A177708 Pentagonal triangle.

1, 6, 12, 18, 57, 51, 40, 156, 209, 145, 75, 330, 531, 534, 330, 126, 600, 1074, 1278, 1122, 651, 196, 987, 1895, 2488, 2559, 2081, 1162, 288, 1512, 3051, 4275, 4824, 4563, 3537, 1926, 405, 2196, 4599, 6750, 8100, 8370, 7506, 5634, 3015

Offset: 1

Author

Jonathan Vos Post, Dec 11 2010

Keywords

• nonn
• easy
• tabl

Comments

This is to A093445 as pentagonal numbers A000326 are to triangular numbers A000217. The n-th row of the triangular table begins by considering A000217(n) pentagonal numbers (starting with 1) in order. Now segregate them into n chunks beginning with n members in the first chunk, n-1 members in the second chunk, and so forth. Now sum each chunk. Thus the first term is the sum of first n numbers = n*(3n-1)/2, the second term is the sum of the next n-1 terms (from n+1 to 2n-1), the third term is the sum of the next n-2 terms (2n to 3n-3)... This triangle can be called the pentagonal triangle. The sequence contains the triangle by rows. The first column is A002411 (Pentagonal pyramidal numbers: n^2*(n+1)/2).

Examples

			The row for n = 4 is (1+5+12+22), (35+51+70), (92+117), 145 => 40, 156, 209, 145.
    1;
    6,   12;
   18,   57,   51;
   40,  156,  209,   145;
   75,  330,  531,   534,   330;
  126,  600, 1074,  1278,  1122,   651;
  196,  987, 1895,  2488,  2559,  2081,  1162;
  288, 1512, 3051,  4275,  4824,  4563,  3537,  1926;
  405, 2196, 4599,  6750,  8100,  8370,  7506,  5634, 3015;
  550, 3060, 6596, 10024, 12570, 13775, 13450, 11631, 8534, 4510;
		

Crossrefs

Cf. A000217, A000326, A002411, A093445, A236770 (right border).

Programs

Maple

A000326 :=proc(n) n*(3*n-1)/2 ; end proc:
A177708 := proc(n,k) kc := 1 ; nsk := n ; ns := 1 ; while kc < k do ns := ns+nsk ; kc := kc+1 ; nsk := nsk-1 ; end do: add(A000326(i),i=ns..ns+nsk-1) ; end proc: # R. J. Mathar, Dec 14 2010

Mathematica

Table[Total/@TakeList[PolygonalNumber[5,Range[60]],Range[n,1,-1]],{n,10}]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Feb 17 2018 *)

Formula

T(n,1) = A002411(n).

T(n,2) = n*(n-1)*(7*n-2)/2.

T(n,3) = (n-2)*(19*n^2-26*n+9)/2 = Sum_{i=2n..3(n-1)} A000326(i).