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%I #55 Nov 26 2021 06:07:25
%S 1,2,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,
%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,
%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131
%N 2 together with the odd numbers (essentially the result of the first stage of the sieve of Eratosthenes).
%C Number of Fibonacci binary words of length n and having no subword 1011. A Fibonacci binary word is a binary word having no 00 subword. Example: a(5) = 9 because of the 13 Fibonacci binary words of length 5 the following do not qualify: 11011, 10110, 10111 and 01011. - _Emeric Deutsch_, May 13 2007
%C a(1) = 1; for n > 1, a(n) = least number > a(n-1) which is a unique sum of two earlier terms, not necessarily distinct. - _Franklin T. Adams-Watters_, Nov 01 2011
%D F. S. Roberts, Applied Combinatorics, Prentice-Hall, 1984, p. 256.
%H G. C. Greubel, <a href="/A004280/b004280.txt">Table of n, a(n) for n = 1..5000</a>
%H Borys Kuca, <a href="https://arxiv.org/abs/1804.09594">Structures in Additive Sequences</a>, arXiv:1804.09594 [math.NT], 2018. See V(1,2).
%H H. B. Meyer, <a href="http://www.hbmeyer.de/eratosiv.htm">Eratosthenes' sieve</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%F From _Paul Barry_, Mar 05 2007: (Start)
%F G.f.: x*(1+x^3)/(1-x)^2;
%F a(n) = 2*n - 3 + C(1, n-1) + C(0, n-1). (End)
%F a(n) = 2*n - 3 + floor(2/n). - _Wesley Ivan Hurt_, May 23 2013
%F E.g.f.: (1/2)*(6 + 4*x + x^2 - 2*(3 - 2*x)*exp(x)). - _G. C. Greubel_, Nov 25 2021
%p 1,2,seq(2*n-1,n=2..70); # _Emeric Deutsch_, May 13 2007
%t Union[ Join[ 2Range[70] - 1, {2}]] (* _Robert G. Wilson v_ *)
%o (PARI) a(n)=2*n + 2\n - 3 \\ _Charles R Greathouse IV_, Nov 01 2011
%o (Sage) [1,2]+[2*n-3 for n in (3..70)] # _G. C. Greubel_, Nov 25 2021
%Y Cf. A002858.
%K easy,nonn
%O 1,2
%A _N. J. A. Sloane_
%E Offset changed to 1 and formulas updated accordingly (at the suggestion of _Michel Marcus_) by _Charles R Greathouse IV_, Sep 03 2013
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