A005379 - OEIS

A005379

The male of a pair of recurrences.
(Formerly M0278)

0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 43, 44, 45, 45

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

M(n) is not equal to F(n) if and only if n+1 is a Fibonacci number (A000045); a(n)=A005379(n)-A192687(n). [Reinhard Zumkeller, Jul 12 2011]

REFERENCES

D. R. Hofstadter, "Goedel, Escher, Bach", p. 137.

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

D. R. Hofstadter, Eta-Lore [Cached copy, with permission]

D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]

D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

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

J. Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.

Th. Stoll, On Hofstadter's married functions, Fib. Q., 46/47 (2008/2009), 62-67. - from N. J. A. Sloane, May 30 2009

Eric Weisstein's World of Mathematics, Hofstadter Male-Female Sequences.

Index entries for Hofstadter-type sequences

Index entries for sequences from "Goedel, Escher, Bach"

FORMULA

F(0) = 1; M(0) = 0; F(n) = n - M(F(n-1)); M(n) = n - F(M(n-1)).

The g.f. -z^2*(-1-z^3-z^6-z-z^4-z^7+z^8)/(z+1)/(z^2+1)/(z^4+1)/(z-1)^2, conjectured by Simon Plouffe in his 1992 dissertation is incorrect: the coefficient of z^33 in the g.f. is 21, but a(33) = 20. (Discovered by Sahand Saba, Jan 14 2013.) - Frank Ruskey, Jan 16 2013

MAPLE

F:= proc(n) option remember; n - M(procname(n-1)) end proc:

M:= proc(n) option remember; n - F(procname(n-1)) end proc:

F(0):= 1: M(0):= 0:

seq(M(n), n=0..100); # Robert Israel, Jun 15 2015

MATHEMATICA

f[0] = 1; m[0] = 0; f[n_] := f[n] = n - m[f[n-1]]; m[n_] := m[n] = n - f[m[n-1]]; Table[m[n], {n, 0, 73}]

(* Jean-François Alcover, Jul 27 2011 *)

PROG

(Haskell) Cf. A005378.

(PARI) f(n) = if(n<1, 1, n - m(f(n - 1)));

m(n) = if(n<1, 0, n - f(m(n - 1)));