Search: a157757 -id:a157757
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A157758
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a(n) = 297754*n - 244754.
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+10
3
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53000, 350754, 648508, 946262, 1244016, 1541770, 1839524, 2137278, 2435032, 2732786, 3030540, 3328294, 3626048, 3923802, 4221556, 4519310, 4817064, 5114818, 5412572, 5710326, 6008080, 6305834, 6603588, 6901342, 7199096
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OFFSET
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1,1
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COMMENTS
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The identity (15780962*n^2-25943924*n+10662963)^2-(2809*n^2-4618*n+1898)*(297754*n-244754)^2=1 can be written as A157759(n)^2-A157757(n)*a(n)^2=1.
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - a(n-2).
G.f.: x*(53000 + 244754*x)/(1-x)^2.
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MATHEMATICA
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LinearRecurrence[{2, -1}, {53000, 350754}, 30]
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PROG
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(Magma) I:=[53000, 350754]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..40]];
(PARI) a(n) = 297754*n - 244754;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A157759
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a(n) = 15780962*n^2 - 25943924*n + 10662963.
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+10
3
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500001, 21898963, 74859849, 159382659, 275467393, 423114051, 602322633, 813093139, 1055425569, 1329319923, 1634776201, 1971794403, 2340374529, 2740516579, 3172220553, 3635486451, 4130314273, 4656704019, 5214655689, 5804169283
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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The identity (15780962*n^2 - 25943924*n + 10662963)^2 - (2809*n^2 - 4618*n+1898)*(297754*n - 244754)^2 = 1 can be written as a(n)^2 - A157757(n)*A157758(n)^2 = 1.
This is the case s=53 and r=2309 of the identity (2*(s^2*n-r)^2+1)^2 - (((s^2*n-r)^2+1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2+1)/s^2 is an integer if r^2 == -1 (mod s^2). Therefore, for s=53, nonnegative r values are: 500, 2309, 3309, 5118, 6118, 7927, 8927, 10736, 11736, ... - Bruno Berselli, Apr 24 2018
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LINKS
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FORMULA
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G.f.: x*(500001 - 20398960*x - 10662963*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {500001, 21898963, 74859849}, 30]
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PROG
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(Magma) I:=[500001, 21898963, 74859849]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..30]];
(PARI) a(n) = 15780962*n^2 - 25943924*n + 10662963;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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