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A177708 Pentagonal triangle. 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..45.

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).

EXAMPLE

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;

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 *)

CROSSREFS

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

Sequence in context: A206038 A205859 A288794 * A100357 A190265 A135358

Adjacent sequences:  A177705 A177706 A177707 * A177709 A177710 A177711

KEYWORD

nonn,easy,tabl

AUTHOR

Jonathan Vos Post, Dec 11 2010

STATUS

approved

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Last modified December 11 18:19 EST 2019. Contains 329925 sequences. (Running on oeis4.)