A066212 -id:A066212 - cp's OEIS frontend

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

Henry Bottomley, Mar 19 2002

sign

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).

			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.

Cf. A066212.

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

A111366 Numbers such that the sum of the digits of floor(phi^n) is also the sum of the digits of the n-th Fibonacci number (in base 10), where phi is the golden ratio.

Original entry on oeis.org

1, 6, 13, 61, 73, 92, 97, 198, 212, 217, 222, 270, 349, 380, 404, 438, 524, 630, 649, 836, 937, 1446, 1477, 1513, 1532, 1729, 2005, 2046, 2060, 2077, 2209, 2348, 2660, 2862, 2934, 3265, 3649, 3889, 4093, 4609, 4686, 4945, 5180, 5444, 5497, 5749, 5929, 6102

Offset: 1

Stefan Steinerberger, Nov 07 2005

Questions: (1) Is this sequence infinite? (2) Are the gaps between the elements of this sequence bounded from above? (3) If this sequence is infinite, what is its asymptotic growth? (4) Consider the definition of this sequence for other values c instead of the golden ratio. What are the properties of this modified sequence?

			trunc(phi^6) = 17, the 6th Fibonacci number is 8; the sum of their digits is the same, thus 6 is in the sequence.

Cf. A066212, A001999.

$MaxExtraPrecision = 10^9; fQ[n_] := Plus @@ IntegerDigits@Floor@(GoldenRatio^n) == Plus @@ IntegerDigits@Fibonacci@n; Select[ Range[6108], fQ[ # ] &] (* Robert G. Wilson v *)

Edited, corrected and extended by Robert G. Wilson v, Nov 16 2005

cp's OEIS Frontend

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

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