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A046729
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G.f.: 4x/((1+x)(1-6x+x^2)).
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9
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0, 4, 20, 120, 696, 4060, 23660, 137904, 803760, 4684660, 27304196, 159140520, 927538920, 5406093004, 31509019100, 183648021600, 1070379110496, 6238626641380, 36361380737780, 211929657785304, 1235216565974040
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Related to Pythagorean triples: alternate terms of A001652 and A046090.
Even-valued legs of nearly isosceles right triangles: legs differ by 1. 0 is smaller leg of degenerate triangle with legs 0 and 1 and hypotenuse 1. - Charlie Marion (charliem(AT)bestweb.net), Nov 11 2003
The complete (nearly isosceles) primitive Pythagorean triple is given by {a(n),a(n)+(-1)^n, A001653(n)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), Feb 19 2004
Note also that A046092 is the even leg of this other class of nearly isosceles Pythagorean triangles {A005408(n), A046092(n), A001844(n)}, i.e. {2n+1, 2n(n+1), 2n(n+1)+1} where longer sides (viz. even leg and hypotenus) are consecutive. - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 22 2004
Union of even entries of A001652 and A046090. Sum of legs of primitive Pythagorean triangles is A002315(n)=2*a(n)+(-1)^n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Apr 30 2004
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REFERENCES
| A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
W. Sierpinski, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 17. MR2002669.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| a(n)=((1+sqrt(2))^(2n+1)+(1-sqrt(2))^(2n+1)+2(-1)^(n+1))/4; a(n)=A089499(n)*A089499(n+1); cf. A084159.
a(n)=4*A084158(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 16 2004
a(n) = ceil((sqrt(2)+1)^(2*n+1)-(sqrt(2)-1)^(2*n+1)-2*(-1)^n)/4. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 12 2004
a(n) is the k-th entry amongst the complete near-isosceles primitive Pythagorean triple A114336(n), where k={3*(2n-1)-(-1)^n}/2, i.e., a(n)=A114336(A047235(n)), for positive n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 04 2006
a(n) = A046727(n)-(-1)^n = 2*A114620(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 14 2006
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EXAMPLE
| [1,0,1]*[1,2,2;2,1,2;2,2,3]^0 gives (degenerate) primitive Pythagorean triple [1, 0, 1], so a(0) = 0. [1,0,1]*[1,2,2;2,1,2;2,2,3]^7 gives primitive Pythagorean triple [137903, 137904, 195025] so a(7) = 137904
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PROG
| (PARI) a(n)=n%2+(real((1+quadgen(8))^(2*n+1))-1)/2
(PARI) a(n)=if(n<0, -a(-1-n), polcoeff(4*x/(1+x)/(1-6*x+x^2)+x*O(x^n), n))
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CROSSREFS
| Cf. A046727, A084159, A084158, A001652, A046090.
Sequence in context: A101055 A013197 A089498 * A093123 A092055 A187848
Adjacent sequences: A046726 A046727 A046728 * A046730 A046731 A046732
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Philip Sung (phil(AT)main.nu), May 05 2001
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