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%I #14 Feb 23 2021 03:45:00
%S 1,1,1,1,2,1,3,4,5,4,4,11,14,18,16,12,28,53,69,88,78,27,91,178,295,
%T 396,489,457,82,291,685,1196,1867,2503,3071,2938,228,1004,2548,5051,
%U 8385,12560,16905,20667,20118,733,3471,9876,21018,38078,60736,89038,119571,146381,144113
%N Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0.
%C "A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.
%C T(n,k) is the number of floor plan arrangements represented by 3-connected trivalent maps with n internal rooms and k+3 rooms adjacent to the outside.
%C "... may be evaluated from the results given by Brown."
%C The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P -c2m2 [n]" will compute values for a diagonal. The '-c2' and '-m2' options indicate graphs must be biconnected and with minimum vertex degree 2. - _Andrew Howroyd_, Feb 22 2021
%D C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
%H Andrew Howroyd, <a href="/A169808/b169808.txt">Table of n, a(n) for n = 0..1325</a>
%H G. Brinkmann and B. McKay, <a href="http://users.cecs.anu.edu.au/~bdm/plantri/">Plantri (program for generation of certain types of planar graph)</a>
%H William G. Brown, <a href="http://dx.doi.org/10.1112/plms/s3-14.4.746">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
%H William G. Brown, <a href="/A002709/a002709.pdf">Enumeration of Triangulations of the Disk</a>, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy].
%H C. F. Earl and L. J. March, <a href="/A005500/a005500_1.pdf">Architectural applications of graph theory</a>, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
%H Andrew Howroyd, <a href="/A169808/a169808.txt">PARI Program</a>
%F T(n,k) = (A262586(n,k) + A169809(n,k)) / 2. - _Andrew Howroyd_, Feb 22 2021
%e Array begins:
%e ============================================================
%e n\k | 0 1 2 3 4 5 6
%e ----+-------------------------------------------------------
%e 0 | 1 1 1 3 4 12 27 ...
%e 1 | 1 2 4 11 28 91 291 ...
%e 2 | 1 5 14 53 178 685 2548 ...
%e 3 | 4 18 69 295 1196 5051 21018 ...
%e 4 | 16 88 396 1867 8385 38078 169918 ...
%e 5 | 78 489 2503 12560 60736 290595 1367374 ...
%e 6 | 457 3071 16905 89038 451613 2251035 11025626 ...
%e 7 | 2938 20667 119571 652198 3429943 17658448 89328186 ...
%e ...
%o (PARI) \\ See link for script file.
%o A169808Array(6) \\ _Andrew Howroyd_, Feb 22 2021
%Y Columns k=0..3 are A002713, A005500, A005501, A005502.
%Y Rows n=0..2 are A000207, A005503, A005504.
%Y Antidiagonal sums give A005027.
%Y Cf. A146305 (rooted), A169809 (achiral), A262586 (oriented).
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, May 25 2010
%E Edited by _Andrew Howroyd_, Feb 22 2021
%E a(29) corrected and terms a(36) and beyond from _Andrew Howroyd_, Feb 22 2021
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