n-1)+procname(floor(log(n)))
end proc:
f(1):= 1: f(2):= 2:
map(f, [$1..100]); # Robert Israel, Sep 28 2017
  • Mathematica
    a[n_] := a[n] = If[n <= 2, n, a[n - 1] + a[Floor@ Log@ n]]; Array[a, 62] (* Michael De Vlieger, Sep 21 2017 *)
  • PARI
    a(n) = if (n<=2, n, a(n-1) + a(floor(log(n)))); \\ Michel Marcus, Sep 21 2017

A292620 a(n) = a(n-1) + a(floor(log_2(n))), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 23, 26, 29, 32, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 220, 227, 234, 241, 248, 255, 262, 269, 276, 283, 290, 297, 304, 311
Offset: 1

Views

Author

Yi Yang, Sep 20 2017

Keywords

Comments

a(n) > c*n*log_2(n)*log_2(log_2(n))*log_2(log_2(log_2(n)))*...*log_2(log_2...(log_2(n))...) (k layers) for any sufficiently large n, any constant c and any positive integer k.
The sum of 1/a(i) for i = 1, 2, 3, ... converges extremely slowly to the limit 5.70....

Links

Crossrefs

Cf. A000523, A292621.

Programs

  • Maple
    f:= proc(n) option remember; procname(n-1)+procname(ilog2(n)) end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Sep 24 2017
  • Mathematica
    a[n_] := a[n] = If[n == 1, 1, a[n - 1] + a[Floor@ Log2@ n]]; Array[a, 59] (* Michael De Vlieger, Sep 21 2017 *)
  • PARI
    a(n) = if (n<=2, n, a(n-1) + a(logint(n, 2))); \\ Michel Marcus, Sep 21 2017

A226239 Minimum m such that there exists an n-row subtractive triangle with distinct integers in 1..m.

Original entry on oeis.org

1, 3, 6, 10, 15, 22, 33, 44, 59, 76, 101, 125, 158
Offset: 1

Views

Author

Yi Yang, Jun 01 2013

Keywords

Comments

In an n-row subtractive triangle, there are n-i+1 integers in the i-th row. The integers in the first row are arbitrary. From the next row, the integers are the absolute difference between adjacent integers in the previous row.

Examples

			a(6)=22 because there is a 6-row subtractive triangle with distinct integers in [1..22] as follows:
1:  6 20 22  3 21 13
2: 14  2 19 18  8
3: 12 17  1 10
4:  5 16  9
5: 11  7
6:  4
However, there is no such triangle with distinct integers in [1..21].

Links

Crossrefs

Cf. A035312, A035313.

Extensions

a(12) from Yi Yang, Mar 04 2015
a(13) from Denis Cazor, Aug 01 2022

A189243 Number of ways to dissect a nonsquare rectangle into n rectangles with equal area.

Original entry on oeis.org

1, 2, 6, 21, 88, 390, 1914
Offset: 1

Views

Author

Yi Yang, Apr 19 2011

Keywords

Comments

Dissections which differ by rotations or reflections are counted as distinct.
Rectangles may have different shapes.
a(1) to a(5) are the same (but not a(6)) as:
A033540 a(n+1) = n*(a(n)+1), n >= 1, a(1) = 1.
If the dissections with a cross (where four squares share a vertex) were counted twice then a(1) to a(5) would be the same as the 'guillotine partitions' counted by A006318. - Geoffrey H. Morley, Dec 31 2012

Examples

			There are 6 ways to form a rectangle from 3 rectangles with same area:
+-----+ +-+-+-+ +-----+ +--+--+ +-+---+ +---+-+
|     | | | | | |     | |  |  | | |   | |   | |
+-----+ | | | | +--+--+ |  |  | | |   | |   | |
|     | | | | | |  |  | |  |  | | +---+ +---+ |
+-----+ | | | | |  |  | +--+--+ | |   | |   | |
|     | | | | | |  |  | |     | | |   | |   | |
+-----+ +-+-+-+ +--+--+ +-----+ +-+---+ +---+-+
So a(3)=6.
From _Geoffrey H. Morley_, Dec 03 2012: (Start)
b(n) in the given formula is the sum of the appropriate tilings from certain 'frames'. A number that appears in a subrectangle in a frame is the number of rectangles into which the subrectangle is to be divided. Tilings are also counted that are from a reflection and/or half-turn of the frame.
For n = 6 there are 3(X2) frames:
+---+-+-+  +-+-----+  +-+-----+
|   | | |  | |     |  | |     |
|   | | |  | +---+-+  | |  2  |
+-+-+ | |  | |   | |  | |     |
| | | | |  | +---+ |  | +---+-+
| | +-+-+  | |   | |  | |   | |
| | |   |  +-+---+ |  +-+---+ |
| | |   |  |     | |  |     | |
+-+-+---+  +-----+-+  +-----+-+
  2 ways     2 ways     8 ways
The only other frames which yield desired tilings are obtained by rotating each frame above by 90 degrees and scaling it to fit a rectangle with the inverse aspect ratio.
So b(6) = 2(2+2+8) = 24, and a(6) = b(6)+4*a(5)+2*a(4)-4*a(3)-2*a(2) = 24+4*88+2*21-4*6-2*2 = 390.
For n = 7 we can use 7(X2) frames:
+---+--+
|   |  |
|   |  |
| 4 |3 |
|   |  |
|   |  |
|   |  |
+---+--+
63 ways [of creating tilings counted by b(7)]
+---+--+  +-+----+  +--+---+  +-----++  +--+---+  +----+-+
|   |  |  | |    |  |  |   |  ++----+|  |  |   |  ++-+-+ |
|   +-++  | +---++  |2 | 2 |  ||    ||  |  +-+-+  || | | |
| 3 | ||  |2|   ||  |  +--++  ||    ||  |2 | | |  || | | |
|   | ||  | | 2 ||  |  |  ||  || 3  ||  |  | | |  || +-+-+
|   | ||  | |   ||  +--+--+|  ||    ||  +--+-+2|  || |   |
+---+-+|  +-+---+|  |     ||  |+----++  |    | |  |+-+---+
+-----++  +-----++  +-----++  ++-----+  +----+-+  ++-----+
24 ways   16 ways   12 ways   10 ways    8 ways    4 ways
As for n = 6, these are only half the frames and tilings.
So b(7) = 2(63+24+16+12+10+8+4) = 274, and a(7) = b(7)+4*a(6)+2*a(5)-4*a(4)-2*a(3) = 274+4*390+2*88-4*21-2*6 = 1914.
(End)

Links

Crossrefs

See the analogous sequences A219861 and A108066 where we count dissections up to symmetry of nonsquare rectangles and squares respectively. - Geoffrey H. Morley, Dec 03 2012

Formula

For n > 4, a(n) = b(n)+
+-------+ +-------+ +-------+ +---+---+ +---+---+
| | | | | | | | | | | |
+-------+ +-------+ +-------+ +---+---+ +---+---+
|[a(n-1)| | | | | |[a(n-2)| | |
|-a(n-2)|*4+| a(n-2)|*2+| a(n-3)|*4+|-a(n-3)|*4+| a(n-4)|*2
|-a(n-3)| +-------+ +---+---+ |-a(n-4)| +---+---+
|] | | | | | | |] | | | |
+-------+ +-------+ +---+---+ +-------+ +---+---+
= b(n)+4*a(n-1)+2*a(n-2)-4*a(n-3)-2*a(n-4) where b(n) is the number of tilings in which no side of the rectangle comprises the side of a tile or the equal sides of two congruent tiles. For example, b(5) = 2. '*2' counts, say, rotation clockwise by 90 degrees (and rescaling the aspect ratio), while '*4' counts all rotations. - Geoffrey H. Morley, Dec 07 2012

Extensions

Edited by N. J. A. Sloane, Apr 21 2011
a(7) added by Geoffrey H. Morley, Dec 03 2012
a(7) corrected by Geoffrey H. Morley, Dec 05 2012

A179594 Y-coordinate of the first n*n non-coprime block above the line x=y in the first quadrant.

Original entry on oeis.org

2, 20, 1308, 10199370, 131487690152
Offset: 1

Views

Author

Yi Yang, Jul 20 2010

Keywords

Comments

a(n) is the minimum y that satisfies x <= y and the GCD of (x,y),(x+1,y), ..., (x+n-1,y), (x,y+1),(x+1,y+1), ..., (x+n-1,y+1), ..., (x,y+n-1),(x+1,y+n-1), ..., (x+n-1,y+n-1) are all larger than 1.

Examples

			a(2)=20 because the GCDs of (14,20), (15,20), (14,21), (15,21) are all larger than 1, and there is no such array within x <= y < 20.

References

  • Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 1093.

Links

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