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A297622 Triangle read by rows: a(n,k) is the number of k X n matrices which are the first k rows of an alternating sign matrix of size n.

1, 1, 1, 1, 2, 2, 1, 3, 7, 7, 1, 4, 16, 42, 42, 1, 5, 30, 149, 429, 429, 1, 6, 50, 406, 2394, 7436, 7436, 1, 7, 77, 938, 9698, 65910, 218348, 218348, 1, 8, 112, 1932, 31920, 403572, 3096496, 10850216, 10850216, 1, 9, 156, 3654, 90576, 1931325, 29020904, 247587252, 911835460, 911835460

Offset: 0

Ben Branman, Jan 01 2018

Comments: An alternating sign matrix of size n is an n X n matrix of 0's, 1's and -1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.  If k < n, we relax the condition on the columns slightly, and require that
(a) If a column is not all zeros, the first nonzero entry is 1;
(b) The nonzero entries in each column alternate in sign.
The second reference gives a sequence of partially ordered sets Phi_n such that the alternating sign matrices of size n are in bijection with the maximal chains of Phi_n.  This sequence gives the number of saturated chains in Phi_n which begin at the root vertex and end at any vertex of height k.

			a(3,3)=7 because there are seven alternating sign matrices of size 3.  Six of these are the permutation matrices, and the seventh is the matrix ((0,1,0),(1,-1,1),(0,1,0)).
a(n,0)=1 because there is only one possible n X 0 matrix: the empty matrix.
a(4,4)=42 because there are 42 4 X 4 alternating sign matrices.  If we look only at the first two rows in each of the 42 4 X 4 alternating sign matrices, we get 16 distinct 2 X 4 matrices, and so a(4,2)=16.  The 16 2 X 4 matrices are
  {{0,  0,  0,  1}, {0,  0,  1,  0}},
  {{0,  0,  0,  1}, {0,  1,  0,  0}},
  {{0,  0,  0,  1}, {1,  0,  0,  0}},
  {{0,  0,  1,  0}, {0,  0,  0,  1}},
  {{0,  0,  1,  0}, {0,  1,  0,  0}},
  {{0,  0,  1,  0}, {1,  0,  0,  0}},
  {{0,  1,  0,  0}, {0,  0,  0,  1}},
  {{0,  1,  0,  0}, {0,  0,  1,  0}},
  {{0,  1,  0,  0}, {1,  0,  0,  0}},
  {{1,  0,  0,  0}, {0,  0,  0,  1}},
  {{1,  0,  0,  0}, {0,  0,  1,  0}},
  {{1,  0,  0,  0}, {0,  1,  0,  0}},
  {{0,  0,  1,  0}, {0,  1, -1,  1}},
  {{0,  0,  1,  0}, {1,  0, -1,  1}},
  {{0,  1,  0,  0}, {1, -1,  0,  1}},
  {{0,  1,  0,  0}, {1, -1,  1,  0}}.
Triangle begins:
=============================================================================================
n\k|  0  1   2    3      4       5         6          7           8            9           10
---|-----------------------------------------------------------------------------------------
_0 |  1
_1 |  1  1
_2 |  1  2   2
_3 |  1  3   7    7
_4 |  1  4  16   42     42
_5 |  1  5  30  149    429     429
_6 |  1  6  50  406   2394    7436      7436
_7 |  1  7  77  938   9698   65910    218348     218348
_8 |  1  8 112 1932  31920  403572   3096496   10850216    10850216
_9 |  1  9 156 3654  90576 1931325  29020904  247587252   911835460    911835460
10 |  1 10 210 6468 229680 7722110 205140540 3586953760 33631201864 129534272700 129534272700
  ...

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999

Robert G. Wilson v, Table of n, a(n) for n = 0..104
Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.
Paul Terwilliger, A Poset Phi_n whose maximal chains are in bijection with the n by n alternating sign matrices, arXiv:1710.04733 [math.CO], 2017.

Cf. A005130.

(* First we compute the Hasse diagram for Terwilliger's poset as a directed graph object. *)
ToAlternatingSignList[list_] :=
Module[{s = 1},
  Table[If[list[[k]] == 0, 0, (s = -s); -s], {k, 1, Length[list]}]]
AllAlternatingSignRows[n_] :=
AllAlternatingSignRows[
   n] = (ToAlternatingSignList /@
    Select[Table[IntegerDigits[q, 2, n], {q, 0, 2^n - 1}],
     OddQ[Total[#]] &])
output[vertex_] :=
Select[Table[
   vertex + li, {li, AllAlternatingSignRows[Length[vertex]]}],
  And[Min[#] >= 0, Max[#] <= 1] &]
elist[vertex_] := ((vertex \[DirectedEdge] #) & /@ output[vertex])
ASMPoset[n_] :=
ASMPoset[n] =
  Graph[Flatten[
    Table[elist[IntegerDigits[k, 2, n]], {k, 0, 2^n - 1}]]]
(*Now we compute the number of paths of length k starting at the root vertex.*)
ASMPosetAdjacencyMatrix[n_] := Normal[AdjacencyMatrix[ASMPoset[n]]]
Table[Total /@
  First /@ NestList[ASMPosetAdjacencyMatrix[n].# &,
    IdentityMatrix[2^n], n], {n, 1, 10}]

a(n,0) = 1;
a(n,1) = n;
a(n,n-1) = a(n,n) = A005130(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.

A208229 Consider Wolfram's universal 2-state 3-symbol Turing machine on a one-way-infinite tape with all but the first n cells initially blank and the head initially in state 1. a(n) is the maximum number of steps the Turing machine can make before halting.

Original entry on oeis.org

3, 3, 15, 19, 37, 51, 69, 97, 111, 161, 167, 241, 247, 327, 341, 435, 451, 547, 571, 665, 709, 801, 849

Offset: 0

Ben Branman, Jan 10 2013

The head starts in state 1 on the leftmost cell (cell 1.) The machine halts if the head moves to the left of the first cell. a(n) gives the maximum halting time for any of the 3^n initial configurations of the first n cells.
Because the rule (rule 596440) is universal, the halting problem is undecidable for arbitrary tapes. However, it is not known whether universal computation can be achieved with a finite number of nonzero starting cells. Thus, this function might be computable.
Since the head starts in state one, it will immediately move left and halt unless the first cell starts out with a 0.
For initial conditions with n<=11, the machine always halts (the Mathematica code given assumes that the machine halts for all finite configurations). Whether this remains true is an open question.

			For n=3, the initial tape 021000000... runs for 19 steps, before terminating in the state 22221200000... with the head one cell to the left of the tape. This is the longest running time without starting with nonzero symbols further to the right, so a(3)=19

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 709, 1120.

Michael S. Branicky, Python program for OEIS A208229
Wikipedia, Wolfram's 2-state 3-symbol Turing machine
Stephen Wolfram, Wolfram 2,3 Turing Machine Research Prize

r = {{1, 2} -> {1, 1, -1}, {1, 1} -> {1, 2, -1}, {1, 0} -> {2, 1, 1}, {2, 2} -> {1, 0, 1}, {2, 1} -> {2, 2, 1}, {2, 0} -> {1, 2, -1}}; len[n_, k_] := Length[NestWhileList[TuringMachine[r, #] &, {{1,2}, {Prepend[IntegerDigits[k, 3, n], 3], 0}}, UnsameQ, All]] - 2; a[n_] := Max[Table[len[n, k], {k, 0, 3^(n - 1) - 1}]]; Join[{3},Table[a[n],{n,1,8}]]

Python
```
# see Branicky link
```

a(12)-a(14) from Robert G. Wilson v, Mar 22 2015

a(15)-a(22) from Michael S. Branicky, Jul 07 2025

A202177 Number of partitions p of n such that each part of p is prime and each part of the conjugate partition of p is also prime.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 1, 1, 2, 0, 2, 2, 2, 2, 3, 3, 0, 4, 2, 5, 2, 4, 3, 8, 2, 6, 4, 11, 0, 10, 4, 14, 2, 14, 4, 21, 2, 20, 5, 25, 0, 28, 6, 30, 2, 38, 5, 46, 0, 44, 4, 54, 0, 56, 6, 67, 2, 72, 4, 93, 2, 74, 7, 113, 0, 100, 8, 131, 0, 128

Offset: 1

Ben Branman, Jan 09 2013

nonn

			For n=17, there are three valid partitions: (7,7,3), its conjugate partition (3,3,3,2,2,2,2), and the self-conjugate partition (5,5,3,2,2).
Thus a(17)=3.

Eric Weisstein's World of Mathematics, Prime Partition.

Cf. A000040, A000041, A000607

ConjugatePartition[l_List] :=
 Module[{i, r = Reverse[l], n = Length[l]},
  Table[n + 1 - Position[r, _?(# >= i &), Infinity, 1][[1, 1]], {i,
    l[[1]]}]];f[n_] := Select[Select[IntegerPartitions[n], And @@ (PrimeQ[#]) &],
  And @@ (PrimeQ[ConjugatePartition[#]]) &];a[n_] := Length[f[n]];Table[a[n],{n,1,40}]

A213012 Trajectory of 26 under the Reverse and Add! operation carried out in base 2.

Original entry on oeis.org

26, 37, 78, 135, 360, 405, 744, 837, 1488, 1581, 3024, 3213, 6048, 6237, 12192, 12573, 24384, 24765, 48960, 49725, 97920, 98685, 196224, 197757, 392448, 393981, 785664, 788733, 1571328, 1574397, 3144192, 3150333

Offset: 0

Ben Branman, Jun 01 2012

26 is the second-smallest number (after 22) whose base 2 trajectory does not contain a palindrome.
lim_{n -> infinity} a(n)/a(n-1) = 2 for n mod 2 = 0.
lim_{n -> infinity} a(n)/a(n-1) = 1 for n mod 2 = 1. - Branman
In 2001, Brockhaus proved that if the binary Reverse and Add trajectory of an integer contains an integer of one of four specific given  forms, then the trajectory never reaches a palindrome. In the case of 26, that would be 3(2^(2k + 1) - 2^k), with k = 3 corresponding to 360. - Alonso del Arte, Jun 02 2012

			In binary, 26 is 11010.
a(1) = 37 because 11010 + 01011 = 100101, or 37.
a(2) = 78 because 100101 + 101001 = 1001110, or 78.

Klaus Brockhaus, On the'Reverse and Add!' algorithm in base 2
Index entries for sequences related to Reverse and Add!

Cf. A035522, A061561, A066059, A077076, A077077.

binRA[n_] := If[Reverse[IntegerDigits[n, 2]] == IntegerDigits[n, 2], n, FromDigits[Reverse[IntegerDigits[n, 2]], 2] + n]; NestList[binRA, 26, 100]

A212554 Products of supersingular primes (A002267).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Ben Branman, May 21 2012

nonn

The initial 1 is included because it has no non-supersingular prime factors.

Many of the early terms divide the order of the monster simple group (see A174670). The first n such that a(n) does not belong to A174670 is a(204)=289.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
A. P. Ogg, Modular functions, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 521-532, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.
Eric Weisstein's World of Mathematics, Supersingular Prime

Cf. A002267, A174670, A108764 (products of exactly two supersingular primes).

ps = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}; fQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, Complement[p, ps] == {}]; Join[{1}, Select[Range[2,1000], fQ]] (* T. D. Noe, May 21 2012 *)

log a(n) ~ k*n^(1/15). - Charles R Greathouse IV, Jul 18 2012

A182503 Engel expansion of the Dottie number, A003957.

Original entry on oeis.org

2, 3, 3, 4, 5, 15, 17, 66, 196, 233, 284, 375, 1613, 2131, 3574, 14122, 24171, 49097, 56871, 69361, 193406, 243145, 289951, 682749, 14501588, 21191773, 121635191, 810759781, 1292785381, 136110231377, 294401497761

Offset: 1

Ben Branman, May 02 2012

Dottie number = 1/2 + 1/2/3 + 1/2/3/3 + 1/2/3/3/4 + 1/2/3/3/4/5 + 1/2/3/3/4/5/15 +...

G. C. Greubel, Table of n, a(n) for n = 1..500
Jean-Christophe Pain, An exact series expansion for the Dottie number, arXiv:2303.17962 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Engel Expansion
Eric Weisstein's World of Mathematics, Dottie Number
Index entries for sequences related to Engel expansions

Cf. A003957, A006784.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; z = FindRoot[x == Cos[x], {x, 1}, WorkingPrecision -> 10000][[1, -1]]; EngelExp[z, 30]

A212113 Denominators of convergents to the Dottie number, A003957.

Original entry on oeis.org

1, 1, 3, 4, 19, 23, 939, 962, 9597, 39350, 88297, 127647, 2003002, 4133651, 51606814, 55740465, 1222156579, 1277897044, 22946406327, 1194491026048, 3606419484471, 4800910510519, 32411882547585

Offset: 0

Ben Branman, May 01 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A212112.

z = FindRoot[Cos[x] == x, {x, 0, 1}, WorkingPrecision -> 10000][[1, -1]]; Denominator[Convergents[z, 100]]

A212112 Numerators of convergents to the Dottie number, A003957.

Original entry on oeis.org

0, 1, 2, 3, 14, 17, 694, 711, 7093, 29083, 65259, 94342, 1480389, 3055120, 38141829, 41196949, 903277758, 944474707, 16959347777, 882830559111, 2665451025110, 3548281584221, 23955140530436

Offset: 0

Ben Branman, May 01 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Dottie Number

Cf. A003957, A212113.

z = FindRoot[Cos[x] == x, {x, 0, 1}, WorkingPrecision -> 10000][[1, -1]]; Numerator[Convergents[z, 100]]

A182101 Random walk determined by the binary digits of the Dottie number, A003957.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 7, 8, 7, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 7, 8, 7, 6, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10

Offset: 0

Ben Branman, Apr 11 2012

Start at a(0)=0. Each 0 in the binary expansion corresponds to a step of -1, while a 1 corresponds to a step of +1.
Partial sums of the sequence 2*A121967(n)-1.
The first time a(n) is negative is n=93.

			a(5)=3, and the sixth bit of the Dottie number is 1, so a(6)=4.
On the other hand, the seventh bit of the Dottie number is 0, so a(7)=3.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A003957, A121967, A166006 (analogous sequence for Pi).

Accumulate[RealDigits[FindRoot[Cos[x] == x, {x, 0}, WorkingPrecision -> 1000][[1, -1]], 2][[1]] 2 - 1]

A209401 Number of noncommutative rings with n elements.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 18, 2, 0, 0, 4, 0, 0, 0, 228, 0, 4, 0, 4, 0, 0, 0, 36, 2, 0, 23, 4, 0, 0, 0

Offset: 1

Ben Branman, Mar 26 2012

a(n)=0 if and only if n is squarefree.

			For n=8, there are 52 rings of order 8, 18 of which are noncommutative, so a(8)=18.

Gregory Dresden, Rings of Size Four, demonstrating that a(4)=2.

Cf. A027623, A037289, A037291.

a(n) = A027623(n) - A037289(n).

cp's OEIS Frontend

User: Ben Branman

A297622 Triangle read by rows: a(n,k) is the number of k X n matrices which are the first k rows of an alternating sign matrix of size n.

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A208229 Consider Wolfram's universal 2-state 3-symbol Turing machine on a one-way-infinite tape with all but the first n cells initially blank and the head initially in state 1. a(n) is the maximum number of steps the Turing machine can make before halting.

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A202177 Number of partitions p of n such that each part of p is prime and each part of the conjugate partition of p is also prime.

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A213012 Trajectory of 26 under the Reverse and Add! operation carried out in base 2.

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A212554 Products of supersingular primes (A002267).

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A182503 Engel expansion of the Dottie number, A003957.

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A212113 Denominators of convergents to the Dottie number, A003957.

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Mathematica

A212112 Numerators of convergents to the Dottie number, A003957.

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