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A161479 Positive numbers y such that y^2 is of the form x^2+(x+113)^2 with integer x. 3
85, 113, 173, 337, 565, 953, 1937, 3277, 5545, 11285, 19097, 32317, 65773, 111305, 188357, 383353, 648733, 1097825, 2234345, 3781093, 6398593, 13022717, 22037825, 37293733, 75901957, 128445857, 217363805, 442389025, 748637317, 1266889097 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-36, a(1)) and (A161478(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 1.

LINKS

Table of n, a(n) for n=1..30.

FORMULA

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=85, a(2)=113, a(3)=173, a(4)=337, a(5)=565, a(6)=953.

G.f.: (1-x)*(85+198*x+371*x^2+198*x^3+85*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 113*A001653(k) for k >= 1.

EXAMPLE

(-36, a(1)) = (-36, 85) is a solution: (-36)^2+(-36+113)^2 = 1296+5929 = 7225 = 85^2.

(A161478(1), a(2)) = (0, 113) is a solution: 0^2+(0+113)^2 = 12769 = 113^2.

(A161478(3), a(4)) = (175, 337) is a solution: 175^2+(175+113)^2 = 30625+82944 = 113569 = 337^2.

PROG

(PARI) {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+226*n+12769, &k), print1(k, ", ")))}

CROSSREFS

Cf. A161478, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Sequence in context: A039487 A157469 A066474 * A027453 A259219 A273123

Adjacent sequences:  A161476 A161477 A161478 * A161480 A161481 A161482

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jun 13 2009

STATUS

approved

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Last modified May 10 15:15 EDT 2021. Contains 343773 sequences. (Running on oeis4.)