%I M3302 #72 Jun 25 2021 06:20:21
%S 4,7,10,16,28,52,100,196,388,772,1540,3076,6148,12292,24580,49156,
%T 98308,196612,393220,786436,1572868,3145732,6291460,12582916,25165828,
%U 50331652,100663300,201326596,402653188,805306372,1610612740,3221225476
%N Bode numbers multiplied by 10: 4 + 3*floor(2^(n-1)).
%C 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.
%D J. R. Newman, The World of Mathematics, Vol. I, p. 221, 1956.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003461/b003461.txt">Table of n, a(n) for n = 0..500</a>
%H M. Haynes and S. Churchman, <a href="http://astrosun.tn.cornell.edu/courses/astro201/bodes_law.htm">Bode's Law</a>
%H Hitchhiker's Guide to the Galaxy, <a href="http://www.h2g2.com/approved_entry/A354412">The Discovery of the Asteroid Belt</a>, Dec 5, 2000, Nov 29, 2011.
%H Hitchhiker's Guide to the Galaxy, <a href="/A003461/a003461_3.pdf">The Discovery of the Asteroid Belt</a>, 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 ...".)
%H S. L. Jaki, <a href="/A003461/a003461_2.pdf">The Titius-Bode law: a strange bicentenary</a>, Sky and Telescope, 43 (No. 5, May 1972), 280-281. [Annotated scanned copy]
%H W. I. McLaughlin, <a href="/A003461/a003461.pdf">Letters to N. J. A. Sloane, 1974</a>
%H W. I. McLaughlin, <a href="/A003461/a003461_1.pdf">Note on a tetranacci alternative to Bode's law</a>, Preprint, 1974 [Annotated scanned copy]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bode's_law">Bode's Law</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2)
%F a(n) = 2*a(n-1) - 4, n > 1.
%p A003461:=-(-4+5*z+3*z**2)/((2*z-1)*(z-1)); [Conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation]
%t Table[4 + 3 Floor[2^(n - 1)], {n, 0, 31}] (* _Robert G. Wilson v_, Mar 19 2008 *)
%t Join[{4},NestList[2#-4&,7,30]] (* _Harvey P. Dale_, Sep 03 2013 *)
%o (PARI) a(n)=4+3*floor(2^(n-1));
%Y Cf. A061654.
%Y First differences of A087009.
%K nonn,easy,nice
%O 0,1
%A _N. J. A. Sloane_, based on correspondence from W. I. McLaughlin, 1974.
%E Description corrected by _Michael Somos_