Walter Trump - cp's OEIS frontend

User: Walter Trump

Walter Trump has authored 5 sequences.

A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

0, 0, 0, 1, 1, 1, 3, 2, 4, 4, 6, 5, 10, 7, 12, 11, 16, 13, 22, 17, 26, 23, 32, 27, 41, 33, 47, 42, 56, 48, 68, 57, 77, 69, 89, 78, 105, 90, 117, 106, 133, 118, 153, 134, 169, 154, 189, 170, 214, 190, 234, 215, 259, 235, 289, 260, 314, 290, 344, 315, 380

Offset: 0

Walter Trump, Aug 22 2025

a(n) also is the number of convex polyiamonds (triangular polyominoes) with perimeter n.

Walter Trump, Table of n, a(n) for n = 0..10000
Walter Trump, Convex Polyiamonds
Walter Trump, Examples for a(3)-a(12)

Cf. A096004 (number of convex polyiamonds with n cells), A284869 (including nonconvex but simply connected polyiamonds with perimeter n), A057729 (including polyiamonds with holes), A036418 (including rotations and reflections but no holes).

A353727 Index in A353715 of the first term divisible by 2^n and no higher power of 2, or -1 if no such term exists.

Original entry on oeis.org

0, 2, 3, 11, 54, 74, 88, 183, 20334, 30938, 21247, 90575, 3913, 124845, 2643790, 5828721, 2469947, 4005550, 19917707

Offset: 0

Walter Trump, May 11 2022

			Table showing initial values of n (column 1) and a(n) (column 3).
The central column shows the corresponding entry of A353715 written in base 2.
The entries in column 2 end in exactly n zeros.
   n                    A353715(a(n))       a(n)
   0                                1         0
   1                              110         2
   2                             1100         3
   3                          1101000        11
   4                         11110000        54
   5                       1111100000        74
   6                      11101000000        88
   7                     110110000000       183
   8              1111110101100000000     20334
   9             10111110101000000000     30938
  10              1111111110000000000     21247
  11           1001111111100000000000     90575
  12                11111000000000000      3913
  13          10011111110000000000000    124845
  14      111110111110100000000000000   2643790
  15    10011111111111000000000000000   5828721
  16      111111111110000000000000000   2469947
  17     1111111101100000000000000000   4005550
  18  1011111111111000000000000000000  19917707

Cf. A353709, A353715, A353724, A353725, A353726.

A343595 a(n) is the number of axially symmetric tilings of the order-n Aztec Diamond by square tetrominoes and Z-shaped tetrominoes, not counting rotations and reflections as distinct.

Original entry on oeis.org

1, 1, 2, 7, 26, 162, 1096, 12210, 149384, 2979716, 65702176, 2347717180, 93123644320, 5962338902536, 424966024145024, 48757525297347464, 6240064849995542656, 1282987881672304949776, 294690971817685508825600, 108580010933558879525595504

Offset: 1

Walter Trump, Apr 21 2021

nonn

No tiling is symmetric to both the x- and the y-axis.
No tiling is symmetric to an oblique symmetry axis of the diamond.
If a tiling is symmetric to the x-axis then a reflection over the y-axis is equal to a rotation by 180 degrees.
The number of tilings is 4 * a(n) if rotations are counted as distinct.
All tilings have exactly the minimum number of square tetrominoes given by ceiling(n/2).

James Propp, A Pedestrian Approach to a Method of Conway, or, A Tale of Two Cities, Mathematics Magazine, Vol. 70, No. 5 (Dec., 1997), 327-340.
Walter Trump, Axially symmetric Aztec diamond tilings

Cf. A342907.

A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 0, 1, 9, 9, 3, 1, 16, 48, 32, 0, 1, 25, 150, 250, 75, 15, 1, 36, 360, 1200, 1224, 288, 0, 1, 49, 735, 4165, 8869, 6321, 931, 133, 1, 64, 1344, 11648, 43136, 64512, 33024, 4096, 0, 1, 81, 2268, 27972, 160866, 423306, 469800

Offset: 1

Walter Trump, Mar 09 2021

T(0,0):=1 for combinatorial reasons.
A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.
Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.

			  1;
  1,  1;
  1,  4,   0;
  1,  9,   9,   3;
  1, 16,  48,  32,  0;
  1, 25, 150, 250, 75, 15;

Walter Trump, Table of n, a(n) for n = 1..222
Walter Trump, Semi-queen problem

Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.

T(n,0) = 1.
T(n,1) = n^2.
T(n,2) = n^2*(n-1)*(n-2)/2.
T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.
T(2n+1,2n+1) = A006717(n).
T(2n,2n) = 0.

A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

Original entry on oeis.org

37, 53, 89, 113, 3821, 3989, 4657, 28661, 29021, 41641, 41669, 44249, 50909, 56053, 57041, 57301, 133981, 16501361, 46178761, 47633441, 47633477, 47722049, 47736121, 47774621, 47803477, 47810209, 47835013, 47835341, 47854969, 47862413, 47865017, 49448573, 49448617

Offset: 1

N. J. A. Sloane, Dec 17 2019, following a suggestion from Eric Angelini. a(5) and a(6) were found by Walter Trump. a(7)-a(17) from N. J. A. Sloane, Dec 17 2019

nonn

Eric Angelini's illustration shows the first 19 rows of the triangle. Each row ends when a prime is reached, and the next row starts directly under this prime but moves in the opposite direction.
The extended illustration from Walter Trump resembles a giant ski run.
Hans Havermann's plots of A330545, linked here, extend Walter Trump's graph to 4*10^8 rows (probably the longest ski run in the world). Only the turns are shown, and the illustration has been turned sideways.
A330545(k) = 0 iff prime(k) is a term of the present sequence. In a sense A330545 and the simpler A330547 are the more fundamental sequences and show the connection between the present problem and the ordinary primes and their alternating sums.
Note that because primes > 2 are odd, a prime can only appear in column 0 at the end of a row that is moving towards the left. A prime appearing in a row moving to the right will always appear in an odd-numbered column (and in particular, not in the zero column).
Furthermore the column number mod 4 uniquely determines the residue class of primes mod 4 in that column. If the column number is 0,1,2,3 mod 4 then the primes in that column are 1,3,3,1 respectively (see the "Notes" link). In particular, a(n) == 1 mod 4. - N. J. A. Sloane, Jan 04 2020
Note that the primes > 2 in column one and two are the primes in A282178.
Note on the links: The illustrations from Angelini and Trump show all the terms 0,1,2,3,4,..., while those of Havermann and Sloane just show the primes (as in A330545).

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 516 terms from Hans Havermann)
Eric Angelini, Illustration of beginning of the triangle in A330339.
Hans Havermann, Plot of 4*10^8 terms of A330545, sampled every 1000 terms, points joined.
Hans Havermann, More detailed view of terms of A330545 from 290 million to 310 million, sampled every 10 terms, points joined.
N. J. A. Sloane, Illustration of first 16 rows of A330545.
N. J. A. Sloane, Notes on the sequence of Bostrophedon primes (A330339) and the "ski-run" A330545.
N. J. A. Sloane, State diagram for columns of A330545.
Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [The zeroth column is just to the right of the vertical red line. Note that after a while the rows extend to the left of the red line. The digits are too small to be read.]
Walter Trump, An extended picture of the triangle in A330339, showing the first 550 rows, down to the row that starts 3989. [Same picture as the previous one, but with 6 red dots added to show the primes in column 0.]

Cf. A282178, A330545, A330547.
A330546 gives the list of indices i such that a(n) = prime(i).
A127596 is another sequence with a similar flavor.
Not to be confused with A000747 = Boustrophedon transform of primes.

More terms from Hans Havermann, Dec 17 2019

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User: Walter Trump

A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

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A353727 Index in A353715 of the first term divisible by 2^n and no higher power of 2, or -1 if no such term exists.

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A343595 a(n) is the number of axially symmetric tilings of the order-n Aztec Diamond by square tetrominoes and Z-shaped tetrominoes, not counting rotations and reflections as distinct.

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A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

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A330339 Boustrophedon primes: write the numbers 0, 1, 2, 3, ... in a triangle on a square grid in the boustrophedon manner, ending a row when a prime is reached; sequence lists primes that appear in the zeroth column.

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