A233327 Distance from 2^n to the nearest triangular number.

0, 1, 1, 2, 1, 4, 2, 8, 3, 16, 11, 32, 1, 64, 87, 128, 167, 256, 306, 512, 500, 1024, 552, 2048, 688, 4096, 3041, 8192, 579, 16384, 20854, 32768, 37075, 65536, 55618, 131072, 37108, 262144, 222296, 524288, 147729, 1048576, 891994, 2097152, 602155, 4194304, 3523022

Offset: 0

Author

Alex Ratushnyak, Feb 23 2014

Keywords

• nonn
• easy

Examples

			Triangular number nearest to 2^8=256 is 253, so a(8)=256-253=3.
		

Crossrefs

Bisections: a(2n) = abs(A238455(n)), a(2n+1) = A000079(n).

Cf. A000217, A006516, A007582.

Programs

Mathematica

a[n_] := Module[{k, k0}, k0 = k /. FindRoot[2^n == k*(k+1)/2, {k, 2^(n/2)}, WorkingPrecision -> 20] // Round; Abs[2^n-k0*(k0+1)/2]]; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Feb 27 2014 *)

Formula

a(2*k+1) = 2^k.

Specifically, both the nearest triangular number below: A006516(n) = A000217((2^n)-1) = 2^(2n-1) - 2^(n-1) and the nearest triangular number above: A007582(n) = A000217(2^n) = 2^(2n-1) + 2^(n-1) are at the same distance from 2^(2n-1). - Antti Karttunen, Feb 26 2014