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A160176 Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x. 4
533, 617, 733, 2465, 3085, 3865, 14257, 17893, 22457, 83077, 104273, 130877, 484205, 607745, 762805, 2822153, 3542197, 4445953, 16448713, 20645437, 25912913, 95870125, 120330425, 151031525, 558772037, 701337113, 880276237 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(-92, a(1)) and (A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.
LINKS
FORMULA
a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.
G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 617*A001653(k) for k >= 1.
EXAMPLE
(-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
(A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2.
(A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {533, 617, 733, 2465, 3085, 3865}, 50] (* G. C. Greubel, May 04 2018 *)
PROG
(PARI) {forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ", ")))}
(PARI) x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, May 04 2018
(Magma) I:=[533, 617, 733, 2465, 3085, 3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 04 2018
CROSSREFS
Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).
Sequence in context: A048055 A067803 A098258 * A077085 A165989 A183598
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, May 18 2009
STATUS
approved

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Last modified April 20 17:12 EDT 2024. Contains 371845 sequences. (Running on oeis4.)