A093445 The triangular triangle.

1, 3, 3, 6, 9, 6, 10, 18, 17, 10, 15, 30, 33, 27, 15, 21, 45, 54, 51, 39, 21, 28, 63, 80, 82, 72, 53, 28, 36, 84, 111, 120, 114, 96, 69, 36, 45, 108, 147, 165, 165, 150, 123, 87, 45, 55, 135, 188, 217, 225, 215, 190, 153, 107, 55, 66, 165, 234, 276, 294, 291, 270, 234

Offset

1,2

Comments

The n-th row of the triangular table begins by considering n triangular numbers (A000217) in order. Now segregate them into n groups beginning with n members in the first group, n-1 members in the second group, etc. Now sum each group. Thus the first term is the sum of first n numbers = n(n+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), etc. and the last term is simply n(n+1)/2. This triangle can be called a triangular triangle. The sequence contains the triangle by rows.

Links

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

Formula

T(n) = A000217(n) is the n-th Triangular number. TT(n, k) is the k-th term of the n-th row, 0 < k <= n.

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

TT(n, 1) = TT(n, n) = T(n) = A000217(n).

Example

Triangle begins:
   1
   3,  3
   6,  9,   6
  10, 18,  17,  10
  15, 30,  33,  27,  15
  21, 45,  54,  51,  39, 21
  28, 63,  80,  82,  72, 53, 28
  36, 84, 111, 120, 114, 96, 69, 36
The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.

Maple

A093445 := proc(n, k)
    A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;
end proc:
seq(seq(A093445(n, k), k=1..n), n=1..10) ; # R. J. Mathar, Dec 09 2015

Mathematica

T[n_] := n(n + 1)/2; TT[n_, k_] := T[k*n - T[k - 1]] - T[(k - 1)*n - T[k - 2]]; Flatten[ Table[ TT[n, k], {n, 1, 11}, {k, 1, n}]] (* Robert G. Wilson v, Apr 24 2004 *)
Table[Total/@TakeList[Range[(n(n+1))/2], Range[n, 1, -1]], {n, 20}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 15 2019 *)

Prog

(Haskell)
a093445 n k = a093445_row n !! (k-1)
a093445_row n = f [n, n - 1 .. 1] [1 ..] where
   f [] _      = []
   f (x:xs) ys = sum us : f xs vs where (us, vs) = splitAt x ys
a093445_tabl = map a093445_row [1 ..]
-- Reinhard Zumkeller, Oct 03 2012

Crossrefs

Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n-1) = A014209. TT(0, k) = A027480.

Cf. A005920 (central terms), A002817 (row sums).

Sequence in context: A104715 A164743 A110769 * A098358 A136289 A128012

Adjacent sequences: A093442 A093443 A093444 * A093446 A093447 A093448

Keyword

nonn,nice,tabl

Author

Amarnath Murthy, Apr 02 2004

Extensions

Edited, corrected and extended by Robert G. Wilson v, Apr 24 2004

Status

approved