%I #16 Jan 01 2020 21:52:24

%S 1,5,4,18,17,10,50,48,37,20,115,110,93,67,35,231,220,194,156,109,56,

%T 420,399,360,306,240,165,84,708,672,615,540,450,348,237,120,1125,1068,

%U 987,885,765,630,483,327,165,1705,1620,1508,1372,1215,1040,850,648,437,220

%N Group the natural numbers so that the n-th group contains n(n+1)/2 = T(n) terms: (1), (2,3,4), (5,6,7,8,9,10), (11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35),... The n-th row of the following triangle is formed from the sum of first n terms, next n-1 terms,next n-2 terms ... of the n-th group; e.g. third row is (5+6+7), (8+9), (10) or 18,17,10. Sequence contains the triangle read by rows.

%C The leading diagonal is A000292 (tetrahedral (or pyramidal) numbers: C(n+3,3) = (n+1)(n+2)(n+3)/6.)

%C The sequence contains very few duplicate terms. In the first 10000 terms, only 12 are duplicates and there are no terms that repeat more than two times. - _Harvey P. Dale_, Jun 10 2018

%H Harvey P. Dale, <a href="/A100791/b100791.txt">Table of n, a(n) for n = 1..10000</a>

%e 1

%e 5 4

%e 18 17 10

%e 50 48 37 20

%e 115 110 93 67 35

%e ...

%t Module[{nn=10,r1,r2},r1=Accumulate[Range[nn]];r2=Total[r1];Total/@ Flatten[ TakeList[#,Range[(Sqrt[8*Length[#]+1]-1)/2,1,-1]]&/@TakeList[ Range[ r2],r1],1]] (* Requires Mathematica version 11 or later *) (* _Harvey P. Dale_, Jun 10 2018 *)

%Y Cf. A100792.

%K easy,nonn,tabl

%O 1,2

%A _Amarnath Murthy_, Dec 01 2004

%E More terms from _Ray Chandler_, Dec 10 2004