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<DIV>Seqfans, <BR> I wish to share a few more
"functional puzzles" with solutions. </DIV>
<DIV>Slight variations could produce more interesting results. </DIV>
<DIV> Paul </DIV>
<DIV>-------------------------------------------------------------
<BR>Consider the systems of simultaneous equations below; for each
case, </DIV>
<DIV>what is the unique solution to variable A as a formal power series in
x? <BR> <BR>(1)<BR>A = 1 + xAB<BR>B = A + xBC<BR>C = B + xCD<BR>D = C +
xDE<BR>E = D + xEF<BR>...<BR> <BR>(2a) <BR>A = (1 + xB)<BR>B = A(1 +
xC)<BR>C = B(1 + xD)<BR>D = C(1 + xE)<BR>E = D(1 + xF)<BR>... <BR>(2b)
<BR>A = 1 + xAB<BR>B = 1 + xABC<BR>C = 1 + xABCD<BR>D = 1 + xABCDE<BR>E = 1 +
xABCDEF <BR>...<BR>(same solution as (2a)).<BR> <BR>(3) <BR>A = 1 +
xB<BR>B = 1 + xAC<BR>C = 1 + xABD<BR>D = 1 + xABCE<BR>E = 1 +
xABCDF<BR>...</DIV>
<DIV> <BR>SOLUTIONS.<BR>(1) A satisfies: A(x) = 1
+ x*A(x)^2*A(xA(x)) = g.f. of A088714: <BR>[1, 1, 3, 13, 69, 419, 2809, 20353,
157199, 1281993, 10963825,...]. <BR> <BR>(2a) and (2b): A
satisfies: A(x) = 1 + xA(x)*A(xA(x)) <BR>where xA(x) = g.f. of A030266,
and coefficients of A begin: <BR>[1, 1, 2, 6, 23, 104, 531, 2982, 18109,
117545, 808764,...]. <BR> <BR>(3) A satisfies: A(x) = 1 + xA(xA(x)) =
g.f. of A087949: <BR>[1, 1, 1, 2, 5, 16, 59, 246, 1131, 5655, 30428,
174835,...]. </DIV>
<DIV> </DIV>
<DIV>END.</DIV></BODY></HTML>