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 A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals. 15
 1, 3, 1, 6, 4, 1, 10, 9, 5, 1, 15, 16, 12, 6, 1, 21, 25, 22, 15, 7, 1, 28, 36, 35, 28, 18, 8, 1, 36, 49, 51, 45, 34, 21, 9, 1, 45, 64, 70, 66, 55, 40, 24, 10, 1, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 78, 121, 145, 153, 148, 133, 111 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n>=3. This is the accumulation array (Cf. A144112) of A144257 (which is the weight array of A086270). [Clark Kimberling, Sep 16 2008] By rows, the sequence beginning (1, N,...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0,...); and is the second partial sum of (1, (N-2), (N-2), (N-2),...). Example: The sequence (1, 4, 9, 16, 25,...) is the binomial transform of (1, 3, 2, 0, 0, 0,...) and the second partial sum of (1, 2, 2, 2,...). - Gary W. Adamson, Aug 23 2015 LINKS Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004. Wikipedia, Polygonal number: Table of values. FORMULA T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k). 2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). [Bruno Berselli, Dec 19 2014] EXAMPLE First 6 rows: ========================================= n\k|  1   2    3    4    5    6     7 ---|------------------------------------- 1  |  1   3    6   10   15   21    28 ... (A000217, triangular numbers) 2  |  1   4    9   16   25   36    49 ... (A000290, squares) 3  |  1   5   12   22   35   51    70 ... (A000326, pentagonal numbers) 4  |  1   6   15   28   45   66    91 ... (A000384, hexagonal numbers) 5  |  1   7   18   34   55   81   112 ... (A000566, heptagonal numbers) 6  |  1   8   21   40   65   96   133 ... (A000567, octagonal numbers) ... The array formed by the complements: A183225. MATHEMATICA t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten PROG (MAGMA) T:=func; [T(k, n-k+1): k in [1..n], n in [1..12]]; // Bruno Berselli, Dec 19 2014 CROSSREFS Cf. A086271, A086272, A086273, A139601. Cf. A144257. [Clark Kimberling, Sep 16 2008] Sequence in context: A133110 A286158 A185915 * A104712 A122177 A255874 Adjacent sequences:  A086267 A086268 A086269 * A086271 A086272 A086273 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 14 2003 EXTENSIONS Extended by Clark Kimberling, Jan 01 2011 STATUS approved

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Last modified December 12 03:27 EST 2018. Contains 318052 sequences. (Running on oeis4.)