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Search: id:A157348
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| A157348 |
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Positive numbers y such that y^2 is of the form x^2+(x+281)^2 with integer x. |
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4
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| 229, 281, 365, 1009, 1405, 1961, 5825, 8149, 11401, 33941, 47489, 66445, 197821, 276785, 387269, 1152985, 1613221, 2257169, 6720089, 9402541, 13155745, 39167549, 54802025, 76677301, 228285205, 319409609, 446908061, 1330543681
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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(-60, a(1)) and (A129626(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+281)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (297+68*sqrt(2))/281 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (130803+73738*sqrt(2))/281^2 for n mod 3 = 1.
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FORMULA
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a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=229, a(2)=281, a(3)=365, a(4)=1009, a(5)=1405, a(6)=1961.
G.f.: (1-x)*(229+510*x+875*x^2+510*x^3+229*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 281*A001653(k) for k >= 1.
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EXAMPLE
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(-60, a(1)) = (-60, 229) is a solution: (-60)^2+(-60+281)^2 = 3600+48841 = 52441 = 229^2.
(A129626(1), a(2)) = (0, 281) is a solution: 0^2+(0+281)^2 = 78961 = 281^2.
(A129626(3), a(4)) = (559, 1009) is a solution: 559^2+(559+281)^2 = 312481+705600 = 1018081 = 1009^2.
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PROGRAM
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(PARI) {forstep(n=-60, 200000000, [3, 1], if(issquare(2*n^2+562*n+78961, &k), print1(k, ", ")))}
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CROSSREFS
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Cf. A129626, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157349 (decimal expansion of (297+68*sqrt(2))/281), A157350 (decimal expansion of (130803+73738*sqrt(2))/281^2).
Sequence in context: A062589 A094612 A112847 this_sequence A142221 A142779 A139512
Adjacent sequences: A157345 A157346 A157347 this_sequence A157349 A157350 A157351
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KEYWORD
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nonn
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AUTHOR
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Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 12 2009
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