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A156835
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Positive numbers y such that y^2 is of the form x^2+(x+833)^2 with integer x.
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1
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593, 595, 623, 637, 697, 707, 733, 833, 965, 1015, 1037, 1225, 1295, 1547, 1585, 1973, 2023, 2443, 2597, 3145, 3227, 3433, 4165, 5057, 5383, 5525, 6713, 7147, 8687, 8917, 11245, 11543, 14035, 14945, 18173, 18655, 19865, 24157, 29377, 31283, 32113
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OFFSET
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1,1
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COMMENTS
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(-368, a(1)), (-357, a(2)), (-273, a(3)), (-245, a(4)), (-153, a(5)), (-140, a(6)), (-108, a(7)) and (A129010(n), a(n+7)) are solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-15) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 1.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {0, 2, 6, 11}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {3, 5, 8, 9, 12, 14}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {4, 7, 10, 13}.
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LINKS
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FORMULA
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a(n) = 6*a(n-15)-a(n-30) for n > 30; a(1) = 593, a(2) = 595, a(3) = 623, a(4) = 637, a(5) = 697, a(6) = 707, a(7) = 733, a(8) = 833, a(9) = 965, a(10) = 1015, a(11) = 1037, a(12) = 1225, a(13) = 1295, a(14) = 1547, a(15) = 1585, a(16) = 1973, a(17) = 2023, a(18) = 2443, a(19) = 2597, a(20) = 3145, a(21) = 3227, a(22) = 3433, a(23) = 4165, a(24) = 5057, a(25) = 5383, a(26) = 5525, a(27) = 6713, a(28) = 7147, a(29) = 8687, a(30) = 8917.
G.f.: (1-x)*(593 +1188*x+1811*x^2+2448*x^3+3145*x^4+3852*x^5+4585*x^6+5418*x^7+6383*x^8+7398*x^9+8435*x^10+9660*x^11+10955*x^12+12502*x^13+14087*x^14+12502*x^15+10955*x^16+9660*x^17+8435*x^18+7398*x^19+6383*x^20+5418*x^21+4585*x^22+3852*x^23+3145*x^24+2448*x^25+1811*x^26+1188*x^27+593*x^28)/(1-6*x^15+x^30).
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EXAMPLE
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(-368, a(1)) = (-368, 593) is a solution: (-368)^2+(-368+833)^2 = 135424+216225 = 351649 = 593^2.
(A129010(1), a(8)) = (0, 833) is a solution: 0^2+(0+833)^2 = 693889 = 833^2.
(A129010(3), a(10)) = (168, 1015) is a solution: (168)^2+(168+833)^2 = 28224+1002001 = 1030225 = 1015^2.
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PROG
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(PARI) {forstep(n=-400, 26000, [3, 1], if(issquare(2*n^2+1666*n+693889, &k), print1(k, ", ")))}.
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CROSSREFS
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Cf. A129010, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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