A052955 -id:A052955 - cp's OEIS frontend

A267667 T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row unequal to the previous repeated value, and in every column equal to the previous repeated value, and new values introduced in row-major sequential order.

1, 2, 2, 3, 8, 4, 5, 18, 32, 7, 7, 50, 108, 98, 12, 11, 98, 500, 453, 288, 20, 15, 242, 1372, 3143, 1800, 800, 33, 23, 450, 5324, 10933, 18432, 6654, 2178, 54, 31, 1058, 13500, 60401, 80404, 98438, 23967, 5832, 88, 47, 1922, 48668, 188301, 624476, 528980

Offset: 1

R. H. Hardin, Jan 19 2016

Table starts
...1.....2.......3........5.........7..........11...........15............23
...2.....8......18.......50........98.........242..........450..........1058
...4....32.....108......500......1372........5324........13500.........48668
...7....98.....453.....3143.....10933.......60401.......188301........951113
..12...288....1800....18432.....80404......624476......2369260......16599404
..20...800....6654....98438....528980.....5663798.....25652956.....245171384
..33..2178...23967...508681...3351233....49212395....264709135....3440837625
..54..5832...84552..2560344..20607770...413258400...2629810220...46354321490
..88.15488..295176.12721832.124968034..3417993766..25695609592..614111962042
.143.40898.1023321.62688891.751679881.28026084373.248940754959.8066250323665

			Some solutions for n=5 k=4
..0..1..0..1....0..1..1..0....0..0..1..0....0..1..0..1....0..1..1..0
..0..0..1..1....0..0..1..1....1..0..1..1....1..1..0..0....0..0..1..1
..1..1..0..0....0..0..1..1....0..1..1..0....0..0..1..1....1..0..1..0
..0..1..0..1....1..0..0..1....1..0..0..1....0..1..0..0....0..0..1..0
..0..1..0..1....0..0..1..1....1..0..1..0....0..0..1..1....0..1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..179

Column 1 is A000071(n+2).
Row 1 is A052955(n-1).
Row 2 is A267638.

Empirical for column k:
k=1: a(n) = 2*a(n-1) -a(n-3)
k=2: a(n) = 4*a(n-1) -2*a(n-2) -6*a(n-3) +4*a(n-4) +2*a(n-5) -a(n-6)
k=3: [order 16]
k=4: [order 80]
Empirical for row n:
n=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
n=2: a(n) = a(n-1) +6*a(n-2) -6*a(n-3) -8*a(n-4) +8*a(n-5)
n=3: a(n) = a(n-1) +14*a(n-2) -14*a(n-3) -56*a(n-4) +56*a(n-5) +64*a(n-6) -64*a(n-7)
n=4: [order 21]
n=5: [order 96]

A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 5, 7, 4, 1, 0, 1, 7, 14, 12, 5, 1, 0, 1, 11, 30, 33, 19, 6, 1, 0, 1, 15, 57, 84, 62, 27, 7, 1, 0, 1, 23, 119, 222, 204, 108, 37, 8, 1, 0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1

Offset: 0

Alois P. Heinz, Jun 29 2016

			T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  3,   3,   1;
  0, 1,  5,   7,   4,   1;
  0, 1,  7,  14,  12,   5,   1;
  0, 1, 11,  30,  33,  19,   6,   1;
  0, 1, 15,  57,  84,  62,  27,   7,  1;
  0, 1, 23, 119, 222, 204, 108,  37,  8, 1;
  0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..35, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
Diagonals include A000012, A001477, A077043.
Row sums give A274547.
T(n,ceiling(n/2)) gives A305785.
Cf. A124419, A274310 (parities alternate within blocks), A305823.

b:= proc(l, i, t) option remember; `if`(l=[], x,
     `if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
       ))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
       ), j=i..nops(l)))
    end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
         b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);

b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
Flatten[Table[T[n], {n, 0, 12}]] (* Jean-François Alcover, May 27 2018, from Maple *)

Sum_{k=0..n} k * T(n,k) = A305823(n).

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

A243571 Irregular triangular array generated as in Comments; contains every positive integer exactly once.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 14, 17, 18, 20, 24, 32, 15, 19, 21, 22, 25, 26, 28, 33, 34, 36, 40, 48, 64, 23, 27, 29, 30, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 65, 66, 68, 72, 80, 96, 128, 31, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 67, 69

Offset: 1

Clark Kimberling, Jun 07 2014

Decree that row 1 is (1) and row 2 is (2). For n >= 3, row n consists of numbers in increasing order generated as follows: 2*x for each x in row n-1 together with 1+2*x for each x in row n-2. It is easy to prove that row n consists of F(n) numbers, where F = A000045 (the Fibonacci numbers), and that every positive integer occurs exactly once. Row n has F(n-1) even numbers and F(n-2) odd numbers.

The least and greatest numbers in row n are A083329(n-1) and 2^(n-1), for n >= 1.

			First 6 rows of the array:
  1
  2
  3 ... 4
  5 ... 6 ... 8
  7 ... 9 ... 10 .. 12 .. 16
  11 .. 13 .. 14 .. 17 .. 18 .. 20 .. 24 .. 32

Clark Kimberling, Table of n, a(n) for n = 1..1500
Index entries for sequences that are permutations of the natural numbers

Cf. A232559, A243572, A243573, A000045.

Cf. A052955 for the first element in each row.

z = 10; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 2 x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[g[n], {n, 1, z}]; v = Flatten[u] (* A243571 *)

A350138 Number of non-weakly alternating patterns of length n.

Original entry on oeis.org

0, 0, 0, 2, 32, 338, 3560, 40058, 492664, 6647666, 98210192, 1581844994, 27642067000, 521491848218, 10572345303576, 229332715217954, 5301688511602448, 130152723055769810, 3381930236770946120, 92738693031618794378, 2676532576838728227352

Offset: 0

Gus Wiseman, Dec 24 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.

			The a(4) = 32 patterns:
  (1,1,2,3)  (2,1,1,2)  (3,1,1,2)  (4,1,2,3)
  (1,2,2,1)  (2,1,1,3)  (3,1,2,3)  (4,2,1,3)
  (1,2,3,1)  (2,1,2,3)  (3,1,2,4)  (4,3,1,2)
  (1,2,3,2)  (2,1,3,4)  (3,2,1,1)  (4,3,2,1)
  (1,2,3,3)  (2,3,2,1)  (3,2,1,2)
  (1,2,3,4)  (2,3,3,1)  (3,2,1,3)
  (1,2,4,3)  (2,3,4,1)  (3,2,1,4)
  (1,3,2,1)  (2,4,3,1)  (3,3,2,1)
  (1,3,3,2)             (3,4,2,1)
  (1,3,4,2)
  (1,4,3,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A274230, complement A052955.
The strong case of compositions is A345192, ranked by A345168.
The strict case is A348615, complement A001250.
For compositions we have A349053, complement A349052, ranked by A349057.
The complement is counted by A349058.
The version for partitions is A349061, complement A349060.
The version for permutations of prime indices: A349797, complement A349056.
The version for ordered factorizations is A350139, complement A349059.
The strong case is A350252, complement A345194. Also the directed case?
A003242 = Carlitz compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A096441, A336103, A344605, A344614, A344615, A344740, A348610, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@allnorm[n],!whkQ[#]&&!whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([0], vector(n,i,1) + sum(k=1, n, (vector(n,i,k^i) - 2*R(n, k))*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(n) = A000670(n) - A349058(n).

a(9) onwards from Andrew Howroyd, Jan 13 2024

A007179 Dual pairs of integrals arising from reflection coefficients.

Original entry on oeis.org

0, 1, 1, 4, 6, 16, 28, 64, 120, 256, 496, 1024, 2016, 4096, 8128, 16384, 32640, 65536, 130816, 262144, 523776, 1048576, 2096128, 4194304, 8386560, 16777216, 33550336, 67108864, 134209536, 268435456, 536854528, 1073741824, 2147450880, 4294967296, 8589869056

Offset: 0

N. J. A. Sloane, Simon Plouffe

			From _Gus Wiseman_, Feb 26 2022: (Start)
Also the number of integer compositions of n with at least one odd part. For example, the a(1) = 1 through a(5) = 16 compositions are:
  (1)  (1,1)  (3)      (1,3)      (5)
              (1,2)    (3,1)      (1,4)
              (2,1)    (1,1,2)    (2,3)
              (1,1,1)  (1,2,1)    (3,2)
                       (2,1,1)    (4,1)
                       (1,1,1,1)  (1,1,3)
                                  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,1,2)
                                  (1,1,2,1)
                                  (1,2,1,1)
                                  (2,1,1,1)
                                  (1,1,1,1,1)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
J. Heading, Theorem relating to the development of a reflection coefficient in terms of a small parameter, J. Phys. A 14 (1981), 357-367.
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Column k=2 of A309748.
Odd bisection is A000302.
Even bisection is A006516 = 2^(n-1)*(2^n - 1).
The complement is counted by A077957, internal version A027383.
The internal case is A274230, even bisection A134057.
A000045(n-1) counts compositions without odd parts, non-singleton A077896.
A003242 counts Carlitz compositions.
A011782 counts compositions.
A034871, A097805, and A345197 count compositions by alternating sum.
A052952 (or A074331) counts non-singleton compositions without even parts.
Cf. A000918, A033484, A052955, A060867, A116406, A138364.

[Floor(2^n/2-2^(n/2)*(1+(-1)^n)/4): n in [0..40]]; // Vincenzo Librandi, Aug 20 2011

f := n-> if n mod 2 = 0 then 2^(n-1)-2^((n-2)/2) else 2^(n-1); fi;

LinearRecurrence[{2,2,-4},{0,1,1},30] (* Harvey P. Dale, Nov 30 2015 *)
Table[2^(n-1)-If[EvenQ[n],2^(n/2-1),0],{n,0,15}] (* Gus Wiseman, Feb 26 2022 *)

Vec(x*(1-x)/((1-2*x)*(1-2*x^2)) + O(x^50)) \\ Michel Marcus, Jan 28 2016

From Paul Barry, Apr 28 2004: (Start)
  Binomial transform is (A000244(n)+A001333(n))/2.
  G.f.: x*(1-x)/((1-2*x)*(1-2*x^2)).
  a(n) = 2*a(n-1)+2*a(n-2)-4*a(n-3).
  a(n) = 2^n/2-2^(n/2)*(1+(-1)^n)/4. (End)
G.f.: (1+x*Q(0))*x/(1-x), where Q(k)= 1 - 1/(2^k - 2*x*2^(2*k)/(2*x*2^k - 1/(1 + 1/(2*2^k - 8*x*2^(2*k)/(4*x*2^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 22 2013
a(n) = A011782(n+2) - A077957(n) - Gus Wiseman, Feb 26 2022

A268956 T(n,k)=Number of length-n 0..k arrays with no repeated value equal to the previous repeated value, with new values introduced in sequential order.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 11, 7, 1, 2, 4, 12, 29, 11, 1, 2, 4, 12, 39, 77, 15, 1, 2, 4, 12, 40, 138, 201, 23, 1, 2, 4, 12, 40, 153, 499, 525, 31, 1, 2, 4, 12, 40, 154, 634, 1830, 1361, 47, 1, 2, 4, 12, 40, 154, 655, 2785, 6723, 3525, 63, 1, 2, 4, 12, 40, 154, 656, 3045

Offset: 1

R. H. Hardin, Feb 16 2016

Table starts
..1....1.....1.....1.....1.....1.....1.....1.....1.....1.....1.....1.....1
..2....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2.....2
..3....4.....4.....4.....4.....4.....4.....4.....4.....4.....4.....4.....4
..5...11....12....12....12....12....12....12....12....12....12....12....12
..7...29....39....40....40....40....40....40....40....40....40....40....40
.11...77...138...153...154...154...154...154...154...154...154...154...154
.15..201...499...634...655...656...656...656...656...656...656...656...656
.23..525..1830..2785..3045..3073..3074..3074..3074..3074..3074..3074..3074
.31.1361..6723.12634.15124.15579.15615.15616.15616.15616.15616.15616.15616
.47.3525.24714.58409.78930.84572.85314.85359.85360.85360.85360.85360.85360

			Some solutions for n=8 k=4
..0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0. .0
..1. .1. .1. .0. .1. .1. .1. .1. .1. .1. .1. .1. .1. .1. .1. .1
..2. .2. .2. .1. .2. .2. .0. .1. .1. .2. .2. .2. .2. .0. .2. .2
..2. .3. .2. .0. .3. .3. .2. .2. .2. .1. .2. .3. .1. .1. .0. .1
..1. .4. .3. .2. .4. .2. .1. .3. .3. .3. .3. .1. .0. .2. .0. .3
..1. .3. .4. .1. .4. .2. .3. .0. .3. .3. .3. .4. .2. .1. .3. .4
..2. .2. .0. .2. .3. .4. .0. .1. .1. .1. .2. .0. .0. .2. .1. .2
..0. .0. .0. .3. .1. .1. .3. .2. .4. .3. .1. .0. .0. .1. .0. .2

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A052955(n-1).
Column 2 is A267912.
Column 3 is A268069.
Diagonal is A268010.

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = 3*a(n-1) +2*a(n-2) -8*a(n-3) for n>5
k=3: a(n) = 6*a(n-1) -3*a(n-2) -30*a(n-3) +28*a(n-4) +36*a(n-5) -36*a(n-6) for n>8
k=4: [order 9] for n>11
k=5: [order 12] for n>14
k=6: [order 15] for n>17
k=7: [order 18] for n>20

A249723 Numbers n such that there is a multiple of 9 on row n of Pascal's triangle with property that all multiples of 4 on the same row (if they exist) are larger than it.

Original entry on oeis.org

9, 10, 13, 15, 18, 19, 21, 27, 29, 31, 37, 39, 43, 45, 46, 47, 54, 55, 59, 63, 75, 79, 81, 82, 83, 85, 87, 90, 91, 93, 95, 99, 103, 109, 111, 117, 118, 119, 123, 126, 127, 135, 139, 151, 153, 154, 157, 159, 162, 163, 165, 167, 171, 175, 181, 183, 187, 189, 190, 191, 198, 199, 207, 219, 223, 225, 226, 229, 231, 234, 235, 237, 239, 243, 245, 247, 251, 253, 255

Offset: 1

Antti Karttunen, Nov 04 2014

nonn

All n such that on row n of A095143 (Pascal's triangle reduced modulo 9) there is at least one zero and the distance from the edge to the nearest zero is shorter than the distance from the edge to the nearest zero on row n of A034931 (Pascal's triangle reduced modulo 4), the latter distance taken to be infinite if there are no zeros on that row in the latter triangle.

A052955 from its eight term onward, 31, 47, 63, 95, 127, ... seems to be a subsequence. See also the comments at A249441.

			Row 13 of Pascal's triangle (A007318) is: {1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1} and the term binomial(13, 5) = 1287 = 9*11*13 occurs before any term which is a multiple of 4. Note that one such term occurs right next to it, as binomial(13, 6) = 1716 = 4*3*11*13, but 1287 < 1716, thus 13 is included.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A249724.
Natural numbers (A000027) is a disjoint union of the sequences A048278, A249722, A249723 and A249726.
Cf. A007318, A034931, A048645, A051382, A095143, A052955, A249441, A249695.
Cf. A048278, A249722, A249726, A249731, A249732, A249733.

A249723list(upto_n) = { my(i=0, n=0); while(i

A267644 T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row and column unequal to the previous repeated value, and new values introduced in row-major sequential order.

Original entry on oeis.org

1, 2, 2, 3, 8, 3, 5, 18, 18, 5, 7, 50, 51, 50, 7, 11, 98, 189, 189, 98, 11, 15, 242, 429, 1015, 429, 242, 15, 23, 450, 1353, 2887, 2887, 1353, 450, 23, 31, 1058, 2829, 12623, 8917, 12623, 2829, 1058, 31, 47, 1922, 8427, 32303, 47715, 47715, 32303, 8427, 1922, 47, 63

Offset: 1

R. H. Hardin, Jan 18 2016

Table starts
..1....2.....3.......5.......7........11........15..........23..........31
..2....8....18......50......98.......242.......450........1058........1922
..3...18....51.....189.....429......1353......2829........8427.......16899
..5...50...189....1015....2887.....12623.....32303......131673......319541
..7...98...429....2887....8917.....47715....128441......647101.....1614281
.11..242..1353...12623...47715....343145...1117207.....7479533....22499181
.15..450..2829...32303..128441...1117207...3696933....30649693....90914453
.23.1058..8427..131673..647101...7479533..30649693...330654951..1225925501
.31.1922.16899..319541.1614281..22499181..90914453..1225925501..4301737251
.47.4418.49443.1277029.8168679.150415627.784186403.13431652713.62439362175

			Some solutions for n=5 k=4
..0..1..0..1....0..1..0..0....0..1..0..0....0..1..0..0....0..1..0..1
..0..0..1..0....1..0..1..0....1..0..1..1....1..1..0..0....1..0..1..0
..1..1..0..1....1..0..0..1....0..1..0..0....0..0..1..1....0..1..1..0
..1..0..0..1....0..1..1..0....1..0..1..0....0..1..0..0....1..0..0..1
..0..0..1..0....0..1..0..1....0..1..0..1....1..0..1..1....1..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..220

Column 1 is A052955(n-1).

Empirical for column k:
k=1: a(n) = a(n-1) +2*a(n-2) -2*a(n-3)
k=2: a(n) = a(n-1) +6*a(n-2) -6*a(n-3) -8*a(n-4) +8*a(n-5)
k=3: [order 11]
k=4: [order 35]
k=5: [order 93]

A249441 a(n) is the smallest prime whose square divides at least one entry in the n-th row of Pascal's triangle, or 0 if there is no such prime.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 2, 0, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Vladimir Shevelev, Oct 28 2014

a(n) = 3 for 15, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, etc.
The above values all occur in A249723 and from 31 onward seem to be given by A052955(n>=8). (Cf. also A249714 & A249715). - Antti Karttunen, Nov 04 2014
Using the Kummer theorem on carries, one can prove that, if a(n)>3 or 0, then n>23 takes the form of either 1...1 or 101...1 in base 2 and simultaneously 212...2 in base 3. However, it is easy to see that this leads to a contradiction. Thus there are no terms greater than 3 and only 8 zeros, i.e., there are only 8 rows in Pascal's triangle that contain all squarefree numbers. It turns out that the latter result has been known for a long time (see A048278).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146.
Mihai Prunescu, Sign-reductions, p-adic valuations, binomial coefficients modulo p^k and triangular symmetries. Preprint 2013.

Cf. A005117, A007913, A048278, A052955, A249695, A249714, A249715, A249723.

a_list := proc(len) local s; s := proc(L,p) local n; seq(max(op(map(b-> padic[ordp](b,p),{seq(binomial(n,k),k=0..n)}))),n=0..L); map(k-> `if`(k<2,0,p),[%]) end: zip((x,y)-> `if`(x=0,y,x),s(len,2),s(len,3)) end: a_list(86); # Peter Luschny, Nov 01 2014
# alternative
A249441 := proc(n)
    local p,wrks,bi,k;
    if n in [0,1,2,3,5,7,11,23] then
        return 0 ;
    end if;
    p :=2 ;
    while true do
        wrks := false;
        bi := 1 ;
        for k from 0 to n do
            if modp(bi,p^2) = 0 then
                wrks := true;
                break;
            end if;
            bi := bi*(n-k)/(1+k) ;
        end do:
        if wrks then
            return p;
        end if;
        p := nextprime(p) ;
    end do:
end proc: # R. J. Mathar, Nov 04 2014

row[n_] := Table[Binomial[n, k], {k, 1, (n-Mod[n, 2])/2}];
a[n_] := If[MemberQ[{0, 1, 2, 3, 5, 7, 11, 23}, n], 0, For[p = 2, True, p = NextPrime[p], If[AnyTrue[row[n], Divisible[#, p^2]&], Return[p]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 30 2018 *)

a(n) = my(o=0); for(k=1,n\2, o+=valuation((n-k+1)/k, 2); if(o>1, return(2))); if(n<24 && n!=15, 0, 3) \\ Charles R Greathouse IV, Nov 03 2014

A249441(n) = { forprime(p=2,3,for(k=0,n\2,if((0==(binomial(n,k)%(p*p))),return(p)))); return(0); } \\ This is more straightforward, but a slower implementation - Antti Karttunen, Nov 03 2014

a(n)=if((n+1)>>valuation(n+1,2)<5, if(n<24 && setsearch([1,2,3,5,7,11,23],n), 0, 3), 2) \\ Charles R Greathouse IV, Nov 06 2014

More terms from Peter J. C. Moses, Oct 28 2014

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