

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
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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 halfplane. Then T(n,k) is the number of regions that are the intersections of k upper halfplanes and nk lower halfplanes. 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 ith letter indicates whether the region is located at the ith upper halfplane (letter 1) or at the lower halfplane (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), 647656.


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 nth 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



