|
Search: id:A157297
|
|
|
| A157297 |
|
Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x. |
|
+0
4
|
|
| 185, 233, 317, 793, 1165, 1717, 4573, 6757, 9985, 26645, 39377, 58193, 155297, 229505, 339173, 905137, 1337653, 1976845, 5275525, 7796413, 11521897, 30748013, 45440825, 67154537, 179212553, 264848537, 391405325, 1044527305, 1543650397
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
(-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1.
|
|
FORMULA
|
a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.
G.f.: (1-x)*(185+418*x+735*x^2+418*x^3+185*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 233*A001653(k) for k >= 1.
|
|
EXAMPLE
|
(-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.
(A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.
(A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.
|
|
PROGRAM
|
(PARI) {forstep(n=-60, 1100000000, [3, 1], if(issquare(2*n^2+466*n+54289, &k), print1(k, ", ")))}
|
|
CROSSREFS
|
Cf. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).
Sequence in context: A060491 A151586 A139265 this_sequence A156059 A129311 A136048
Adjacent sequences: A157294 A157295 A157296 this_sequence A157298 A157299 A157300
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 11 2009
|
|
|
Search completed in 0.002 seconds
|
