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A118673
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Positive solutions x to the equation x^2+(x+71)^2=y^2.
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13
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0, 13, 160, 213, 280, 1113, 1420, 1809, 6660, 8449, 10716, 38989, 49416, 62629, 227416, 288189, 365200, 1325649, 1679860, 2128713, 7726620, 9791113, 12407220, 45034213, 57066960, 72314749, 262478800, 332610789, 421481416, 1529838729, 1938597916, 2456573889
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Consider all Pythagorean triples (x,x+71,y) ordered by increasing y; sequence gives x values.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2 the associated value in A066049, the x values are given by the sequence defined by: a(n)=6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009]
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FORMULA
| a(n)=6*a(n-3)-a(n-6)+142 with a(0)=0, a(1)=13, a(2)=160, a(3)=213, a(4)=280, a(5)=1113.
O.g.f.: x(13+147x+53x^2-11x^3-49x^4-11*x^5)/((1-x)(1-6x^3+x^6)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
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MATHEMATICA
| Select[Range[0, 100000], IntegerQ[Sqrt[#^2+(#+71)^2]]&] (* or *) LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 13, 160, 213, 280, 1113, 1420}, 100] (* From Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
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CROSSREFS
| Cf. A076296 (p=7), A118120 (p=17), A118674 (p=31), A129836 (p=97), A129992 (p=127), A129993 (p=199), A129991 (p=241), A129999 (p=337), A130004 (p=449), A130005 (p=577), A130013 (p=647), A130014 (p=881), A130017 (p=967).
Sequence in context: A000830 A205170 A205163 * A133180 A090134 A087400
Adjacent sequences: A118670 A118671 A118672 * A118674 A118675 A118676
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KEYWORD
| nonn,changed
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 19 2006
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EXTENSIONS
| Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2008
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