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A128227 Right border (1,1,1,...) added to A002260. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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)

LINKS

Table of n, a(n) for n=0..90.

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.

FORMULA

"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<=k<n, and T(n,n) = 1. - Hartmut 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)

EXAMPLE

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)

MATHEMATICA

(* 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 *)

PROG

(Python)

def T(n, k): return 1 if n==k else k

for n in xrange(1, 11): print [T(n, k) for k in xrange(1, n + 1)] # Indranil Ghosh, Jun 10 2017

(Maxima)

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 */

CROSSREFS

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

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

Sequence in context: A124769 A176484 A144328 * A306727 A228107 A140207

Adjacent sequences:  A128224 A128225 A128226 * A128228 A128229 A128230

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Feb 19 2007

STATUS

approved

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Last modified July 18 01:05 EDT 2019. Contains 325109 sequences. (Running on oeis4.)