A024206 -id:A024206 - cp's OEIS frontend

A058482 Number of 3 X n binary matrices with no zero rows or columns.

1, 25, 265, 2161, 16081, 115465, 816985, 5745121, 40294561, 282298105, 1976795305, 13839692881, 96884227441, 678208723945, 4747518463225, 33232801429441, 232630126566721, 1628412435648985, 11398891698588745, 79792255837258801, 558545832702224401

Offset: 1

Vladeta Jovovic, Nov 26 2000

Index entries for linear recurrences with constant coefficients, signature (11,-31,21).

Cf. A055602, A024206, A055609 (unlabeled case), A058481, column 3 of A183109 and A218695.

LinearRecurrence[{11,-31,21},{1,25,265},30] (* Harvey P. Dale, Aug 15 2014 *)

a(n) = 7^n-3*3^n+3 \\ Charles R Greathouse IV, Feb 10 2017

Number of m X n binary matrices with no zero rows or columns is Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.
a(n) = 7^n-3*3^n+3.
a(n) = 11*a(n-1)-31*a(n-2)+21*a(n-3). G.f.: -x*(21*x^2+14*x+1) / ((x-1)*(3*x-1)*(7*x-1)). - Colin Barker, Jul 10 2013

More terms from Larry Reeves (larryr(AT)acm.org), Dec 04 2000

More terms from Colin Barker, Jul 10 2013

A135841 A000012 * A135839 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 5, 2, 2, 1, 1, 6, 3, 2, 2, 1, 1, 7, 3, 3, 2, 2, 1, 1, 8, 4, 3, 3, 2, 2, 1, 1, 9, 4, 4, 3, 3, 2, 2, 1, 1, 10, 5, 4, 4, 3, 3, 2, 2, 1, 1, 11, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 12, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 13, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Dec 01 2007

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...).

			First few rows of the triangle:
  1;
  2, 1;
  3, 1, 1;
  4, 2, 1, 1;
  5, 2, 2, 1, 1;
  6, 3, 2, 2, 1, 1;
  7, 3, 3, 2, 2, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for n = 1..1275

Cf. A135839, A024206.

T[1, 1] := 1; T[n_, 1] := n; T[n_, n_] := 1; T[n_, k_] := Floor[(n - k + 2)/2]; Table[T[n, k], {n, 1, 15}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 06 2016 *)

T(1, 1) = 1, T(n, 1) = n, T(n, n) = 1, T(n, k) = floor((n - k + 2)/2). - G. C. Greubel, Dec 06 2016

Terms a(56) and beyond from G. C. Greubel, Dec 06 2016

A347970 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_3)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 16, 8, 1, 1, 11, 39, 39, 11, 1, 1, 15, 87, 168, 87, 15, 1, 1, 19, 176, 644, 644, 176, 19, 1, 1, 24, 338, 2348, 4849, 2348, 338, 24, 1, 1, 29, 613, 8137, 37159, 37159, 8137, 613, 29, 1, 1, 35, 1071, 27047, 286747, 679054, 286747, 27047, 1071

Offset: 0

Álvar Ibeas, Sep 21 2021

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

			Triangle begins:
  k:  0   1   2   3   4   5   6   7
      -----------------------------
n=0:  1
n=1:  1   1
n=2:  1   3   1
n=3:  1   5   5   1
n=4:  1   8  16   8   1
n=5:  1  11  39  39  11   1
n=6:  1  15  87 168  87  15   1
n=7:  1  19 176 644 644 176  19   1
There are 4 = A022167(2, 1) one-dimensional subspaces in (F_3)^2, namely, those generated by (0, 1), (1, 0), (1, 1), and (1, 2). The first two are related by coordinate swap, while the remaining two are invariant. Hence, T(2, 1) = 3.

Álvar Ibeas, Entries up to T(16, 7)
H. Fripertinger, Isometry classes of codes
H. Fripertinger, Number of the isometry classes of all ternary (n,k)-codes
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100
Álvar Ibeas, Column k=5 up to n=100
Álvar Ibeas, Column k=6 up to n=100
Álvar Ibeas, Column k=7 up to n=100

Cf. A022167, A024206(n+1) (column k=1), A076831.

A128221 A128174 * A127701.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 1, 1, 3, 1, 5, 1, 2, 1, 4, 1, 6, 1, 1, 3, 1, 5, 1, 7, 1, 2, 1, 4, 1, 6, 1, 8, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13

Offset: 1

Gary W. Adamson, Feb 19 2007

Row sums = A024206: (1, 3, 5, 8, 11, 15, 19, ...). A128222 = A127701 * A128174.

Table T(n,k) = n, if k is odd, 1 if k is even; n, k > 0, read by antidiagonals. -Boris Putievskiy, Jan 30 2013

			From _Boris Putievskiy_, Jan 30 2013: (Start)
The start of the sequence as a table:
  1, 1, 1, 1, 1, 1, 1, ...
  2, 1, 2, 1, 2, 1, 2, ...
  3, 1, 3, 1, 3, 1, 3, ...
  4, 1, 4, 1, 4, 1, 4, ...
  5, 1, 5, 1, 5, 1, 5, ...
  6, 1, 6, 1, 6, 1, 6, ...
  7, 1, 7, 1, 7, 1, 7, ...
  ...
(End)
First few rows of the triangle are:
  1;
  1, 2;
  1, 1, 3;
  1, 2, 1, 4;
  1, 1, 3, 1, 5;
  1, 2, 1, 4, 1, 6;
  1, 1, 3, 1, 5, 1, 7;
  ...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A128174, A127701, A024206, A128222.

a128221[n_, k_] := If[EvenQ[n-k], k, 1]/;1<=k<=n
a128221[r_] := Table[a128221[n, k], {n, 1, r}, {k, 1, n}]
TableForm[a128221[7]] (* triangle *)
Flatten[a128221[10]] (* data *) (* Hartmut F. W. Hoft, Mar 08 2017 *)
t[r_, c_] := If[ OddQ@ c, r, 1]; Table[t[k, n - k + 1], {n, 13}, {k, n}] // Flatten (* Robert G. Wilson v, Mar 09 2017 *)

A128174 * A127701 as infinite lower triangular matrices. By columns, k-th column = k, 1, k, ...; k=1,2,3,...
From Boris Putievskiy, Jan 30 2013: (Start)
As table T(n,k) = (1+(-1)^k)/2 - (-1+(-1)^k)*n/2.
As linear sequence a(n) = (1+(-1)^A004736(n))/2 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)/2 - (-1+(-1)^j)*i/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

More terms from Robert G. Wilson v, Mar 09 2017

A234305 Irregular triangle read by rows. Theoretical distribution of electrons based on the Janet's sequence A167268.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 5, 2, 2, 6, 2, 2, 6, 1, 2, 2, 6, 2, 2, 2, 6, 2, 1, 2, 2, 6, 2, 2, 2, 2, 6, 2, 3, 2, 2, 6, 2, 4, 2, 2, 6, 2, 5, 2, 2, 6, 2, 6, 2, 2, 6, 2, 6, 1, 2, 2, 6, 2, 6, 2, 2, 2, 6, 2, 6, 2, 1, 2, 2, 6, 2, 6, 2, 2, 2, 2, 6, 2, 6, 2, 3, 2, 2, 6, 2, 6, 2, 4

Offset: 1

Paul Curtz, Jan 02 2014

a(n) is not A173642, a compact Bohr-Stoner model (1924), modified by Charles Janet in 1930. The good distribution is A168208.
Only sequences N16(n) in A234398 are used:
N16(1)= 1 followed by 2's              = A040000,
N16(2)= 1, 2, 3, 4, 5, followed by 6's = A101272,
N16(3)= 1 to  9, followed by 10's,
N16(4)= 1 to 13, followed by 14's, etc.
The distribution by rows are in the example.
The N16(n)'s are respectively on columns (hence triangle T)
1, 2, 4,  6,  9, 12, 16, 20, 25, 30, 36,   A002620(n+2)
   3, 5,  8, 11, 15, 19, 24, 29, 35,       A024206(n+2)
      7, 10, 14, 18, 23, 28, 34,           A014616(n+3)
         13, 17, 22, 27, 33,               A004116(n+4)
             21, 26, 32,
                 31, etc.
See A163255.
Antidiagonals give the natural numbers A000027, like rows sums in the example.
A033638=1, 1, 2, 3, 5, 7,... is upon the triangle T.

			1,      H
2,       He
2, 1,    Li
2, 2,    Be
2, 2, 1,
2, 2, 2,
2, 2, 3,
2, 2, 4,
2, 2, 5,
2, 2, 6,
2, 2, 6, 1,
2, 2, 6, 2,
2, 2, 6, 2, 1,
2, 2, 6, 2, 2,
2, 2, 6, 2, 3,
2, 2, 6, 2, 4,
2, 2, 6, 2, 5,
2, 2, 6, 2, 6,
2, 2, 6, 2, 6, 1,
2, 2, 6, 2, 6, 2,
2, 2, 6, 2, 6, 2, 1,
2, 2, 6, 2, 6, 2, 2,
2, 2, 6, 2, 6, 2, 3, etc.

Cf. A002061, A002522 (or A160457), A014206, A059100, diagonals of the triangle T. A004526.

A265282 Number of triangles in a certain geometric structure: see "Illustration of initial terms" link for precise definition.

Original entry on oeis.org

0, 1, 3, 5, 10, 13, 22, 26, 41, 46, 68, 74, 105, 112, 153, 161, 214, 223, 289, 299, 380, 391, 488, 500, 615, 628, 762, 776, 931, 946, 1123, 1139, 1340, 1357, 1583, 1601, 1854, 1873, 2154, 2174, 2485, 2506, 2848, 2870, 3245, 3268, 3677, 3701, 4146, 4171, 4653

Offset: 0

Luce ETIENNE, Dec 06 2015

In words: This sequence gives the number of triangles of all sizes in a (2*n^2+8*n-1+(-1)^n)/8-polyiamond with a (7*n-2-(n-2)*(-1)^n)/4-gon: we have (2*n^3+9*n^2+31*n+21+3*(n^2-5*n-7)*(-1)^n)/96 triangles in a direction and (2*n^3+27*n^2+109*n-66+3*(n^2+9*n+18)*(-1)^n+12*(-1)^((2*n-1+(-1)^n)/4))/192 triangles in the other direction. (But the Illustration link is far more informative. - N. J. A. Sloane, Jan 23 2016)
At stage n, we count (2*n^2 + 6*n + 3 - (2*n+3)*(-1)^n)/16 triangles of size 1 in one direction and (2*n^2 + 10*n - 5 + (2*n+5)*(-1)^n)/16 triangles of size 1 in the opposite direction. The total number of triangles of size 1 in both directions is A024206(n+1).
We observe that a(4)=10 strengthens the Pythagorean relation between 4 and 10 (Tetraktys): cf. triangular numbers, A000217; and that it is from n = 4 we can see and count hexagonal and dodecagonal forms, for example, in a reticular system (incomplete with hexagonal holes) by opposition to the compact shape obtained from A002717.
We can obtain this reticular system from A248851.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Luce ETIENNE, Illustration of initial terms
Luce ETIENNE, A265282 from A248851
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,0,0,-2,2,1,-1).

Cf. A000217, A000292, A001477, A001859, A002620, A002717, A004006, A004526, A045947, A132337.

Cf. A248851.

[(2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n + 4*(-1)^((2*n - 1 + (-1)^n) div 4)) / 64: n in [0..50]]; // Vincenzo Librandi, Dec 07 2015

Table[(2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n +
    4*(-1)^((2*n - 1 + (-1)^n)/4))/64, {n, 0, 100}] (* G. C. Greubel, Dec 20 2015 *)
LinearRecurrence[{1,2,-2,0,0,-2,2,1,-1},{0,1,3,5,10,13,22,26,41},60] (* Harvey P. Dale, Aug 07 2019 *)

vector(100, n, n--; (2*n^3+15*n^2+57*n-8+(3*n^2-n+4)*(-1)^n+4*(-1)^((2*n-1+(-1)^n)/4))/64) \\ Altug Alkan, Dec 06 2015

concat(0, Vec(x*(1+2*x+x^3-x^4-x^5+x^7)/((1-x)^4*(1+x)^3*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 07 2015

a(n) = A045947(floor(n/2)) + A024206(n+1). Note that A045947(floor(n/2)) = (2*n^3-n^2-7*n+(3*n^2-n-4)*(-1)^n+4*(-1)^((2*n-1+(-1)^n)/4))/64.
a(n) = (2*n^3 + 15*n^2 + 57*n - 8 + (3*n^2 - n + 4)*(-1)^n + 4*(-1)^((2*n - 1 + (-1)^n)/4))/64.
G.f.: x*(1+2*x+x^3-x^4-x^5+x^7) / ((1-x)^4*(1+x)^3*(1+x^2)). - Colin Barker, Dec 07 2015

a(26) corrected by Altug Alkan, Dec 06 2015

A342472 T(n,k) is the maximum sum of products of adjacent parts in all compositions of n into k parts: triangle read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 4, 4, 3, 0, 6, 6, 5, 4, 0, 9, 9, 8, 6, 5, 0, 12, 12, 11, 9, 7, 6, 0, 16, 16, 15, 12, 10, 8, 7, 0, 20, 20, 19, 16, 13, 11, 9, 8, 0, 25, 25, 24, 20, 17, 14, 12, 10, 9, 0, 30, 30, 29, 25, 21, 18, 15, 13, 11, 10, 0, 36, 36, 35, 30, 26, 22, 19, 16, 14, 12, 11, 0, 42, 42, 41

Offset: 1

R. J. Mathar, Mar 13 2021

Denote compositions of n into k parts by n = p_1 +p_2 + .... +p_k, p_i>0. For these compositions let S(n,k,c) = p_1*p_2 +p_2*p_3 +.. +p_{k-1}*p_k. Then T(n,k) = max_c S(n,k,c), where c runs through all A007318(n-1,k-1) compositions.
Background: Let p_i be the number of elements in level i of a poset of n points. Connect all points on level i with all points on level i+1 "maximally" with p_i*p_{i+1} arcs in the Hasse diagram. So T(n,k) is a lower bound on the maximum number of arcs in a Hasse diagram with k levels, and the maximum T(n,k)  (+1 to add the diagrams of n disconnected elements) of a row is a lower bound of the row lengths of A342447.
T(n,2) = A002620(n) has the standard interpretation of maximizing the area p_1*p_2 of a rectangle given the semiperimeter p_1+p_2=n. [S=p_1*p_2=p_1*(n-p_1) is a quadratic function of p_1 with well defined maximum.] - R. J. Mathar, Mar 14 2021
T(n,3) maximizes S = +p_1*p_2+p_2*p_3 = p_1*p_2+p_2*(n-p_1-p_2) = p_2*(n-p_2) which again is a quadratic function of p_2 with well defined maximum. - R. J. Mathar, Mar 14 2021
For k>=4 and odd n-k consider p_1=1, p_2=(n-k+1)/2, p_3=p_2+1, p_4=p_5=..=p_k=1 which gives S= n+(n-k)+[(n-k)^2-5]/4, a lower bound (apparently strict). For k>=4 and even n-k consider p_1=1, p_2=p_3=(n-k+2)/2, p_4=p_5=...=p_k=1 which gives S=n-2+(n-k+2)^2/4, a lower bound (apparently strict). - R. J. Mathar, Mar 14 2021

			For n=6 and k=3 for example 6 = 2+3+1 = 1+3+2 obtain 2*3+3*1 = 9 = T(6,3).
For n=6 and k=4 for example 6 = 1+2+2+1 obtains 1*2+2*2+2*1=8 =T(6,4).
For n=7 and k=4 for example 7 = 1+3+2+1 = 1+2+3+1 obtains 1*2+2*3+3*1 = 11 = T(7,4).
For n=7 and k=5 for example 7 = 1+1+2+2+1 = 1+2+2+1+1 obtains 1*2+2*2+2*1+1*1 = 9 = T(7,5).
The triangle starts with n>=1 and 1<=k<=n as:
  0
  0   1
  0   2   2
  0   4   4   3
  0   6   6   5   4
  0   9   9   8   6   5
  0  12  12  11   9   7   6
  0  16  16  15  12  10   8   7
  0  20  20  19  16  13  11   9   8
  0  25  25  24  20  17  14  12  10   9
  0  30  30  29  25  21  18  15  13  11  10
  0  36  36  35  30  26  22  19  16  14  12  11
  0  42  42  41  36  31  27  23  20  17  15  13  12
  0  49  49  48  42  37  32  28  24  21  18  16  14  13
  0  56  56  55  49  43  38  33  29  25  22  19  17  15  14

Cf. A002620 (columns 2,3,5 ?), A024206 (column 4?), A033638 (column 6?), A290743 (column 7?), A342447.

# Maximum of Sum_i  p_i*p(i+1) over all combinations n=p_1+p_2+..p_k
A342472 := proc(n,k)
    local s,c;
    s := 0 ;
    for c in combinat[composition](n,k) do
        add( c[i]*c[i+1],i=1..nops(c)-1) ;
        s := max(s,%) ;
    end do:
    s ;
end proc:
for n from 1 to 15 do
    for k from 1 to n do
        printf("%3d ",A342472(n,k)) ;
    end do:
    printf("\n") ;
end do:

T(n,n) = n-1; where all p_i=1.
T(n,2) = T(n,3) = A002620(n).
T(n,k) >= 2*n-k+((n-k)^2-5)/4, n-k odd, k>=4. - R. J. Mathar, Mar 14 2021
T(n,k) >= n-2+(n-k+2)^2/4, n-k even, k>=4. - R. J. Mathar, Mar 14 2021

A377802 Triangle read by rows: T(n, k) = (2 * (n+1)^2 + 7 - (-1)^n) / 8 - k.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 6, 5, 4, 3, 9, 8, 7, 6, 5, 12, 11, 10, 9, 8, 7, 16, 15, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 14, 13, 25, 24, 23, 22, 21, 20, 19, 18, 17, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31

Offset: 1

Werner Schulte, Nov 07 2024

The natural numbers, based on quarter-squares (A002620 and A033638); every natural number occurs exactly twice.

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :   1   2   3   4   5   6   7   8   9  10  11  12  13
==========================================================
   1 :   1
   2 :   2   1
   3 :   4   3   2
   4 :   6   5   4   3
   5 :   9   8   7   6   5
   6 :  12  11  10   9   8   7
   7 :  16  15  14  13  12  11  10
   8 :  20  19  18  17  16  15  14  13
   9 :  25  24  23  22  21  20  19  18  17
  10 :  30  29  28  27  26  25  24  23  22  21
  11 :  36  35  34  33  32  31  30  29  28  27  26
  12 :  42  41  40  39  38  37  36  35  34  33  32  31
  13 :  49  48  47  46  45  44  43  42  41  40  39  38  37
  etc.

A002620 (column 1), A024206 (column 2), A014616 (column 3), A004116 (column 4), A033638 (main diagonal), A290743 (1st subdiagonal).

Cf. A006003, A026741, A002061, A000290, A002378, A005563, A246694.

PARI
```
T(n,k)=(2*(n+1)^2+7-(-1)^n)/8-k
```

T(n, k) = A002620(n+1) + 1 - k.
T(2*n-1, n) = n^2 - n + 1 = A002061(n); T(2*n-2, n) = (n-1)^2 = A000290(n-1) for n > 1; T(2*n-3, n) = (n-1) * (n-2) = A002378(n-2) for n > 2; T(2*n-4, n) = (n-1) * (n-3) = A005563(n-3) for n > 3.
Row sums are (2 * n^3 + 5 * n - n * (-1)^n) / 8 = (A006003(n) + A026741(n)) / 2.
G.f.: x*y*(1 - x*y + x^2*y + x^4*y^2 - x^5*y^3 + x^6*y^3 - x^3*y*(1 + 2*y - y^2))/((1 - x)^3*(1 + x)*(1 - x*y)^3*(1 + x*y)). - Stefano Spezia, Nov 08 2024

A173261 Array T(n,k) read by antidiagonals: T(n,2k)=1, T(n,2k+1)=n, n>=2, k>=0.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 5, 1, 3, 1, 1, 6, 1, 4, 1, 2, 1, 7, 1, 5, 1, 3, 1, 1, 8, 1, 6, 1, 4, 1, 2, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2, 1, 13, 1, 11, 1, 9, 1, 7, 1, 5, 1, 3, 1, 1, 14, 1, 12, 1, 10, 1, 8, 1, 6, 1, 4, 1, 2

Offset: 2

Paul Curtz, Feb 14 2010

One may define another array B(n,0) = -1, B(n,k) = T(n,k-1) + 2*B(n,k-1), n>=2, which also starts in columns k>=0, as follows:
  -1, -1, 0, 1,  4,  9, 20,  41,  84, 169,  340,  681, 1364 ...: A084639;
  -1, -1, 1, 3,  9, 19, 41,  83, 169, 339,  681, 1363, 2729;
  -1, -1, 2, 5, 14, 29, 62, 125, 254, 509, 1022, 2045, 4094;
  -1, -1, 3, 7, 19, 39, 83, 167, 339, 679, 1363, 2727, 5459 ...: -A173114;
B(n,k) = (n-1)*A001045(k) - T(n,k).
First differences are B(n,k+1) - B(n,k) = (n-1)*A001045(k).

			The array T(n,k) starts in row n=2 with columns k>=0 as:
  1,  2, 1,  2, 1,  2, 1,  2, 1,  2, 1,  2 ... A000034;
  1,  3, 1,  3, 1,  3, 1,  3, 1,  3, 1,  3 ... A010684;
  1,  4, 1,  4, 1,  4, 1,  4, 1,  4, 1,  4 ... A010685;
  1,  5, 1,  5, 1,  5, 1,  5, 1,  5, 1,  5 ... A010686;
  1,  6, 1,  6, 1,  6, 1,  6, 1,  6, 1,  6 ... A010687;
  1,  7, 1,  7, 1,  7, 1,  7, 1,  7, 1,  7 ... A010688;
  1,  8, 1,  8, 1,  8, 1,  8, 1,  8, 1,  8 ... A010689;
  1,  9, 1,  9, 1,  9, 1,  9, 1,  9, 1,  9 ... A010690;
  1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10 ... A010691.
Antidiagonal triangle begins as:
  1;
  1,  2;
  1,  3,  1;
  1,  4,  1,  2;
  1,  5,  1,  3,  1;
  1,  6,  1,  4,  1,  2;
  1,  7,  1,  5,  1,  3,  1;
  1,  8,  1,  6,  1,  4,  1,  2;
  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;
  1, 13,  1, 11,  1,  9,  1,  7,  1,  5,  1,  3,  1;
  1, 14,  1, 12,  1, 10,  1,  8,  1,  6,  1,  4,  1,  2;

G. C. Greubel, Antidiagonals n = 0..50 of the array, flattened

Cf. A000034, A010684, A010685, A010686, A010687, A010688, A010689, A010690, A010691.

Cf. A001045, A024206, A084639, A093178, A173114.

T[n_, k_]:= (1/2)*((n+3) - (n+1)*(-1)^k);
Table[T[n-k, k], {n,2,17}, {k,2,n}]//Flatten (* G. C. Greubel, Dec 03 2021 *)

flatten([[(1/2)*((n-k+3) - (n-k+1)*(-1)^k) for k in (2..n)] for n in (2..17)]) # G. C. Greubel, Dec 03 2021

From G. C. Greubel, Dec 03 2021: (Start)
T(n, k) = (1/2)*((n+3) - (n+1)*(-1)^k).
Sum_{k=0..n} T(n-k, k) = A024206(n).
Sum_{k=0..floor((n+2)/2)} T(n-2*k+2, k) = (1/16)*(2*n^2 4*n -5*(1 +(-1)^n) + 4*sin(n*Pi/2)) (diagonal sums).
T(2*n-2, n) = A093178(n). (End)

A227779 Least splitter of s(n) and s(n+1), where s(n) = sum{(k + 1/2)^(-1/2), k >= 1}.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 4, 1, 3, 2, 4, 1, 3, 2, 3, 5, 1, 3, 2, 3, 5, 1, 4, 3, 2, 3, 6, 1, 4, 3, 2, 3, 5, 1, 5, 3, 2, 3, 4, 7, 1, 4, 3, 2, 3, 4, 6, 1, 5, 3, 5, 2, 3, 4, 7, 1, 5, 3, 5, 2, 3, 4, 6, 1, 6, 4, 3, 2, 5, 3, 5, 8, 1, 5, 4, 3, 2, 5, 3

Offset: 1

Clark Kimberling, Jul 30 2013

Suppose that x < y. The least splitter of x and y is introduced at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. It appears that d=1 (i.e., c/d is an integer) for rationals c/d in positions given by A024206.

			The first 15 splitting rationals are 1, 3/2, 2, 5/2, 3, 7/2, 11/3, 4, 9/2, 14/3, 5, 16/3, 11/2, 23/4, 6.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227631.

r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = Sum[(k + 1/2)^(-1/2), {k, 1, n}]; t = Table[r[s[n], s[n + 1]], {n, 1, 220}]; Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)

cp's OEIS Frontend

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