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 A046729 Expansion of 4*x/((1+x)*(1-6*x+x^2)). 15
 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; text; internal format)
 OFFSET 0,2 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, Nov 11 2003 The complete (nearly isosceles) primitive Pythagorean triple is given by {a(n), a(n)+(-1)^n, A001653(n)}. - Lekraj Beedassy, 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, 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, Apr 30 2004 REFERENCES A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964. W. Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 17. MR2002669. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (5,5,-1). 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). a(n) = 4*A084158(n). - Lekraj Beedassy, Jul 16 2004 a(n) = ceiling((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, Jun 04 2006 a(n) = A046727(n) - (-1)^n = 2*A114620(n). - Lekraj Beedassy, Aug 14 2006 From George F. Johnson, Aug 29 2012: (Start) 2*a(n)*(a(n) + (-1)^n) + 1 = (A000129(2*n+1))^2; n > 0, 2*a(n)*(a(n) + (-1)^n) + 1 = ((a(n+1) - a(n-1))/4)^2, a perfect square. a(n+1) = (3*a(n) + 2*(-1)^n) + 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1). a(n-1) = (3*a(n) + 2*(-1)^n) - 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1). a(n+1) = 6*a(n) - a(n-1) + 4*(-1)^n. a(n+1) = 5*a(n) + 5*a(n-1) - a(n-2). a(n+1) *a(n-1) = a(n)*(a(n) + 4*(-1)^n). a(n) = (sqrt(1 + 8*A029549(n)) - (-1)^n)/2. a(n) = A002315(n) - A084159(n) = A084159(n) - (-1)^n. a(n) = A001652(n) + (1 - (-1)^n)/2 = A046090(n) - (1 + (-1)^n)/2. Limit_{n->infinity} a(n)/a(n-1) = 3 + 2*sqrt(2). Limit_{n->infinity} a(n)/a(n-2) = 17 + 12*sqrt(2). Limit_{n->infinity} a(n)/a(n-r) = (3 + 2*sqrt(2))^r. Limit_{n->infinity} a(n-r)/a(n) = (3 - 2*sqrt(2))^r. (End) From G. C. Greubel, Feb 11 2023: (Start) a(n) = (A001333(2*n+1) - 2*(-1)^n)/4. a(n) = (1/2)*(A001109(n+1) + A001109(n) - (-1)^n). (End) 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. G.f. = 4*x + 20*x^2 + 120*x^3 + 696*x^4 + 4060*x^5 + 23660*x^6 + ... MATHEMATICA LinearRecurrence[{5, 5, -1}, {0, 4, 20}, 25] (* Vincenzo Librandi, Jul 29 2019 *) 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)) (Magma) [4*Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)-2*(-1)^n) / 16): n in [0..30]]; // Vincenzo Librandi, Jul 29 2019 (SageMath) [(lucas_number2(2*n+1, 2, -1) -2*(-1)^n)/4 for n in range(41)] # G. C. Greubel, Feb 11 2023 CROSSREFS Cf. A000129, A001109, A001333, A001652, A002315, A029549, A046090. Cf. A046727, A047235, A084158, A084159, A089499, A114336. Sequence in context: A013197 A319788 A089498 * A277920 A093123 A092055 Adjacent sequences: A046726 A046727 A046728 * A046730 A046731 A046732 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified May 27 19:17 EDT 2023. Contains 362985 sequences. (Running on oeis4.)