A085368 Sum of numerators and denominators of convergents to 1/e.

3, 4, 11, 15, 26, 119, 145, 264, 1729, 1993, 3722, 31769, 35491, 67260, 708091, 775351, 1483442, 18576655, 20060097, 38636752, 560974625, 599611377, 1160586002, 19168987409, 20329573411, 39498560820, 731303668171, 770802228991, 1502105897162, 30812920172231

Offset: 1

Author

Gary W. Adamson, Jun 26 2003

Keywords

• nonn

Comments

Through a(n) natural numbers 1,2,3...a(n), A007677(n-1) of those terms are members of the upper level Beatty sequence A000572; while A007676(n) of those terms are in the lower level Beatty sequence A006594.

Check: a(5) = 26, which has 7 (= A007677(4)) terms in A000572: 3, 7, 11, 14, 18, 22 and 26; while the remaining 19 (= A007676(5)) are members of the lower level Beatty sequence A006594.

A085368(n)/A007677(n-1) converge upon (1 + e), as n approaches infinity. Check: A085368(6)/A007677(5) = 119/32 = 3.71875... where (1 + e) =3.718281828... A085368(n)/A007676(n) converge upon (1 + 1/e). Check: A085368(5)/A007676(5) = 119/87 = 1.3678.., where (1 + 1/e) = 1.367879441... A006594 and A000572 form Beatty pairs, with floor n*(1 + e) being the generator for A000572(n) and floor n*(1 + 1/e) the generator for A006594(n).

The cutting sequence for y = (1/e)x is generated from the line starting at (0,0), passing through an array of squares, giving "1" to an intersection with a vertical line and "0" to an intersection with a horizontal line. The cutting sequence for y = (1/e)x is 0, then (terms 1 through 26): 1 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0. In this sequence, n's for 0's are all members of the upper Beatty pair: A000572 (check: n's for the 0's are 3, 7, 11, 14, 18, 22 and 26 (the 7 being A007677(4)); while 19 terms (19 = A007676(5)) are members of the lower Beatty pair A006594, being denoted by "1" and thus intersecting vertical lines.

Examples

			a(6) = 119 = 32 + 87 where 32/87 is the 6th convergent to 1/e: [2,1,2,1,1,4]= 32/87 = .367816...& 1/e = .3678794...
a(6) = 119 = 32 + 87 = A007677(5) + A007676(6).
		

Crossrefs

Cf. A007677, A007676, A006594, A000572.

Formula

Convergents to 1/e are generated from the partial quotients of the continued fraction form of 1/e: [2, 1, 2, 1, 1, 4, 1, 1, 6...], where below each partial quotient, the first 9 convergents are 1/2, 1/3, 3/8...(i.e. 1/2 = [2], 1/3 = [2, 1], 3/8 = [2, 1, 2], etc;...then 4/11, 7/19, 32/87, 39/106, 71/193, 465/1264, where a(n) = sum of numerator and denominator of n-th convergent to 1/e with 1/2 = first convergent.

a(n) = A007676(n) + A007677(n-1) where A007676 = 2, 3, 8, 11, 19, 87...(numerators to convergents to e); and A007677 = 1, 1, 3, 4, 7, 32, 39, 71...(denominators of convergents to e).

Extensions

More terms from Colin Barker, Mar 11 2014