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A118674
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+31)^2 = y^2.
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15
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0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076
(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+31, y).
Corresponding values y of solutions (x, y) are in A157646.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 0.
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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)+62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
G.f.: x*(9+51*x+33*x^2-7*x^3-17*x^4-7*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 31*A001652(k) for k >= 0.
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MATHEMATICA
| ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* From Jean-François Alcover, Dec 27 2011, after given formula *)
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PROG
| (PARI) {forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ", ")))}
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CROSSREFS
| cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).
Sequence in context: A039929 A099333 A098327 * A074431 A081904 A085373
Adjacent sequences: A118671 A118672 A118673 * A118675 A118676 A118677
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KEYWORD
| nonn
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AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 19 2006
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EXTENSIONS
| Edited by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 11 2009
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