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

0,4

Links

Table of n, a(n) for n=0..46.

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

Example

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

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 *)

Crossrefs

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

Cf. A000217, A006516, A007582.

Sequence in context: A117000 A082392 A377377 * A307107 A085086 A274623

Adjacent sequences: A233324 A233325 A233326 * A233328 A233329 A233330

Keyword

nonn,easy

Author

Alex Ratushnyak, Feb 23 2014

Status

approved