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%I #45 Mar 27 2017 22:02:29
%S 1,2,3,5,6,9,11,16,17,23,26,35,37,48,53,69,70,87,93,116,119,145,154,
%T 189,191,228,239,287,292,345,361,430,431,501,518,605,611,704,727,843,
%U 846,965,991,1136,1145,1299,1334,1523,1525,1716,1753,1981,1992,2231,2279,2566
%N The "semi-Fibonacci numbers": a(n) = A030067(2n - 1), where A030067 is the semi-Fibonacci sequence.
%C Also the unique values of A030067 sorted. - _Ralf Stephan_, Oct 28 2013
%C Also, the subsequence of record values of the semi-Fibonacci sequence A030067.
%C The first differences of this sequence give back A030067. - It is more natural to use offset 1 and a(n) = A060037(2n-1), rather than 0 and a(n) = A060037(2n+1): First, a set should have this offset, and this is indeed the set of values or the range of A030067, i.e., the set of semi-Fibonacci numbers. Second, A060037 also starts at index 1. Third, the sequence A284282(n) = (k such that A030067(2k-1)=n or 0 if there's no such k) is then the characteristic function of this sequence, with nonzero values read as 1. - _M. F. Hasler_, Mar 24 2017
%H N. J. A. Sloane, <a href="/A030068/b030068.txt">Table of n, a(n) for n = 1..10001</a>
%F G.f.: x*(r(x) * r(x^2) * r(x^4) * r(x^8) * ...) where r(x) is (1 + 2x + x^2 + x^3 + x^4 + x^5 + ...). - _Gary W. Adamson_, Sep 02 2016
%F a(n+1) = a(n) + A060037(n). The above g.f. can be written as x*Product_{k=0,oo} (1/(1-x^2^k)+x^2^k). - _M. F. Hasler_, Mar 27 2017
%t f[1] = 1; f[n_?EvenQ] := f[n] = f[n/2]; f[n_?OddQ] := f[n] = f[n-1] + f[n-2]; a[n_] := f[2*n+1]; Table[a[n], {n, 1, 55}] (* _Jean-François Alcover_, Jul 16 2015 *)
%o (PARI) A030068_vec=[1,2,3]; A030068(n)={n>#A030068_vec&&for(n=#A030068_vec,-1+#A030068_vec=concat(A030068_vec,vector(n-#A030068_vec)),A030068_vec[n+1]=A030068_vec[n]+A030067(n));A030068_vec[n]} \\ _M. F. Hasler_, Mar 24 2017
%o (PARI) Vec(prod(k=0,5,1/Ser(x^2^k)+x^2^k)) \\ Correct for n < 2*2^5. - _M. F. Hasler_, Mar 27 2017
%Y Cf. A030067. Bisections: A169739, A169740.
%K nonn,nice
%O 1,2
%A _David W. Wilson_
%E Offset changed to 1 by _N. J. A. Sloane_, Mar 27 2017
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