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A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.

%I A380580 #89 Aug 19 2025 16:37:18
%S A380580 1,1,2,2,1,1,1,1,4,4,3,1,1,3,2,2,2,2,5,5,4,1,1,4,3,2,2,3,2,2,1,1,2,2,
%T A380580 7,7,6,1,1,6,5,2,2,5,4,2,1,1,2,4,3,3,1,1,3,3,8,8,7,1,1,7,6,2,2,6,5,2,
%U A380580 1,1,2,5,4,3,1,1,3,4,3,3,2,2,3,3,10,10,9,1,1,9,8,2,2,8,7,2,1,1,2,7,6,3,1,1,3,6,5,3,2,2,3,5
%N A380580 Irregular tetrahedron T(s,r,k) read by rows in which the slice s is an irregular triangle, itself read by rows, in which row r lists the r-th row of A237593 sandwiched between two A380579(s+1,r+1), with s >= 0; 0 <= r <= s; k >= 0. Assume that row 0 of A237593 is empty.
%C A380580 The discussion of this sequence was too long to be included here, and can be found in the attached "Discussion" text file (see the first link). - _N. J. A. Sloane_, Jul 31 2025
%H A380580 Paolo Xausa, <a href="/A380580/b380580.txt">Table of n, a(n) for n = 0..14441</a> (slices 0..50 of the tetrahedron, flattened).
%H A380580 Omar E. Pol, <a href="/A380580/a380580.txt">Discussion of the Sequence A380580.</a>
%H A380580 Omar E. Pol, <a href="/A000203/a000203_1.jpg">Illustration of initial terms of A000203 in the pyramid.</a>
%H A380580 Omar E. Pol, <a href="/A001065/a001065.jpg">Illustration of initial terms of A001065 in the pyramid.</a>
%H A380580 Omar E. Pol, <a href="/A048050/a048050.jpg">Illustration of initial terms of A048050 in the pyramid.</a>
%H A380580 Omar E. Pol, <a href="/A067742/a067742.jpg">Illustration of initial terms of A067742 in the pyramid.</a>
%H A380580 Omar E. Pol, <a href="/A237593/a237593_3.png">Illustration of initial terms of A224613 (black spiders).</a>
%H A380580 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Illustration of initial terms of A237593 (essentially a template for the Pop-Up pyramid).</a>
%H A380580 Omar E. Pol, <a href="/A221529/a221529.jpg">Prism of partitions of 10 and its companion tower (both have the same volume).</a>
%H A380580 Omar E. Pol, <a href="/A237593/a237593_1.png">The symmetric representation of sigma(n), n = 1..64 (the top view of the pyramid and of the tower).</a>
%t A380580 A237593row[n_] := Join[#, Reverse[#]] & [Table[Ceiling[(n+1)/k - (k+1)/2] + Quotient[k*(k+3) - 2*n, 2*(k+1)], {k, Quotient[Sqrt[8*n + 1] - 1, 2]}]];
%t A380580 A380580slice[s_] := Table[Join[#, A237593row[r], #] & [{Quotient[3*s, 2] - r + 1}], {r, 0, s}];
%t A380580 Array[A380580slice, 10, 0] (* _Paolo Xausa_, Aug 19 2025 *)
%Y A380580 See the "Discussion" text file for the cross-references.
%K A380580 nonn,tabf
%O A380580 0,3
%A A380580 _Omar E. Pol_, Jan 27 2025
%E A380580 Edited by _N. J. A. Sloane_, Jul 31 2025