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Search: id:A118611
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| A118611 |
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Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2. |
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+0
4
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| 0, 77, 132, 245, 392, 585, 728, 1029, 1428, 1725, 2352, 3185, 4292, 5117, 6860, 9177, 10904, 14553, 19404, 25853, 30660, 40817, 54320, 64385, 85652, 113925, 151512, 179529, 238728, 317429, 376092, 500045, 664832, 883905, 1047200, 1392237
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OFFSET
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1,2
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COMMENT
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Also values x of Pythagorean triples (x, x+343, y); 343=7^3.
Corresponding values y of solutions (x, y) are in A157246.
lim_{n -> infinity} a(n)/a(n-7) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {1, 2, 4, 5, 6}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {0, 3}.
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FORMULA
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a(n) = 6*a(n-7)-a(n-14)+686 for n > 14; a(1)=0, a(2)=77, a(3)=132, a(4)=245, a(5)=392, a(6)=585, a(7)=728, a(8)=1029, a(9)=1428, a(10)=1725, a(11)=2352, a(12)=3185, a(13)=4292, a(14)=5117.
G.f.: x*(77+55*x+113*x^2+147*x^3+193*x^4+143*x^5+301*x^6-63*x^7 -33*x^8-51*x^9-49*x^10-51*x^11-33*x^12-63*x^13)/((1-x)*(1-6*x^7+x^14)).
a(7*k+1) = 343*A001652(k) for k >= 0.
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EXAMPLE
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132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
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PROGRAM
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(PARI) {forstep(n=0, 1400000, [1, 3], if(issquare(n^2+(n+343)^2), print1(n, ", ")))}
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CROSSREFS
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Cf. A157246, A001652, A118576, A118554, A118611, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).
Sequence in context: A127335 A105998 A039444 this_sequence A072431 A046435 A004964
Adjacent sequences: A118608 A118609 A118610 this_sequence A118612 A118613 A118614
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KEYWORD
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nonn
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AUTHOR
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Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 08 2006
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EXTENSIONS
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Edited by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 25 2009
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