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A177708
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Pentagonal triangle.
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4
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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
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OFFSET
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1,2
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COMMENTS
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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).
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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