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A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2*n^2/2 - (3*k-4)*n/2 + 1 (k >= 1, n >= 0).

%I A386478 #64 Aug 11 2025 23:09:52
%S A386478 1,1,1,1,2,1,1,2,4,1,1,3,7,7,1,1,5,14,16,11,1,1,8,25,34,29,16,1,1,12,
%T A386478 40,61,63,46,22,1,1,17,59,97,113,101,67,29,1,1,23,82,142,179,181,148,
%U A386478 92,37,1,1,30,109,196,261,286,265,204,121,46,1,1,38,140,259,359,416,418,365,269,154,56,1,1,47,175,331,473,571,607,575,481,343,191,67,1
%N A386478 Array read by upward antidiagonals: T(k,n) = 1 (k = 0, n >= 0), T(k,n) = k^2*n^2/2 - (3*k-4)*n/2 + 1 (k >= 1, n >= 0).
%C A386478 A k-chain is a planar graph consisting of a continuous path made up of k straight segments. T(k,n) is the maximum number of regions the plane can be divided into by drawing n k-chains.
%C A386478 The array is almost symmetric: the difference between T(k,n) and T(n,k) is 2*|k-n| (which is exactly the difference between the numbers of infinite regions). All the rows and columns satisfy the recurrence u(n) = 3*u(n-1) - 3*u(n-2) + u(n-3).
%D A386478 David O. H. Cutler and N. J. A. Sloane, paper in preparation, August 1 2025.
%H A386478 Paolo Xausa, <a href="/A386478/b386478.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).
%e A386478 Array begins (the rows are T(0,n>=0),, T(1,n>=0), T(2,n>=0), ...):
%e A386478   1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A386478   1, 2, 4, 7, 11, 16, 22, 29, 37, ...
%e A386478   1, 2, 7, 16, 29, 46, 67, 92, 121, ...
%e A386478   1, 3, 14, 34, 63, 101, 148, 204, 269, ...
%e A386478   1, 5, 25, 61, 113, 181, 265, 365, 481, ...
%e A386478   1, 8, 40, 97, 179, 286, 418, 575, 757, ...
%e A386478   1, 12, 59, 142, 261, 416, 607, 834, 1097, ...
%e A386478   1, 17, 82, 196, 359, 571, 832, 1142, 1501, ...
%e A386478   1, 23, 109, 259, 473, 751, 1093, 1499, 1969, ...
%e A386478   ...
%e A386478 The first few antidiagonals are:
%e A386478   1,
%e A386478   1, 1,
%e A386478   1, 2, 1,
%e A386478   1, 2, 4, 1,
%e A386478   1, 3, 7, 7, 1,
%e A386478   1, 5, 14, 16, 11, 1,
%e A386478   1, 8, 25, 34, 29, 16, 1,
%e A386478   1, 12, 40, 61, 63, 46, 22, 1,
%e A386478   ...
%t A386478 A386478[k_, n_] := If[k == 0, 1, ((k*n - 3)*k + 4)*n/2 + 1];
%t A386478 Table[A386478[k - n, n], {k, 0, 12}, {n, 0, k}] (* _Paolo Xausa_, Jul 26 2025 *)
%Y A386478 The first few rows are A000124, A130883, A140064, A080856, A383465.
%Y A386478 The n=1 and 2 columns are A152948 and A386479.
%K A386478 nonn,tabl
%O A386478 0,5
%A A386478 _N. J. A. Sloane_, Jul 24 2025
%E A386478 Row 0 added by _N. J. A. Sloane_, Jul 26 2025