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A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals. 14
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

Table of n, a(n) for n=1..73.

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<h, i | h*Binomial(i, 2)+i>; [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: A201904 A133110 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 April 29 22:12 EDT 2017. Contains 285615 sequences.