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A093445 The triangular triangle.

%I A093445 #22 Feb 15 2019 13:53:10
%S A093445 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 A093445 72,53,28,36,84,111,120,114,96,69,36,45,108,147,165,165,150,123,87,45,
%U A093445 55,135,188,217,225,215,190,153,107,55,66,165,234,276,294,291,270,234
%N A093445 The triangular triangle.
%C A093445 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 A093445 Reinhard Zumkeller, <a href="/A093445/b093445.txt">Rows n = 1..100 of triangle, flattened</a>
%F A093445 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 A093445 TT(n, k) = T(k*n - T(k - 1)) - T((k - 1)*n - T(k - 2)).
%F A093445 TT(n, 1) = TT(n, n) = T(n) = A000217(n).
%e A093445 Triangle begins:
%e A093445    1
%e A093445    3,  3
%e A093445    6,  9,   6
%e A093445   10, 18,  17,  10
%e A093445   15, 30,  33,  27,  15
%e A093445   21, 45,  54,  51,  39, 21
%e A093445   28, 63,  80,  82,  72, 53, 28
%e A093445   36, 84, 111, 120, 114, 96, 69, 36
%e A093445 The row for n = 4 is (1+2+3+4), (5+6+7), (8+9), 10 => 10 18 17 10.
%p A093445 A093445 := proc(n,k)
%p A093445     A000217(k*n-A000217(k-1))-A000217((k-1)*n-A000217(k-2)) ;
%p A093445 end proc:
%p A093445 seq(seq(A093445(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Dec 09 2015
%t A093445 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 A093445 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 (Haskell)
%o A093445 a093445 n k = a093445_row n !! (k-1)
%o A093445 a093445_row n = f [n, n - 1 .. 1] [1 ..] where
%o A093445    f [] _      = []
%o A093445    f (x:xs) ys = sum us : f xs vs where (us,vs) = splitAt x ys
%o A093445 a093445_tabl = map a093445_row [1 ..]
%o A093445 -- _Reinhard Zumkeller_, Oct 03 2012
%Y A093445 Cf. A000217, A093446. TT(n, 2) = A045943. TT(n, n-1) = A014209. TT(0, k) = A027480.
%Y A093445 Cf. A005920 (central terms), A002817 (row sums).
%K A093445 nonn,nice,tabl
%O A093445 1,2
%A A093445 _Amarnath Murthy_, Apr 02 2004
%E A093445 Edited, corrected and extended by _Robert G. Wilson v_, Apr 24 2004