A003461 Bode numbers multiplied by 10: 4 + 3*floor(2^(n-1)).
4, 7, 10, 16, 28, 52, 100, 196, 388, 772, 1540, 3076, 6148, 12292, 24580, 49156, 98308, 196612, 393220, 786436, 1572868, 3145732, 6291460, 12582916, 25165828, 50331652, 100663300, 201326596, 402653188, 805306372, 1610612740, 3221225476
Offset: 0
References
- J. R. Newman, The World of Mathematics, Vol. I, p. 221, 1956.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 23-25.
Links
- T. D. Noe, Table of n, a(n) for n = 0..500
- M. Haynes and S. Churchman, Bode's Law
- Hitchhiker's Guide to the Galaxy, The Discovery of the Asteroid Belt, Dec 5, 2000, Nov 29, 2011.
- Hitchhiker's Guide to the Galaxy, The Discovery of the Asteroid Belt, Dec 5, 2000, Nov 29, 2011 [Cached copy of pdf version] (The sentence that is illegible at the bottom of the second page begins "This theory is often wrongly attributed to Bode (and often cited as Bode's Law), but it was Titius that first discovered the number series for the planets, and Bode (two years later in 1778) that ...".)
- S. L. Jaki, The Titius-Bode law: a strange bicentenary, Sky and Telescope, 43 (No. 5, May 1972), 280-281. [Annotated scanned copy]
- W. I. McLaughlin, Letters to N. J. A. Sloane, 1974
- W. I. McLaughlin, Note on a tetranacci alternative to Bode's law, Preprint, 1974 [Annotated scanned copy]
- 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
- Wikipedia, Bode's Law
- Index entries for linear recurrences with constant coefficients, signature (3,-2)
Programs
-
Maple
A003461:=-(-4+5*z+3*z**2)/((2*z-1)*(z-1)); [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation]
-
Mathematica
Table[4 + 3 Floor[2^(n - 1)], {n, 0, 31}] (* Robert G. Wilson v, Mar 19 2008 *) Join[{4},NestList[2#-4&,7,30]] (* Harvey P. Dale, Sep 03 2013 *) -
PARI
a(n)=4+3*floor(2^(n-1));
Formula
a(n) = 2*a(n-1) - 4, n > 1.
E.g.f.: (3*exp(2*x) + 8*exp(x) - 3)/2. - Stefano Spezia, Jul 04 2025
Extensions
Description corrected by Michael Somos
Comments