A128228 -id:A128228 - cp's OEIS frontend

A128229 A natural number transform, inverse of signed A094587.

1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 0, 4, 1, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1

Offset: 1

Gary W. Adamson, Feb 19 2007

Signed version of the transform (with -1, -2, -3, ... in the subdiagonal) gives A094587 having row sums A000522: (1, 2, 5, 16, 65, 236, ...). Unsigned inverse gives signed A094587 (with alternate signs); giving row sums = a signed variation of A094587 as follows: (1, 0, 1, -2, 9, -44, 265, -1854, ...). Binomial transform of the triangle = A093375.

Eigensequence of the triangle = A000085 starting (1, 2, 4, 10, 26, 76, ...). - Gary W. Adamson, Dec 29 2008

			First few rows of the triangle are:
1;
1, 1;
0, 2, 1;
0, 0, 3, 1;
0, 0, 0, 4, 1;
0, 0, 0, 0, 5, 1;
0, 0, 0, 0, 0, 6, 1;
0, 0, 0, 0, 0, 0, 7, 1;
...

Cf. A094587, A000522, A094587, A093375, A128228, A128227.

Cf. A000085. - Gary W. Adamson, Dec 29 2008

a128229[n_] := Table[Which[r==q, 1, r-1==q, q, True, 0], {r, 1, n}, {q, 1, r}]
Flatten[a128229[13]] (* data *)
TableForm[a128229[8]] (* triangle *)
(* Hartmut F. W. Hoft, Jun 10 2017 *)

def T(n, k): return 1 if n==k else n - 1 if k==n - 1 else 0
for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 10 2017

Infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal.

T(n,n)=1, T(n,n-1)=n-1 and T(n,k)=0 for 1<=k<=n, 1<=n. - Hartmut F. W. Hoft, Jun 10 2017

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

Original entry on oeis.org

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)

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A128229 A natural number transform, inverse of signed A094587.

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

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