A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 12, 16, 15, 1, 7, 15, 22, 25, 21, 1, 8, 18, 28, 35, 36, 28, 1, 9, 21, 34, 45, 51, 49, 36, 1, 10, 24, 40, 55, 66, 70, 64, 45, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66, 1, 13, 33, 58, 85, 111, 133, 148, 153, 145, 121, 78

Offset

1,3

Comments

The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

Links

G. C. Greubel, Table of n, a(n) for the first 50 diagonals, flattened

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Formula

T(n, k) = k*C(n,2) + n.

From Stefano Spezia, Sep 02 2022: (Start)

G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).

G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)

Example

Columns 1,2,3 are the triangular, square and pentagonal numbers.
Northwest corner:
       k=1 k=2 k=3 k=4 k=5
  n=1:   1   1   1   1   1 ...
  n=2:   3   4   5   6   7 ...
  n=3:   6   9  12  15  18 ...
  n=4:  10  16  22  28  34 ...
  n=5:  15  25  35  45  55 ...
  ...

Mathematica

T[n_, k_] := PolygonalNumber[k+2, n]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 04 2016 *)

Crossrefs

Cf. A006522, A086270, A086272, A086273.

Main diagonal gives A006000(n-1).

Sequence in context: A194540 A351153 A193043 * A345229 A080851 A209518

Adjacent sequences: A086268 A086269 A086270 * A086272 A086273 A086274

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Jul 14 2003

Status

approved