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A118337
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+23)^2 = y^2.
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13
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0, 12, 33, 69, 133, 252, 460, 832, 1525, 2737, 4905, 8944, 16008, 28644, 52185, 93357, 167005, 304212, 544180, 973432, 1773133, 3171769, 5673633, 10334632, 18486480, 33068412, 60234705, 107747157, 192736885, 351073644, 627996508, 1123352944, 2046207205
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also values x of Pythagorean triples (x, x+23, y).
Corresponding values y of solutions (x, y) are in A156567.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number in A028871, m>=5, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+2, a(3)=3*m^2-10m+8, a(4)=3p, a(5)=3*m^2+10m+8, a(6)=20*m^2-58m+42. Pairs (p, m) are (23, 5), (47, 7), (79, 9), (167, 13), (223, 15), (359, 19), (439, 21), (727, 27), (839, 29), ...
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/((27+10*sqrt(2))/23)^2 for n mod 3 = 0.
For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=m^2+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009]
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REFERENCES
| Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278. Solution appeared in Vol. 38, No. 2, May 2000, pp. 183-184.
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FORMULA
| a(n) = 6*a(n-3)-a(n-6)+46 for n > 6; a(1)=0, a(2)=12, a(3)=33, a(4)=69, a(5)=133, a(6)=252.
G.f.: x*(12+21*x+36*x^2-8*x^3-7*x^4-8*x^5)/((1-x)*(1-6*x^3+x^6)).
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MATHEMATICA
| Select[Range[0, 100000], IntegerQ[Sqrt[#^2+(#+23)^2]]&] (* or *) LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 12, 33, 69, 133, 252, 460}, 50] (* From Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
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PROG
| (PARI) {forstep(n=0, 1124000000, [1, 3], if(issquare(2*n*(n+23)+529), print1(n, ", ")))}
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CROSSREFS
| Cf. A156567, A028871 (primes of form n^2 - 2), A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23).
Cf. A118675 (p=47), A118676 (p=79), A130608 (p=167), A130609 (p=223), A130610 (p=359), A130645 (p=439), A130646 (p=727), A130647 (p=839).
Sequence in context: A063296 A051624 A039338 * A032604 A043161 A043941
Adjacent sequences: A118334 A118335 A118336 * A118338 A118339 A118340
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KEYWORD
| nonn,changed
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 14 2006
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EXTENSIONS
| Edited by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 10 2009
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