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Right border (1,1,1,...) added to A002260.
7

%I #37 Nov 09 2024 19:20:32

%S 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,

%T 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,

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

%N Right border (1,1,1,...) added to A002260.

%C 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, ...).

%C Eigensequence of the triangle = A005425: (1, 2, 5, 14, 43, ...). - _Gary W. Adamson_, Aug 27 2010

%C From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)

%C 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.

%C 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).

%C (End)

%H A. Bogolmony, <a href="http://www.cut-the-knot.org/proofs/LinesDividePlane.shtml">Number of Regions N Lines Divide Plane</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PlaneDivisionbyLines.html">Plane Division by Lines</a>.

%H J. E. Wetzel, <a href="http://dx.doi.org/10.2307/2320333">On the division of the plane by lines</a>, The American Mathematical Monthly Vol. 85 (1978), 647-656.

%F "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;...).

%F T(n,k) = k if 1<=k<n, and T(n,n) = 1. - _Hartmut F. W. Hoft_, Jun 10 2017

%F From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)

%F The n-th row are the coefficients in the expansion of ((x^2 + (n - 2)*x - n)*x^n + 1)/(x - 1)^2.

%F G.f. for column k: ((k*x + 1)*x^k)/(1 - x). (End)

%e First few rows of the triangle are:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 2, 3, 1;

%e 1, 2, 3, 4, 1;

%e 1, 2, 3, 4, 5, 1;

%e 1, 2, 3, 4, 5, 6, 1;

%e 1, 2, 3, 4, 5, 6, 7, 1;

%e 1, 2, 3, 4, 5, 6, 7, 8, 1;

%e ...

%e From _Franck Maminirina Ramaharo_, Aug 25 2018: (Start)

%e For n = 5, the binary words are

%e (k = 0) 00000;

%e (k = 1) 10000, 00001;

%e (k = 2) 11000, 10001, 00011;

%e (k = 3) 11100, 11001, 10011, 00111;

%e (k = 4) 11110, 11101, 11011, 10111, 01111;

%e (k = 5) 11111.

%e (End)

%t (* first n rows of the triangle *)

%t a128227[n_] := Table[If[r==q, 1, q], {r, 1, n}, {q, 1, r}]

%t Flatten[a128227[13]] (* data *)

%t TableForm[a128227[5]] (* triangle *)

%t (* _Hartmut F. W. Hoft_, Jun 10 2017 *)

%o (Python)

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

%o for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jun 10 2017

%o (Python)

%o from math import comb, isqrt

%o 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

%o (Maxima)

%o T(n, k) := if n = k then 1 else k + 1$

%o for n:0 thru 10 do print(makelist(T(n, k), k, 0, n)); /* _Franck Maminirina Ramaharo_, Aug 25 2018 */

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

%Y Cf. A005425. - _Gary W. Adamson_, Aug 27 2010

%K nonn,tabl,changed

%O 0,5

%A _Gary W. Adamson_, Feb 19 2007