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A157247
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Positive numbers y such that y^2 is of the form x^2+(x+2401)^2 with integer x.
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1
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1715, 1781, 1855, 2009, 2401, 2989, 3451, 3821, 4459, 5831, 6865, 7679, 9065, 12005, 15925, 18851, 21145, 25039, 33271, 39409, 44219, 52381, 69629, 92561, 109655, 123049, 145775, 193795, 229589, 257635, 305221, 405769, 539441, 639079, 717149
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| (-1029, a(1)), (-820, a(2)), (-672, a(3)), (-441, a(3)) and (A118630(n), a(n+4)) are solutions (x, y) to the Diophantine equation x^2+(x+2401)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-9) = 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 9 = {1, 5, 6}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 9 = {0, 2, 4, 7}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 9 = {3, 8}.
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FORMULA
| a(n)=6*a(n-9)-a(n-18) for n > 18; a(1)=1715, a(2)=1781, a(3)=1855, a(4)=2009, a(5)=2401, a(6)=2989, a(7)=3451, a(8)=3821, a(9)=4459, a(10)=5831, a(11)=6865, a(12)=7679, a(13)=9065, a(14)=12005, a(15)=15925, a(16)=18851, a(17)=21145, a(18)=25039.
G.f.: x * (1-x) * (1715 +3496*x +5351*x^2 +7360*x^3 +9761*x^4 +12750*x^5 +16201*x^6 +20022*x^7 +24481*x^8 +20022*x^9 +16201*x^10 +12750*x^11 +9761*x^12 +7360*x^13 +5351*x^14 +3496*x^15 +1715*x^16) / (1 -6*x^9 +x^18).
a(9*k-4) = 2401*A001653(k) for k >= 1.
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EXAMPLE
| (-1029, a(1)) = (-1029, 1715) is a solution: (-1029)^2+(-1029+2401)^2 = 1058841+1882384 = 2941225 = 1715^2.
(A118630(1), a(5)) = (0, 2401) is a solution: 0^2+(0+2401)^2 = 5764801 = 2401^2.
(A118630(3), a(7)) = (924, 3451) is a solution: 924^2+(924+2401)^2 = 853776+11055625 = 11909401 = 3451^2.
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MATHEMATICA
| Sqrt[#]&/@Select[Table[2x^2+4802x+5764801, {x, -1200, 510000}], IntegerQ[ Sqrt[ #]]&] (* From Harvey P. Dale, Jul 21 2011 *)
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PROG
| (PARI) {forstep(n=-1032, 540000, [3 , 1], if(issquare(n^2+(n+2401)^2, &k), print1(k, ", ")))}
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CROSSREFS
| Cf. A118630, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).
Sequence in context: A129540 A008744 A184090 * A024408 A185795 A194719
Adjacent sequences: A157244 A157245 A157246 * A157248 A157249 A157250
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KEYWORD
| nonn
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AUTHOR
| Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Feb 25 2009
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EXTENSIONS
| G.f. adapted to the offset by Bruno Berselli, Apr 01 2011
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