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A003461 Bode numbers multiplied by 10: 4 + 3*floor(2^(n-1)).
(Formerly M3302)
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Bode's law is that the average distance of the n-th planet from the sun is (4 + 3*floor(2^(n-1)))/10 astronomical units.

The Titius-Bode Law is a rough rule that predicts the spacing of the planets in the Solar System. The relationship was first pointed out by Johann Titius in 1766 and was formulated as a mathematical expression by J. E. Bode in 1778. It lead Bode to predict the existence of another planet between Mars and Jupiter in what is now called the asteroid belt. - Robert G. Wilson v, Mar 19 2008

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).

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 Galazy, The Discovery of the Asteroid Belt, Dec 5, 2000, Nov 29, 2011.

Hitchhiker's Guide to the Galazy, 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.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Wikipedia, Bode's Law

Index entries for linear recurrences with constant coefficients, signature (3,-2)

FORMULA

a(n) = 2*a(n-1) - 4, n > 1.

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 *)

PROG

(PARI) a(n)=4+3*floor(2^(n-1));

CROSSREFS

Cf. A061654.

First differences of A087009.

Sequence in context: A191154 A209257 A131500 * A023375 A310714 A310715

Adjacent sequences:  A003458 A003459 A003460 * A003462 A003463 A003464

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, based on correspondence from W. I. McLaughlin, 1974.

EXTENSIONS

Description corrected by Michael Somos

STATUS

approved

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Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)