Search: id:A055167
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%I A055167
%S A055167 1,1,1,1,2,3,4,5,7,9,13,17,23,30,39,50,65,83,107,136,174,219,278,
%T A055167 348,437,544,678,839,1039,1279,1574,1929,2362,2881,3511,4264,5174,6258,
               7560,9107,
%U A055167 10959,13152,15766,18855,22522,26844,31960,37973,45066,53386,63167,74615,
               88038
%N A055167 Number of optimal binary prefix-free codes with n words all ending in 
               1.
%D A055167 Z. Kukorelly, Optimal binary one-ended codes. IEEE Trans. Inform. Theory, 
               Vol. 48 (2002), no. 7, 2125-2132.
%F A055167 a(n)=sum_{0 <= a < n/3} sum_{0 <= b < n/3} g(n, a, b) with for 1 <= n 
               <= 4, g(n, a, b)=1 if a=b=1 and g(n, a, b)=0 otherwise; for n >= 
               5: g(n, b, 1)=sum_{a = 0..b, b-a even } g(n-1, a, b), (b >= 1); g(n, 
               a, b)=g(n-1, a, b-1), (2 <= b <= a); g(n, a, b)=g(n-1, a, b-1), (b-a 
               >= 2 text{even}, a >= 0); g(n, a, b)=g(n-1, a+1, b), (b-a >= 1 text{odd}, 
               a >= 0); g(n, a, b)=0 (otherwise).
%t A055167 g[n_, a_, b_] := 0; g[n_, 1, 1] := 1/;(n >= 1 && n<=4); g[n_, a_, b_] 
               := 0/;(n >= 1 && n<=4 && (a!=1 || b!=1)); g[n_, b_, 1] := (g[n, b, 
               1] = Sum[g[n-1, 2 a, b], {a, 0, b/2}])/;(b>=1 && EvenQ[b]); g[n_, 
               b_, 1] := (g[n, b, 1] = Sum[g[n-1, 2 a + 1, b], {a, 0, (b-1)/2}])/
               ;(b>=1 && OddQ[b]); g[n_, a_, b_] := (g[n, a, b] = g[n-1, a, b-1]_/
               ;(b>=2 && a>=b); g[n_, a_, b_] := (g[n, a, b] = g[n-1, a, b-1])/;
               (a>=0 && b>=a+2 && EvenQ[b-a]); g[n_, a_, b_] := (g[n, a, b] = g[n-1, 
               a+1, b])/;(a>=0 && b>=a+1 && OddQ[b-a]);
%t A055167 a[n_] := Sum[g[n, a, b], {a, 0, Floor[n/3]}, {b, 0, Floor[n/3]}]
%Y A055167 Sequence in context: A026450 A039863 A036802 this_sequence A064628 A017834 
               A050763
%Y A055167 Adjacent sequences: A055164 A055165 A055166 this_sequence A055168 A055169 
               A055170
%K A055167 nonn,easy,nice
%O A055167 1,5
%A A055167 N. J. A. Sloane (njas(AT)research.att.com), Jul 04 2000

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