login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A093445 The triangular triangle. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 10 01:36 EDT 2021. Contains 343747 sequences. (Running on oeis4.)