A318274 -id:A318274 - cp's OEIS frontend

A128227 Right border (1,1,1,...) added to A002260.

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

Offset: 0

Gary W. Adamson, Feb 19 2007

Row sums = A000124: (1, 2, 4, 7, 11, 16, ...). n* each term of the triangle gives A128228, having row sums A006000: (1, 4, 12, 28, 55, ...).
Eigensequence of the triangle = A005425: (1, 2, 5, 14, 43, ...). - Gary W. Adamson, Aug 27 2010
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
T(n,k) is the number of binary words of length n having k letters 1 such that no 1's lie between any pair of 0's.
Let n lines with equations y = (i - 1)*x - (i - 1)^2, i = 1..n, be drawn in the Cartesian plane. For each line, call the half plane containing the point (-1,1) the upper half plane and the other half the lower half-plane. Then T(n,k) is the number of regions that are the intersections of k upper half-planes and n-k lower half-planes. Here, T(0,0) = 1 corresponds to the plane itself. A region obtained from this arrangement of lines can be associated with a length n binary word such that the i-th letter indicates whether the region is located at the i-th upper half-plane (letter 1) or at the lower half-plane (letter 0).
(End)

			First few rows of the triangle are:
1;
1, 1;
1, 2, 1;
1, 2, 3, 1;
1, 2, 3, 4, 1;
1, 2, 3, 4, 5, 1;
1, 2, 3, 4, 5, 6, 1;
1, 2, 3, 4, 5, 6, 7, 1;
1, 2, 3, 4, 5, 6, 7, 8, 1;
...
From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)
For n = 5, the binary words are
(k = 0) 00000;
(k = 1) 10000, 00001;
(k = 2) 11000, 10001, 00011;
(k = 3) 11100, 11001, 10011, 00111;
(k = 4) 11110, 11101, 11011, 10111, 01111;
(k = 5) 11111.
(End)

A. Bogolmony, Number of Regions N Lines Divide Plane.
Eric Weisstein's World of Mathematics, Plane Division by Lines.
J. E. Wetzel, On the division of the plane by lines, The American Mathematical Monthly Vol. 85 (1978), 647-656.

Cf. A002260, A128228, A000124, A006000, A318274.

Cf. A005425. - Gary W. Adamson, Aug 27 2010

(* first n rows of the triangle *)
a128227[n_] := Table[If[r==q, 1, q], {r, 1, n}, {q, 1, r}]
Flatten[a128227[13]] (* data *)
TableForm[a128227[5]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

T(n, k) := if n = k then 1 else k + 1$
for n:0 thru 10 do print(makelist(T(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, Aug 25 2018 */

def T(n, k): return 1 if n==k else k
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 10 2017

from math import comb, isqrt
def A128227(n): return n-comb(r:=(m:=isqrt(k:=n+1<<1))+(k>m*(m+1))+1,2)+(2 if k==m*(m+1) else r) # Chai Wah Wu, Nov 09 2024

"1" added to each row of "start counting again": (1; 1,2; 1,2,3,...) such that a(1) = 1, giving: (1; 1,1; 1,2,1;...).
T(n,k) = k if 1<=kHartmut F. W. Hoft, Jun 10 2017
From Franck Maminirina Ramaharo, Aug 25 2018: (Start)
The n-th row are the coefficients in the expansion of ((x^2 + (n - 2)*x - n)*x^n + 1)/(x - 1)^2.
G.f. for column k: ((k*x + 1)*x^k)/(1 - x). (End)

A319840 Table read by antidiagonals: T(n, k) is the number of elements on the perimeter of an n X k matrix.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 8, 8, 5, 6, 10, 10, 10, 10, 6, 7, 12, 12, 12, 12, 12, 7, 8, 14, 14, 14, 14, 14, 14, 8, 9, 16, 16, 16, 16, 16, 16, 16, 9, 10, 18, 18, 18, 18, 18, 18, 18, 18, 10, 11, 20, 20, 20, 20, 20, 20, 20, 20, 20, 11, 12, 22, 22, 22

Offset: 1

Stefano Spezia, Sep 29 2018

The table T(n, k) can be indifferently read by ascending or descending antidiagonals.

			The table T starts in row n=1 with columns k >= 1 as:
   1   2   3   4   5   6   7   8   9  10 ...
   2   4   6   8  10  12  14  16  18  20 ...
   3   6   8  10  12  14  16  18  20  22 ...
   4   8  10  12  14  16  18  20  22  24 ...
   5  10  12  14  16  18  20  22  24  26 ...
   6  12  14  16  18  20  22  24  26  28 ...
   7  14  16  18  20  22  24  26  28  30 ...
   8  16  18  20  22  24  26  28  30  32 ...
   9  18  20  22  24  26  28  30  32  34 ...
  10  20  22  24  26  28  30  32  34  36 ...
  ...
The triangle X(n, k) begins
  n\k|   1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
   1 |   1
   2 |   2   2
   3 |   3   4   3
   4 |   4   6   6   4
   5 |   5   8   8   8   5
   6 |   6  10  10  10  10   6
   7 |   7  12  12  12  12  12   7
   8 |   8  14  14  14  14  14  14   8
   9 |   9  16  16  16  16  16  16  16   9
  10 |  10  18  18  18  18  18  18  18  18  10
  ...

Cf. A000027 (1st column/right diagonal of the triangle or 1st row/column of the table), A005843 (2nd row/column of the table, or 2nd column of the triangle), A008574 (main diagonal of the table), A005893 (row sum of the triangle).
Cf. A003991 (the number of elements in an n X k matrix).
Cf. A131821, A318274, A154325.

[[k lt 3 or n+1-k lt 3 select (n+1-k)*k else 2*n-2: k in [1..n]]: n in [1..10]]; // triangle output

a := (n, k) -> (n+1-k)*k-(n-1-k)*(k-2)*(limit(Heaviside(min(n+1-k, k)-3+x), x = 0, right)): seq(seq(a(n, k), k = 1 .. n), n = 1 .. 20)

Flatten[Table[(n + 1 - k) k-(n-1-k)*(k-2)Limit[HeavisideTheta[Min[n+1-k,k]-3+x], x->0, Direction->"FromAbove"  ],{n, 20}, {k, n}]] (* or *)
f[n_] := Table[SeriesCoefficient[(x y - x^3 y^3)/((-1 + x)^2 (-1 + y)^2), {x, 0, i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f,20]]

T(n, k) = if ((n+1-k<3) || (k<3), (n+1-k)*k, 2*n-2);
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(20) \\ triangle output

T(n, k) = n*k - (n - 2)*(k - 2)*H(min(n, k) - 3), where H(x) is the Heaviside step function, taking H(0) = 1.
G.f. as rectangular array: (x*y - x^3*y^3)/((-1 + x)^2*(-1 + y)^2).
X(n, k) = A131821(n, k)*A318274(n - 1, k)*A154325(n - 1, k). - Franck Maminirina Ramaharo, Nov 18 2018

cp's OEIS Frontend

A128227 Right border (1,1,1,...) added to A002260.

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