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A003461 Bode numbers multiplied by 10: 4 + 3*floor(2^(n-1)).

%I A003461 M3302 #81 Jul 08 2025 16:27:40
%S A003461 4,7,10,16,28,52,100,196,388,772,1540,3076,6148,12292,24580,49156,
%T A003461 98308,196612,393220,786436,1572868,3145732,6291460,12582916,25165828,
%U A003461 50331652,100663300,201326596,402653188,805306372,1610612740,3221225476
%N A003461 Bode numbers multiplied by 10: 4 + 3*floor(2^(n-1)).
%C A003461 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 A003461 J. R. Newman, The World of Mathematics, Vol. I, p. 221, 1956.
%D A003461 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003461 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 23-25.
%H A003461 T. D. Noe, <a href="/A003461/b003461.txt">Table of n, a(n) for n = 0..500</a>
%H A003461 M. Haynes and S. Churchman, <a href="http://astrosun.tn.cornell.edu/courses/astro201/bodes_law.htm">Bode's Law</a>
%H A003461 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 A003461 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 A003461 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 A003461 W. I. McLaughlin, <a href="/A003461/a003461.pdf">Letters to N. J. A. Sloane, 1974</a>
%H A003461 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 A003461 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 A003461 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A003461 Wikipedia, <a href="http://en.wikipedia.org/wiki/Bode&#39;s_law">Bode's Law</a>
%H A003461 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2)
%F A003461 a(n) = 2*a(n-1) - 4, n > 1.
%F A003461 E.g.f.: (3*exp(2*x) + 8*exp(x) - 3)/2. - _Stefano Spezia_, Jul 04 2025
%p A003461 A003461:=-(-4+5*z+3*z**2)/((2*z-1)*(z-1)); [Conjectured (correctly) by _Simon Plouffe_ in his 1992 dissertation]
%t A003461 Table[4 + 3 Floor[2^(n - 1)], {n, 0, 31}] (* _Robert G. Wilson v_, Mar 19 2008 *)
%t A003461 Join[{4},NestList[2#-4&,7,30]] (* _Harvey P. Dale_, Sep 03 2013 *)
%o A003461 (PARI) a(n)=4+3*floor(2^(n-1));
%Y A003461 Cf. A061654.
%Y A003461 First differences of A087009.
%K A003461 nonn,easy,nice
%O A003461 0,1
%A A003461 _N. J. A. Sloane_, based on correspondence from W. I. McLaughlin, 1974
%E A003461 Description corrected by _Michael Somos_