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 A093445 The triangular triangle. 6

%I

%S 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,

%T 72,53,28,36,84,111,120,114,96,69,36,45,108,147,165,165,150,123,87,45,

%U 55,135,188,217,225,215,190,153,107,55,66,165,234,276,294,291,270,234

%N The triangular triangle.

%C 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.

%H Reinhard Zumkeller, <a href="/A093445/b093445.txt">Rows n = 1..100 of triangle, flattened</a>

%F 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.

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

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

%e Triangle begins:

%e 1

%e 3, 3

%e 6, 9, 6

%e 10, 18, 17, 10

%e 15, 30, 33, 27, 15

%e 21, 45, 54, 51, 39, 21

%e 28, 63, 80, 82, 72, 53, 28

%e 36, 84, 111, 120, 114, 96, 69, 36

%e The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.

%p A093445 := proc(n,k)

%p A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;

%p end proc:

%p seq(seq(A093445(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Dec 09 2015

%t 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 *)

%t 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 *)

%o a093445 n k = a093445_row n !! (k-1)

%o a093445_row n = f [n, n - 1 .. 1] [1 ..] where

%o f [] _ = []

%o f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys

%o a093445_tabl = map a093445_row [1 ..]

%o -- _Reinhard Zumkeller_, Oct 03 2012

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

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

%K nonn,nice,tabl

%O 1,2

%A _Amarnath Murthy_, Apr 02 2004

%E Edited, corrected and extended by _Robert G. Wilson v_, Apr 24 2004

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)