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%I #57 May 31 2024 06:13:36
%S 1,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,
%T 8362,13530,21892,35422,57314,92736,150050,242786,392836,635622,
%U 1028458,1664080,2692538,4356618,7049156,11405774,18454930,29860704,48315634,78176338
%N a(0) = 1, then twice the Fibonacci sequence.
%C a(n) is the number of sequences over the alphabet {0,1} such that all maximal blocks (of both 0's and 1's) have odd length. E.g., a(4) = 6 because we have 0001, 0101, 0111, 1000, 1010, 1110. - _Geoffrey Critzer_, Mar 06 2012
%H Michael De Vlieger, <a href="/A055389/b055389.txt">Table of n, a(n) for n = 0..4785</a>
%H Hung Viet Chu and Zachary Louis Vasseur, <a href="https://arxiv.org/abs/2405.19352">Weighted Schreier-type Sets and the Fibonacci Sequence</a>, arXiv:2405.19352 [math.CO], 2024. See p. 2.
%H Yuhong Guo, <a href="https://doi.org/10.3770/j.issn:2095-2651.2018.02.003">Some Identities for Palindromic Compositions Without 2's</a>, Journal of Mathematical Research with Applications 38.2 (2018): 130-136.
%H Yu-hong Guo, <a href="https://www.emis.de/journals/JIS/VOL21/Guo/guo19.html">Some Identities for Palindromic Compositions</a>, J. Int. Seq., Vol. 21 (2018), Article 18.6.6.
%H Augustine O. Munagi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Munagi/munagi10.html">Integer Compositions and Higher-Order Conjugation</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F G.f.: (1 + x - x^2)/(1 - x - x^2).
%F E.g.f.: 1 + 4*exp(x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - _Stefano Spezia_, Apr 18 2022
%t Join[{1}, Table[2*Fibonacci[n], {n, 70}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 10 2012 *)
%t CoefficientList[Series[(1 + x - x^2)/(1 - x - x^2), {x, 0, 38}], x] (* _Michael De Vlieger_, Jun 14 2018 *)
%o (PARI) a(n)=if(n,2*fibonacci(n),1) \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Magma) [1] cat [2*Fibonacci(n): n in [1..40]]; // _G. C. Greubel_, Apr 28 2021
%o (Sage) [1]+[2*fibonacci(n) for n in (1..40)] # _G. C. Greubel_, Apr 28 2021
%Y Essentially the same as A006355.
%Y Cf. A000045.
%K easy,nonn,changed
%O 0,2
%A _Robert G. Wilson v_, Jul 05 2000
%E More terms from _James A. Sellers_, Jul 07 2000
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