A075276 Generalized Markoff numbers: union of numbers a, b, c, d satisfying the Markoff(4) equation a^2 + b^2 + c^2 + d^2 = 4*a*b*c*d.
1, 3, 11, 41, 131, 153, 571, 1561, 1803, 2131, 5761, 7953, 17291, 18601, 25091, 29681, 79291, 110771, 221651, 253353, 295681, 349451, 413403, 817961, 1542841, 2282281, 2453891, 2641211, 3018753, 3252611, 3487001, 4114771, 4867203, 5757961, 11141771
Offset: 1
Examples
Some solutions to Markoff(4) equation a^2 + b^2 + c^2 + d^2 = 4abcd are (1,1,1,1), (1,1,1,3), (1,1,3,11), (1,3,11,131), (3,11,131,17291), (1,1,11,41), (1,11,41,1803), (1,1,41,153), (1,3,131,1561), (1,11,131,5761), (1,1,153,571), (1,1,571,2131), (1,1,2131,7953), (1,3,1561,18601).
Links
- Xianwen Wang, Table of n, a(n) for n = 1..5555
Crossrefs
Cf. A002559.
Programs
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Mathematica
MAX=10^10; data = NestWhile[Select[Union[Sort/@Flatten[Table[{a,b,c,4 a b c-d}/.MapThread[Rule,{{a,b,c,d},#}]&/@Map[RotateLeft[ii,#]&,Range[4]],{ii,#}],1]], Max[#] < MAX&]&,{{1,1,1,1},{1,1,1,3}}, UnsameQ,2]; Take[data//Flatten//Union, 30] (* Xianwen Wang, Feb 23 2013 *)
Formula
If (a, b, c, d) satisfies Markoff(4) equation, then so does (a, b, c, 4abc - d) and 4abc - d = (a^2 + b^2 + c^2)/d.
Extensions
Sequence corrected by Xianwen Wang, Feb 23 2013
Comments