A068515 A measure of how close the square root of 2 is to rational numbers.

2, -12, 12, -12, 70, 12, -70, 26, -33, 70, -25, -408, 34, -70, 70, -43, 408, 39, -146, 70, -70, 195, -49, -408, 70, -113, 147, -70, 2378, 70, -195, 126, -100, 408, 70, -408, 114, -146, 253, -93, -2378, 106, -228, 195, -125, 855, 100, -408, 165, -173, 408, -113, -1135, 147, -252, 286, -146, 2378, 135, -408

Offset

1,1

Comments

New peaks (in absolute terms) occur when n is a Pell number (1,2,5,12,29,70,... A000129) and take alternate Pell values with alternating signs (2,-12,70,-408,2378,-13860,... A001542). Each new peak (after the first) appears twice (with different signs) before the next peak, when n is a numerator of a continued fraction convergent to sqrt(2) (3,7,17,41,99,... A001333) and when n is twice a Pell number (4,10,24,58,140,... A052542).

Links

Table of n, a(n) for n=1..60.

Formula

a(n) =round[1/(sqrt(2)-round[sqrt(2)*n]/n)] =round[1/(sqrt(2)-A022846(n)/n)] where sqrt(2)=1.41421356...

Example

a(5) = round[1/(sqrt(2)-round[sqrt(2)*5]/5)] = round[1/(sqrt(2)-7/5)] = round[70.355] = 70, i.e. sqrt(2) is about 1/70 more than the nearest multiple of 1/5.

Crossrefs

Cf. A066212.

Sequence in context: A216478 A181060 A171446 * A088240 A168457 A045895

Adjacent sequences: A068512 A068513 A068514 * A068516 A068517 A068518

Keyword

sign

Author

Henry Bottomley, Mar 19 2002

Status

approved