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A030068 The "semi-Fibonacci numbers": a(n) = A030067(2n - 1), where A030067 is the semi-Fibonacci sequence. 5
1, 2, 3, 5, 6, 9, 11, 16, 17, 23, 26, 35, 37, 48, 53, 69, 70, 87, 93, 116, 119, 145, 154, 189, 191, 228, 239, 287, 292, 345, 361, 430, 431, 501, 518, 605, 611, 704, 727, 843, 846, 965, 991, 1136, 1145, 1299, 1334, 1523, 1525, 1716, 1753, 1981, 1992, 2231, 2279, 2566 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the unique values of A030067 sorted. - Ralf Stephan, Oct 28 2013

Also, the subsequence of record values of the semi-Fibonacci sequence A030067.

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10001

FORMULA

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

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

MATHEMATICA

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-Fran├žois Alcover, Jul 16 2015 *)

PROG

(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

(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

CROSSREFS

Cf. A030067. Bisections: A169739, A169740.

Sequence in context: A008771 A008770 A230515 * A239958 A008769 A115270

Adjacent sequences:  A030065 A030066 A030067 * A030069 A030070 A030071

KEYWORD

nonn,nice

AUTHOR

David W. Wilson

EXTENSIONS

Offset changed to 1 by N. J. A. Sloane, Mar 27 2017

STATUS

approved

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Last modified March 20 21:49 EDT 2019. Contains 321352 sequences. (Running on oeis4.)