A342719 - OEIS

A342719

Array read by ascending antidiagonals: T(k, n) is the sum of the consecutive positive integers from 1 to (n - 1)*k placed along the perimeter of an n-th order perimeter-magic k-gon.

21, 36, 45, 55, 78, 78, 78, 120, 136, 120, 105, 171, 210, 210, 171, 136, 231, 300, 325, 300, 231, 171, 300, 406, 465, 465, 406, 300, 210, 378, 528, 630, 666, 630, 528, 378, 253, 465, 666, 820, 903, 903, 820, 666, 465, 300, 561, 820, 1035, 1176, 1225, 1176, 1035, 820, 561

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,1

LINKS

Table of n, a(n) for n=3..57.

Terrel Trotter, Perimeter-Magic Polygons, Journal of Recreational Mathematics Vol. 7, No. 1, 1974, pp. 14-20 (see equation 3).

FORMULA

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

E.g.f.: exp(x+y)*x*(x - x*y + y^2 + x*y^2)/2.

T(k, n) = (n - 1)*k*((n - 1)*k + 1)/2.

EXAMPLE

The array begins:

k\n| 3 4 5 6 7 ...

---+------------------------

3 | 21 45 78 120 171 ...

4 | 36 78 136 210 300 ...

5 | 55 120 210 325 465 ...

6 | 78 171 300 465 666 ...

7 | 105 231 406 630 903 ...

...

MATHEMATICA

T[k_, n_]:=(n-1)k((n-1)k+1)/2; Table[T[k+3-n, n], {k, 3, 12}, {n, 3, k}]//Flatten

CROSSREFS

Cf. A014105 (n = 3), A033585 (n = 5), A037270 (1st superdiagonal), A081266 (n = 4), A083374 (1st subdiagonal), A110450 (diagonal), A144312 (n = 6), A144314 (n = 7), A342757, A342758.

Sequence in context: A301963 A219919 A219215 * A155710 A001491 A112352

Adjacent sequences: A342716 A342717 A342718 * A342720 A342721 A342722

KEYWORD

nonn,tabl

AUTHOR

Stefano Spezia, Mar 19 2021

STATUS

approved