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%I #27 Mar 07 2024 11:13:09
%S 0,0,1,0,1,3,0,1,4,6,0,1,5,9,10,0,1,6,12,16,15,0,1,7,15,22,25,21,0,1,
%T 8,18,28,35,36,28,0,1,9,21,34,45,51,49,36,0,1,10,24,40,55,66,70,64,45,
%U 0,1,11,27,46,65,81,91,92,81,55,0,1,12,30,52,75,96,112,120,117,100,66
%N Square array T(n,k) = (n+1)*(k-1)*k/2+k, of polygonal numbers, read by antidiagonals upwards.
%C A general formula for polygonal numbers is P(n,k) = (n-2)(k-1)k/2 + k, where P(n,k) is the k-th n-gonal number. - _Omar E. Pol_, Dec 21 2008
%H G. C. Greubel, <a href="/A139601/b139601.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber">Figurate number — a very short introduction</a>. With plots from Stefan Friedrich Birkner.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polnum01.jpg">Polygonal numbers</a>, An alternative illustration of initial terms.
%F T(n,k) = A086270(n,k), k>0. - _R. J. Mathar_, Aug 06 2008
%F T(n,k) = (n+1)*(k-1)*k/2+k, n>=0, k>=0. - _Omar E. Pol_, Jan 07 2009
%e The square array of polygonal numbers begins:
%e ========================================================
%e Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28,
%e Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49,
%e Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70,
%e Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91,
%e Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112,
%e Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133,
%e 9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154,
%e 10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175,
%e 11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196,
%e 12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217,
%e And so on ..............................................
%e ========================================================
%t T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jul 12 2009 *)
%Y Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
%Y Cf. A000007, A000012, A000027, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A057145, A086271, A139600.
%Y Cf. A086270, A139617, A139618, A139619, A139620.
%K nonn,tabl,easy
%O 0,6
%A _Omar E. Pol_, Apr 27 2008
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