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Number of initial segments of signature sequences of length n.
1

%I #2 Mar 30 2012 17:35:23

%S 1,2,4,6,8,12,14,16,20,24,26,32,34,36,42,46,48,54,56,60,66,70,72,80,

%T 84,86,92,98,100,110,112,114,118,122,126,136,138,140,146,154,156,166,

%U 168,172,182,186,188,196,200,204,210,216,218,228,234,238,244,248,250,262

%N Number of initial segments of signature sequences of length n.

%C The initial segments of signature sequences correspond to ranges of theta from a1/b1 to a2/b2 (including 1/0 as infinity); a/b appears as a bound, splitting two previous segments, at n = (a+1)(b+1)/2.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SignatureSequence.html">Signature Sequence</a>

%F a(n) = 1 + sum( 2<=k<=n, A163822(k)).

%e For n = 6, the sequences and corresponding ranges for theta are:

%e 1,1,1,1,1,1 0/1..1/5

%e 1,1,1,1,1,2 1/5..1/4

%e 1,1,1,1,2,1 1/4..1/3

%e 1,1,1,2,1,2 1/3..1/2

%e 1,1,2,1,2,1 1/2..2/3

%e 1,1,2,1,2,3 2/3..1/1

%e 1,2,1,3,2,1 1/1..3/2

%e 1,2,1,3,2,4 3/2..2/1

%e 1,2,3,1,4,2 2/1..3/1

%e 1,2,3,4,1,5 3/1..4/1

%e 1,2,3,4,5,1 4/1..5/1

%e 1,2,3,4,5,6 5/1..1/0

%Y Cf. A163822.

%K nonn

%O 1,2

%A _Franklin T. Adams-Watters_, Aug 04 2009