A003410 -id:A003410 - cp's OEIS frontend

A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

1, 2, 3, 3, 5, 7, 10, 15, 22, 32, 47, 69, 101, 148, 217, 318, 466, 683, 1001, 1467, 2150, 3151, 4618, 6768, 9919, 14537, 21305, 31224, 45761, 67066, 98290, 144051, 211117, 309407, 453458, 664575, 973982, 1427440, 2092015, 3065997, 4493437, 6585452, 9651449

Offset: 0

M. F. Hasler, Jun 01 2013

nonn

The sequence produced by repeatedly applying t->(1+t,-1/t), starting from {1} and discarding numbers produced earlier, might be called Fibonacci or rabbit ordering of the rationals, in analogy to that ordering of the positive rationals, with t -> (1+t,1/t).

			The terms produced as described above are (grouped by iteration, including the starting value 1 = iteration 0): [1], [2, -1], [3, -1/2, 0], [4, -1/3, 1/2], [5, -1/4, 2/3, 3/2, -2], [6, -1/5, 3/4, 5/3, -3/2, 5/2, -2/3],[7, -1/6, 4/5, 7/4, -4/3, 8/3, -3/5, 7/2, -2/5, 1/3],[8, -1/7, 5/6, 9/5, -5/4, 11/4, -4/7, 11/3, -3/8, 2/5, 9/2, -2/7, 3/5, 4/3, -3], ...

Cf. A226271, A226274, A226080, A226081, A226130.

Essentially (up to initial terms) the same as A003410, A058278, A097333 and, in particular, A226136.

o.g.f. = (1 + x + x^2 - x^3 - x^5)/(1 - x - x^3)

A003412 From a nim-like game.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 32, 43, 54, 68, 86, 110, 142, 185, 239, 307, 393, 503, 645, 830, 1069, 1376, 1769, 2272, 2917, 3747, 4816, 6192, 7961, 10233, 13150, 16897, 21713, 27905, 35866, 46099, 59249, 76146, 97859, 125764, 161630, 207729, 266978

Offset: 0

N. J. A. Sloane

nonn

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 1).

Cf. A005708.

Join[{1, 2, 3, 4, 6, 8}, LinearRecurrence[{1, 0, 0, 0, 0, 1}, {11, 14, 18, 24, 32, 43}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

Recurrence: a(n) = a(n-1) + a(n-6) for n >= 12.

O.g.f.: -(1+x+x^2+x^3+2*x^4+2*x^5+2*x^6+x^7+x^8+2*x^9+2*x^10+3*x^11) / (-1+x+x^6). - R. J. Mathar, Dec 05 2007

A003413 From a nim-like game.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 24, 31, 40, 52, 67, 86, 110, 141, 181, 233, 300, 386, 496, 637, 818, 1051, 1351, 1737, 2233, 2870, 3688, 4739, 6090, 7827, 10060, 12930, 16618, 21357, 27447, 35274, 45334, 58264, 74882, 96239, 123686, 158960, 204294, 262558

Offset: 0

N. J. A. Sloane

nonn

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. K. Guy, Letter to N. J. A. Sloane, Apr 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 1).

Cf. A005708.

A003413:=-(z**5+z**3+1)*(z**2+z+1)/(z**6+z-1); # Simon Plouffe in his 1992 dissertation

Join[{1, 2}, LinearRecurrence[{1, 0, 0, 0, 0, 1}, {3, 4, 5, 7, 9, 12}, 80]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

Recurrence: a(n) = a(n-1) + a(n-6) for n >= 8.

O.g.f.: -(x^2+x+1)*(x^5+x^3+1)/(-1+x+x^6) = -x-1+(-2-x-x^3-x^4-2*x^5)/(-1+x+x^6). - R. J. Mathar, Dec 05 2007

A170877 Number of binary words of length n with properties that there is no pair of adjacent 1's and no subword of the form X^4 for any string X.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 22, 30, 43, 61, 88, 123, 173, 246, 348, 487, 688, 972, 1371, 1928, 2714, 3822, 5387, 7582, 10681, 15046, 21194, 29835, 42009, 59159, 83305, 117292, 165170, 232593, 327530, 461198, 649431, 914493, 1287747, 1813281, 2553346, 3595465

Offset: 0

Benjamin Chaffin and N. J. A. Sloane, Jan 07 2010

nonn

The subword 01010101 (corresponding to X = 01) for example cannot occur.

			a(3) = 5: 000, 001, 010, 100, 101.
a(4) = 7: 0001, 0010, 0100, 1000, 0101, 1010, 1001.

Lars Blomberg, Table of n, a(n) for n = 0..43

Cf. A003410, A028445, A135491.

a(24)-a(42) from Lars Blomberg, Aug 22 2013

cp's OEIS Frontend

A226275 Number of new rationals produced at the n-th iteration by applying the map t -> {t+1, -1/t} to nonzero terms, starting with S[0] = {1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A003412 From a nim-like game.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A003413 From a nim-like game.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A170877 Number of binary words of length n with properties that there is no pair of adjacent 1's and no subword of the form X^4 for any string X.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions