A393443 - OEIS

%I #30 Apr 22 2026 07:41:41

%S 1,2,1,4,4,3,7,8,8,7,11,13,14,14,13,16,19,21,22,22,21,22,26,29,31,32,

%T 32,31,29,34,38,41,43,44,44,43,37,43,48,52,55,57,58,58,57,46,53,59,64,

%U 68,71,73,74,74,73,56,64,71,77,82,86,89,91,92,92,91,67,76,84,91,97,102,106,109,111,112,112,111,79,89,98,106,113,119,124,128,131,133,134,134,133

%N Array read by downward antidiagonals: T(n,k) (n>=0, k>=0) = 1+k+2*n*k+k*(k-1)/2+n*(n-1).

%C Note that this is not quite the same as A393442, which is the maximum number of regions the plane can be divided into by drawing n circles and k lines (the two arrays differ in the k=0 column).

%D Jacob Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349-364. See Eq. (23). [Note that the title uses the old spelling of Teilung. This is not a typo.]

%H Paolo Xausa, <a href="/A393443/b393443.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).

%H David O. H. Cutler, Jonas Karlsson, and Neil J. A. Sloane, <a href="https://arxiv.org/abs/2511.15864">Cutting a Pancake with an Exotic Knife</a>, arXiv:2511.15864[math.CO], v3, April 19 2026.

%F G.f.: (1 - y + y^2 - x*(2 - 4*y + 4*y^2) + x^2*(3 - 7*y + 5*y^2))/((1 - x)^3*(1 - y)^3). - _Stefano Spezia_, Mar 30 2026

%e The array begins:

%e   1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, ...

%e   1, 4, 8, 13, 19, 26, 34, 43, 53, 64, 76, ...

%e   3, 8, 14, 21, 29, 38, 48, 59, 71, 84, 98, ...

%e   7, 14, 22, 31, 41, 52, 64, 77, 91, 106, 122, ...

%e   13, 22, 32, 43, 55, 68, 82, 97, 113, 130, 148, ...

%e   21, 32, 44, 57, 71, 86, 102, 119, 137, 156, 176, ...

%e   31, 44, 58, 73, 89, 106, 124, 143, 163, 184, 206, ...

%e   43, 58, 74, 91, 109, 128, 148, 169, 191, 214, 238, ...

%e   57, 74, 92, 111, 131, 152, 174, 197, 221, 246, 272, ...

%e   73, 92, 112, 133, 155, 178, 202, 227, 253, 280, 308, ...

%e   91, 112, 134, 157, 181, 206, 232, 259, 287, 316, 346, ...

%e   ...

%e The first few downward antidiagonals are:

%e   1,

%e   2, 1,

%e   4, 4, 3,

%e   7, 8, 8, 7,

%e   11, 13, 14, 14, 13,

%e   16, 19, 21, 22, 22, 21,

%e   22, 26, 29, 31, 32, 32, 31,

%e   29, 34, 38, 41, 43, 44, 44, 43,

%e   37, 43, 48, 52, 55, 57, 58, 58, 57,

%e   46, 53, 59, 64, 68, 71, 73, 74, 74, 73,

%e   ...

%t A393443[n_, k_] := 1 + k + 2*n*k + k*(k-1)/2 + n*(n-1) ;

%t Table[A393443[k, n - k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Mar 31 2026 *)

%Y Cf. A393442.

%Y Row 0 is A000124, row 1 is essentially A034856; column 0 is A002061, columns 1 and 2 are (apart from their initial values) the same as A014206 (& A386480), and column 3 is essentially the same as column 0.

%Y Cf. A140063 (main diagonal).

%K nonn,tabl

%O 0,2

%A _Jonas Karlsson_ and _N. J. A. Sloane_, Mar 30 2026

%E Data corrected by _Paolo Xausa_, Mar 31 2026