Douglas Boffey - cp's OEIS frontend

A376702 Numbers k whose nonzero digits are strictly decreasing when written in factoradic.

1, 2, 4, 5, 6, 12, 13, 14, 18, 19, 20, 22, 23, 24, 48, 49, 50, 54, 72, 73, 74, 76, 77, 78, 84, 85, 86, 96, 97, 98, 100, 101, 102, 108, 109, 110, 114, 115, 116, 118, 119, 120, 240, 241, 242, 246, 264, 360, 361, 362, 364, 365, 366, 372, 373, 374, 384, 408, 409, 410

Offset: 1

Douglas Boffey, Oct 02 2024

Base factorial is defined as the right hand digit being the units, the next left being the 2's, then the 6's, and so on.

			50, being 2010 (base !), is included, whereas 51, being 2011 (base !), is not included.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Wikipedia, Factorial number system

Cf. A007623, A108731.

isok(n)={my(k=1,p=0); while(n, k++; my(r=n%k); if(r, if(r<=p, return(0)); p=r); n\=k); 1} \\ Andrew Howroyd, Oct 04 2024

def f(n, i=2): return [n] if n < i else [*f(n//i, i=i+1), n%i]
def ok(n):
    fnz = [d for d in f(n) if d != 0]
    return len(fnz) == len(set(fnz)) and fnz == sorted(fnz, reverse=True)
print([k for k in range(1, 411) if ok(k)]) # Michael S. Branicky, Oct 02 2024

# faster for initial segment of sequence
from math import factorial
from itertools import count, islice
def bgen(d, i): # strictly decreasing non-zero elmts <= i and dth digit from left <= d
    if d < 1: yield tuple(); return
    yield from ((j,) + t for j in range(0, min(i+1, d+1)) for t in bgen(d-1, i if j == 0 else j-1))
def agen(): # generator of terms
    for digits in count(1):
        for first in range(1, digits+1):
            for rest in bgen(digits-1, first-1):
                t = (first, ) + rest
                yield sum(factorial(i)*d for i, d in enumerate(t[::-1], 1))
print(list(islice(agen(), 60))) # Michael S. Branicky, Oct 02 2024

A374624 a(n) is the number of irreducible finite Coxeter groups in n dimensions, or -1 if there are an infinite number.

Original entry on oeis.org

1, -1, 3, 5, 3, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Douglas Boffey, Jul 14 2024

For n > 8, the Coxeter groups are exactly A(n), B(n) = C(n), and D(n), hence a(n) = 3.

			For n = 4, there are five finite groups, denoted A(4) (symmetry group of the simplex), B(4) (= C(4)) (symmetry group of the tesseract and the 4-dimensional cross polytope), D(4) (symmetry group of the demitesseract), F(4) (symmetry group of the 24-cell) and H(4) (symmetry group of the 120-cell and the 600-cell).

H. S. M. Coxeter, Regular Polytopes, Dover Publications, Inc., 1973.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A060296, A358241.

PadRight[{1, -1, 3, 5, 3, 4, 4, 4}, 100, 3] (* Paolo Xausa, Dec 07 2024 *)

a(n)=if(n>8,3,[1,-1,3,5,3,4,4,4][n]) \\ Charles R Greathouse IV, Jul 15 2024

G.f.: (1 - 2*x + 4*x^2 + 2*x^3 - 2*x^4 + x^5 - x^8)/(1 - x). - Stefano Spezia, Jul 15 2024

A371704 Heights at which a Minecraft boat-drop will break up.

Original entry on oeis.org

12, 13, 49, 51, 111, 114, 198, 202, 310, 315

Offset: 1

Douglas Boffey, Apr 03 2024

Mojang Studios, Boats crash/break and can kill their passengers when falling certain distances contains the bug report relating to this list.
Matt Parker, The Minecraft boat-drop mystery.

Table[{k*(25 k - 1)/2, k*(25 k + 1)/2}, {k, 1, 5}] // Flatten (* Robert P. P. McKone, Apr 03 2024 *)

a(n) = (50*n^2 + 50*n + 23 - 23*(-1)^n*(1+2*n)) / 16. - Robert P. P. McKone, Apr 03 2024

a(7)-a(10) corrected by Robert P. P. McKone, Apr 03 2024

A371163 Numbers that remain unchanged when converted to their compressed fibbinary numbers.

Original entry on oeis.org

0, 1, 2, 9, 10, 115544, 13568075, 13568077, 13568078, 13568083, 13568085, 13568086

Offset: 1

Douglas Boffey, Mar 13 2024

			9 is a term since 9 = 8 + 1 = F(6) + F(2), where F(i) is the i-th Fibonacci number, is Fibbinary A003714(9) = 10001_2, but then all '01's are compressed to '1', leaving A048679(9) = 1001_2, which is 9 itself again.

Fixed points of A048679 and A048680.

Cf. A000045, A003714.

from itertools import count, islice
def A371163_gen(): # generator of terms
    c = 0
    for n in count(0):
        if not (n<<1)&n:
            if  int(bin(n)[2:].replace('01','1'),2) == c:
                yield c
            c += 1
A371163_list = list(islice(A371163_gen(),6)) # Chai Wah Wu, Mar 18 2024

A366503 Triangle read by rows: T(n,k) = number of permutations of (1, 2, ..., n) with longest monotonic subsequence of length k (1<=k<=n).

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 0, 4, 18, 2, 0, 0, 86, 32, 2, 0, 0, 306, 362, 50, 2, 0, 0, 882, 3242, 842, 72, 2, 0, 0, 1764, 24564, 12210, 1682, 98, 2, 0, 0, 1764, 163872, 161158, 32930, 3026, 128, 2, 0, 0, 0, 985032, 1969348, 592652, 76562, 5042, 162, 2

Offset: 1

Douglas Boffey, Oct 12 2023

			Triangle begins:
  1;
  0, 2;
  0, 4,    2;
  0, 4,   18,     2;
  0, 0,   86,    32,     2;
  0, 0,  306,   362,    50,    2;
  0, 0,  882,  3242,   842,   72,  2;
  0, 0, 1764, 24564, 12210, 1682, 98, 2;
  ...
The T(4, 2) = 4 permutations are: 2,1,4,3; 2,4,1,3; 3,1,4,2; 3,4,1,2.

Douglas Boffey, C++ program used to generate the sequence

Row sums are A000142.

Cf. A047874.

A366411 a(n) is the total number of Hamiltonian paths on rectangular grids of size n X k for 1 <= k <= n.

Original entry on oeis.org

1, 5, 29, 353, 5485, 260323, 15277779, 3211933481, 790502793321, 751204032996623, 808973740240335345, 3499358510266725179221, 16872857815987642169925097, 333905047994165804391613820789, 7308582206982080422190512432243395

Offset: 1

Douglas Boffey, Oct 09 2023

			For n = 2, the a(2) = 5 solutions are:
  +---+  +---+---+  +---+---+  +---+---+  +---+---+
  |   |  |   |   |  |   |   |  |   |   |  |   |   |
  | * |  | **|** |  | * | * |  | **|** |  | **|** |
  | * |  |   | * |  | * | * |  | * |   |  | * | * |
  +---+  +---+---+  +---+---+  +---+---+  +---+---+
  | * |  |   | * |  | * | * |  | * |   |  | * | * |
  | * |  | **|** |  | **|** |  | **|** |  | * | * |
  |   |  |   |   |  |   |   |  |   |   |  |   |   |
  +---+  +---+---+  +---+---+  +---+---+  +---+---+

Row sums of triangle A366399.

More terms (using A332307) from Pontus von Brömssen, Oct 09 2023

A366399 Triangle read by rows: T(n,k) is the number of paths traveling orthogonally on an n X k grid that visit every cell.

Original entry on oeis.org

1, 1, 4, 1, 8, 20, 1, 14, 62, 276, 1, 22, 132, 1006, 4324, 1, 32, 336, 3610, 26996, 229348, 1, 44, 688, 12010, 109722, 1620034, 13535280, 1, 58, 1578, 38984, 602804, 12071462, 175905310, 3023313284, 1, 74, 3190, 122188, 2434670, 82550864, 1449655468, 43551685370, 745416341496

Offset: 1

Douglas Boffey, Oct 09 2023

			T(n,k) is a triangular array read by rows:
  1,
  1,  4,
  1,  8, 20,
  1, 14, 62, 276,
  ...
T(2,2) = 4:
  +---+---+  +---+---+  +---+---+  +---+---+
  |   |   |  |   |   |  |   |   |  |   |   |
  | **|** |  | * | * |  | **|** |  | **|** |
  |   | * |  | * | * |  | * |   |  | * | * |
  +---+---+  +---+---+  +---+---+  +---+---+
  |   | * |  | * | * |  | * |   |  | * | * |
  | **|** |  | **|** |  | **|** |  | * | * |
  |   |   |  |   |   |  |   |   |  |   |   |
  +---+---+  +---+---+  +---+---+  +---+---+

See A332307 for another version.

Cf. A120443 (T(n,n)), A366411 (row sums).

More terms (using A332307) from Pontus von Brömssen, Oct 09 2023

A364440 Triangle T(n,k) (n >= 1 and 1 <= k <= n) read by rows, arising from the Mosaic Problem.

Original entry on oeis.org

0, 0, 1, 0, 73, 31998, 0, 3960, 10414981, 20334816290, 0, 190475

Offset: 1

Douglas Boffey, Aug 02 2023

Fill an n X k array of cells with tiles taken from a set of six (each one connecting two sides of the cell). T(n,k) is the number of tilings containing at least one loop.

There are 6 tiles, all of size 1 X 1, one for each way of joining two sides of the cell.

			Triangle begins:
        k=1    k=2       k=3          k=4
  n=1:   0;
  n=2:   0,      1;
  n=3:   0,     73,    31998;
  n=4:   0,   3960, 10414981, 20334816290;
  n=5:   0, 190475, ...
  ...
For T(3, 2), there are 73 solutions (squares marked with an asterisk can take any of the six different tiles):
.
1. (36 tilings)   2. (36 tilings)   3. (1 tiling)
  +---+---+---+     +---+---+---+     +---+---+---+
  |   |   |   |     |   |   |   |     |   |   |   |
  |   |   | * |     | * |   |   |     |   |---|   |
  |  /|\  |   |     |   |  /|\  |     |  /|   |\  |
  +---+---+---+     +---+---+---+     +---+---+---+
  |  \|/  |   |     |   |  \|/  |     |  \|   |/  |
  |   |   | * |     | * |   |   |     |   |---|   |
  |   |   |   |     |   |   |   |     |   |   |   |
  +---+---+---+     +---+---+---+     +---+---+---+

Jack Hanke, The Mosaic Problem - How and Why to do Math for Fun, Youtube video.

T(n,1) = 0 for all n.

T(n,2) = 36^n - ((36*beta - 35)*beta^(1 - n) - (36*alpha - 35)*alpha^(1 - n))/(beta - alpha), where alpha = (1 + sqrt(33/37))/2 and beta = (1 - sqrt(33/37))/2.

A356134 Triangular array giving total number of legal Go positions on an n X k board.

Original entry on oeis.org

1, 5, 57, 15, 489, 12675, 41, 4125, 321689, 24318165, 113, 35117, 8180343, 1840058693, 414295148741, 313, 299681, 208144601, 139304759213, 93332304864173, 62567386502084877, 867, 2557605, 5296282323, 10546705714473, 21026744638200555, 41945191530093646965, 83677847847984287628595

Offset: 1

Douglas Boffey, Jul 27 2022

			Triangle T(n,k) begins:
    1;
    5,    57;
   15,   489,   12675;
   41,  4125,  321689,   24318165;
  113, 35117, 8180343, 1840058693, 414295148741;
  ...

Douglas Boffey, Table of n, a(n) for n = 1..190 (rows 1..19 flattened).
J. Tromp, Number of Legal Go Positions

Columns give: A102620, A266278.
Main diagonal gives A094777.
A356049 gives the table by antidiagonals.

a(27) corrected by Sidney Cadot, Jan 05 2023.

A356049 Symmetric array read by antidiagonals: T(n,k) is the number of legal positions in Go on an n X k board.

Original entry on oeis.org

1, 5, 5, 15, 57, 15, 41, 489, 489, 41, 113, 4125, 12675, 4125, 113, 313, 35117, 321689, 321689, 35117, 313, 867, 299681, 8180343, 24318165, 8180343, 299681, 867, 2401, 2557605, 208144601, 1840058693, 1840058693, 208144601, 2557605, 2401

Offset: 1

Douglas Boffey, Jul 24 2022

A Go position is a grid containing white and black stones with the condition that every orthogonally connected group of stones of a single color has liberties, i.e., is orthogonally adjacent to an empty cell.

			Array begins:
   1,   5,  15,  41, ...
   5,  57, 489, ...
  15, 489, ...
  41, ...
  ...
T(3,1) = 15 from
  ... ..w ..b .w. .ww  .b. .bb w.. w.w w.b  ww. b.. b.w b.b bb.

Douglas Boffey, Table of n, a(n) for n = 1..528
J. Tromp, Number of Legal Go Positions

Columns (or rows) give: A102620, A266278.
Main diagonal gives A094777.
This as triangle gives A356134.

cp's OEIS Frontend

User: Douglas Boffey

A376702 Numbers k whose nonzero digits are strictly decreasing when written in factoradic.

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A374624 a(n) is the number of irreducible finite Coxeter groups in n dimensions, or -1 if there are an infinite number.

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Mathematica

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A371704 Heights at which a Minecraft boat-drop will break up.

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A371163 Numbers that remain unchanged when converted to their compressed fibbinary numbers.

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Python

A366503 Triangle read by rows: T(n,k) = number of permutations of (1, 2, ..., n) with longest monotonic subsequence of length k (1<=k<=n).

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A366411 a(n) is the total number of Hamiltonian paths on rectangular grids of size n X k for 1 <= k <= n.

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A366399 Triangle read by rows: T(n,k) is the number of paths traveling orthogonally on an n X k grid that visit every cell.

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A364440 Triangle T(n,k) (n >= 1 and 1 <= k <= n) read by rows, arising from the Mosaic Problem.

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Formula

A356134 Triangular array giving total number of legal Go positions on an n X k board.

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