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%I #25 Apr 28 2014 12:45:28
%S 1,0,1,1,2,2,3,5,7,9,13,19,27,37,53,75,106,150,212,300,424,600,848,
%T 1200,1697,2399,3393,4799,6786,9598,13573,19195,27146,38390,54292,
%U 76780,108584,153560,217167,307121,434334,614242,868669,1228483,1737338,2456966
%N Number of triangular numbers T(k) between powers of 2, 2^(n-1) < T(k) <= 2^n.
%C Count of triangular numbers between powers of 2.
%C a(n)/a(n-1) converges to sqrt(2) (A002193). [_John W. Nicholson_, May 16 2011]
%C Essentially first differences of A017911. - _Jeremy Gardiner_, Aug 11 2013. Also second differences of A001521. - _N. J. A. Sloane_, Apr 27 2014
%H Vincenzo Librandi, <a href="/A190660/b190660.txt">Table of n, a(n) for n = 0..1000</a>
%e Between 2^(6-1)=32 and 2^6=64 are T(8)=36, T(9)=45, T(10)=55 so A190660(6)=3.
%t TriangularIndex[n_] := (-1 + Sqrt[1 + 8*n])/2; Differences[Table[Floor[TriangularIndex[2^n]], {n, -1, 50}]] (* _T. D. Noe_, May 19 2011 *)
%o (PARI) a(n) = if (n==0, 1, sum(i=2^(n-1)+1, 2^n, ispolygonal(i, 3))); \\ _Michel Marcus_, Apr 28 2014
%Y Cf. A001521, A002193, A017911.
%K nonn
%O 0,5
%A _John W. Nicholson_, May 16 2011
%E Extended by _T. D. Noe_, May 19 2011
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