cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A000572 A Beatty sequence: [ n(e+1) ].

Original entry on oeis.org

3, 7, 11, 14, 18, 22, 26, 29, 33, 37, 40, 44, 48, 52, 55, 59, 63, 66, 70, 74, 78, 81, 85, 89, 92, 96, 100, 104, 107, 111, 115, 118, 122, 126, 130, 133, 137, 141, 145, 148, 152, 156, 159, 163, 167, 171, 174, 178, 182, 185, 189, 193, 197, 200, 204, 208, 211, 215, 219
Offset: 1

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Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A006594.

Programs

  • Maple
    for n from 1 to 200 do printf(`%d,`,floor( n*(exp(1)+1))) od:
  • Mathematica
    Table[Floor[n*(E + 1)], {n, 100}] (* T. D. Noe, Jun 20 2012 *)

Extensions

More terms from James Sellers, Feb 19 2001

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

Original entry on oeis.org

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

Views

Author

Gary W. Adamson, Jun 26 2003

Keywords

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

A085369 Cutting sequence for 1/e.

Original entry on oeis.org

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
Offset: 1

Views

Author

Gary W. Adamson, Jun 26 2003

Keywords

Comments

Through any A085368(n) number of terms in the cutting sequence, A007677(n-1) of those terms are zeros and A007676(n) are ones. Check: A085368(5) = 26, the sequence being 3, 4, 11, 15, 26, ... (sum of numerators and denominators of convergents to 1/e). Then through n=26, A085369(n) is 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, with 7 zeros and 19 ones, (7/19 being the 5th convergent to 1/e): 7/19 = [2, 1, 2, 1, 1]. Numerator and denominator sum = 26, with 7 zeros and 19 ones, with the zeros occupying positions n = 3, 7, 11, 14, 18, 22 and 26 (also being the first 7 terms of A000572). Positions of the cutting sequence occupied by ones (1, 2, 4, 5, 6, ...) are consecutive terms of the lower Beatty sequence A006594, being generated by floor(n*(1 + 1/e)).

Examples

			a(6) = 1, where 1's correspond to members of the lower Beatty pair A006594 which is generated from floor(n*(1 + 1/e)). Check: floor(5*(1 + 1/e)) = 6. All terms not in A006594 are 0's.
a(7) = 0, where 7 is not a member of A006594, but is a member of the upper Beatty pair sequence A000572 which has the generator floor(n*(e + 1)). Check: floor(2*(1 + e)) = 7.

References

  • Manfred R. Schroeder, "Fractals, Chaos, Power Laws", Freeman, 1996, p. 56.

Crossrefs

Cf. A085368, A007677, A007676, A006594, A000572.

Formula

Given the line y = (1/e)x starting from (0, 0) and passing through an array of squares, a "1" denotes an intersection with a vertical line, while an "0" denotes an intersection with a horizontal line.
n for 0's are consecutive terms of upper Beatty pair terms A000572: 3, 7, 11, 14, 18, 22, 26, ..., while n's for all 1's are paired lower Beatty terms of A006594: 1, 2, 4, 5, 6, 8, ...
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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