A190660 Number of triangular numbers T(k) between powers of 2, 2^(n-1) < T(k) <= 2^n.

1, 0, 1, 1, 2, 2, 3, 5, 7, 9, 13, 19, 27, 37, 53, 75, 106, 150, 212, 300, 424, 600, 848, 1200, 1697, 2399, 3393, 4799, 6786, 9598, 13573, 19195, 27146, 38390, 54292, 76780, 108584, 153560, 217167, 307121, 434334, 614242, 868669, 1228483, 1737338, 2456966

Offset: 0

Author

John W. Nicholson, May 16 2011

Keywords

• nonn

Comments

Count of triangular numbers between powers of 2.

a(n)/a(n-1) converges to sqrt(2) (A002193). [John W. Nicholson, May 16 2011]

Essentially first differences of A017911. - Jeremy Gardiner, Aug 11 2013. Also second differences of A001521. - N. J. A. Sloane, Apr 27 2014

Examples

			Between 2^(6-1)=32 and 2^6=64 are T(8)=36, T(9)=45, T(10)=55 so A190660(6)=3.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001521, A002193, A017911.

Programs

Mathematica

TriangularIndex[n_] := (-1 + Sqrt[1 + 8*n])/2; Differences[Table[Floor[TriangularIndex[2^n]], {n, -1, 50}]] (* T. D. Noe, May 19 2011 *)

PARI

a(n) = if (n==0, 1, sum(i=2^(n-1)+1, 2^n, ispolygonal(i, 3))); \\ Michel Marcus, Apr 28 2014

Extensions

Extended by T. D. Noe, May 19 2011