A081461
Consider the mapping f(a/b) = (a^2+b^3)/(a^3+b^2) from rationals to rationals. Starting with 1/2 (a=1, b=2) and applying the mapping to each new (reduced) rational number gives 1/2, 9/5, 103/377, ... . Sequence gives values of the numerators.
Original entry on oeis.org
1, 9, 103, 26796621, 236092315725004393, 3561970421302126514421966146019939188025056477849165490630219227287
Offset: 1
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nxt[{a_,b_}]:=Module[{frac=(a^2+b^3)/(a^3+b^2)},{Numerator[frac], Denominator[ frac]}]; Transpose[NestList[nxt,{1,2},5]][[1]] (* Harvey P. Dale, Nov 09 2011 *)
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{r=1/2; for(n=1,7,a=numerator(r); b=denominator(r); print1(a,","); r=(a^2+b^3)/(a^3+b^2))}
A081466
Consider the mapping f(a/b) = (a^2+b^2)/(a^2-b^2) from rationals to rationals. Starting with 2/1 (a=2, b=1) and applying the mapping to each new (reduced) rational number gives 2/1, 5/3, 17/8, 353/225, ... Sequence gives values of the denominators.
Original entry on oeis.org
1, 3, 8, 225, 36992, 6308330625, 21009822254496776192, 3255818067933293622186199316985612890625, 3264008661830516310447364816658205121507617681188862393654856638929469798612992
Offset: 1
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{r=2; for(n=1,9,a=numerator(r); b=denominator(r); print1(b,","); r=(a^2+b^2)/(a^2-b^2))}
A358210
Congruent number sequence starting from the Pythagorean triple (3,4,5).
Original entry on oeis.org
6, 15, 34, 353, 175234, 9045146753, 121609715057619333634, 4138643330264389621194448797227488932353, 27728719906622802548355602700962556264398170527494726660553210068191276023007234
Offset: 1
Starting with the triple (3,4,5) and choosing the b side we obtain by the recurrence the right triangles: (15/2, 4, 17/2), (136/15, 15/2, 353/30), (5295/136, 272/15, 87617/2040), (47663648/79425, 79425/136, 9045146753/10801800), ...
So a(4) = (5295/136) * (272/15) / 2 = 353.
Cf.
A081465 (numerators of hypotenuses).
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nxt[{n_, p_, q_}] := Module[{n1 = Sqrt[p^4 + 4 n^2 q^4], p1 = p Sqrt[p^4 + 4 n^2 q^4], q1 = q^2 n},
a = p1/q1; b = 2 n1 q1/p1; c = Sqrt[p1^4 + 4 n1^2 q1^4]/(p1 q1);
Return [{ a b/2, Numerator[b], Denominator[b]}];]
l = NestList[nxt, {6, 3, 1}, 8] ;
l[[All, 1]]
Showing 1-3 of 3 results.
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