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A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.

%I A127739 #51 Nov 07 2024 20:37:18
%S A127739 1,3,3,6,6,6,10,10,10,10,15,15,15,15,15,21,21,21,21,21,21,28,28,28,28,
%T A127739 28,28,28,36,36,36,36,36,36,36,36,45,45,45,45,45,45,45,45,45,55,55,55,
%U A127739 55,55,55,55,55,55,55,66,66,66,66,66,66,66,66,66,66,66
%N A127739 Triangle read by rows, in which row n contains the triangular number T(n) = A000217(n) repeated n times; smallest triangular number greater than or equal to n.
%C A127739 Seen as a sequence, these are the triangular numbers applied to the Kruskal-Macaulay function A123578. - _Peter Luschny_, Oct 29 2022
%H A127739 Reinhard Zumkeller, <a href="/A127739/b127739.txt">Rows n=1..100 of triangle, flattened</a>
%H A127739 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A127739 Central terms: T(2*n-1,n) = A000384(n). - _Reinhard Zumkeller_, Mar 18 2011
%F A127739 a(n) = A003057(n)*A002024(n)/2; a(n) = (t+2)*(t+1)/2, where t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Feb 08 2013
%F A127739 Sum_{n>=1} 1/a(n)^2 = 8 - 2*Pi^2/3. - _Amiram Eldar_, Aug 15 2022
%F A127739 a(n) = k(n)*(1 + k(n))/2 = A000217(A123578(n)), where k = A123578. - _Peter Luschny_, Oct 29 2022
%e A127739 First few rows of the triangle are:
%e A127739    1;
%e A127739    3,  3;
%e A127739    6,  6,  6;
%e A127739   10, 10, 10, 10;
%e A127739   15, 15, 15, 15, 15;
%e A127739   ...
%p A127739 A127739 := proc(n) local t, s; t := 1; s := 0;
%p A127739 while t <= n do s := s + 1; t := t + s od; s*(1 + s)/2 end:
%p A127739 seq(A127739(n), n = 1..66); # _Peter Luschny_, Oct 29 2022
%t A127739 Table[n(n+1)/2,{n,100},{n}]//Flatten (* _Zak Seidov_, Mar 19 2011 *)
%o A127739 (Haskell)
%o A127739 a127739 n k = a127739_tabl !! (n-1) !! (k-1)
%o A127739 a127739_row n = a127739_tabl !! (n-1)
%o A127739 a127739_tabl = zipWith ($) (map replicate [1..]) $ tail a000217_list
%o A127739 -- _Reinhard Zumkeller_, Feb 03 2012, Mar 18 2011
%o A127739 (PARI) A127739=n->binomial((sqrtint(8*n)+3)\2,2) \\ _M. F. Hasler_, Mar 09 2014
%o A127739 (Python)
%o A127739 from math import isqrt
%o A127739 def A127739(n): return (r:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(r+1)>>1 # _Chai Wah Wu_, Nov 07 2024
%Y A127739 Cf. A000217, A000384, A002024, A002411 (row sums), A003057, A057944, A123578.
%K A127739 nonn,tabl
%O A127739 1,2
%A A127739 _Gary W. Adamson_, Jan 27 2007
%E A127739 Name edited by _Michel Marcus_, Apr 30 2020