%I #35 Jul 13 2024 09:39:10

%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 of polygonal numbers read by ascending antidiagonals: T(n, k) = (n + 1)*(k - 1)*k/2 + k.

%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

%F From _G. C. Greubel_, Jul 12 2024: (Start)

%F t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.

%F t(2*n, n) = A006003(n).

%F t(2*n+1, n) = A002411(n).

%F t(2*n-1, n) = A006000(n-1).

%F Sum_{k=0..n} t(n, k) = A006522(n+2).

%F Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).

%F Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)

%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 *)

%o (Magma)

%o T:= func< n,k | k*((n+1)*(k-1) +2)/2 >;

%o A139601:= func< n,k | T(n-k, k) >;

%o [A139601(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2024

%o (SageMath)

%o def T(n,k): return k*((n+1)*(k-1)+2)/2

%o def A139601(n,k): return T(n-k, k)

%o flatten([[A139601(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 12 2024

%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, A002411, A006000, A006003, A006522.

%Y Cf. A008585, A016957, A017329, A086270, A117142, A139606, A139607.

%Y Cf. A139608, A139609, A139610, A139611, A139612, A139613, A139614.

%Y Cf. A139615, A139616, A057145, A086271, A139600, A139617, A139618.

%Y Cf. A139619, A139620.

%K nonn,tabl,easy

%O 0,6

%A _Omar E. Pol_, Apr 27 2008