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A161487 Positive numbers y such that y^2 is of the form x^2+(x+191)^2 with integer x. 3
149, 191, 269, 625, 955, 1465, 3601, 5539, 8521, 20981, 32279, 49661, 122285, 188135, 289445, 712729, 1096531, 1687009, 4154089, 6391051, 9832609, 24211805, 37249775, 57308645, 141116741, 217107599, 334019261, 822488641, 1265395819 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-51, a(1)) and (A161486(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 1.

LINKS

Table of n, a(n) for n=1..29.

FORMULA

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=149, a(2)=191, a(3)=269, a(4)=625, a(5)=955, a(6)=1465.

G.f.: (1-x)*(149+340*x+609*x^2+340*x^3+149*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 191*A001653(k) for k >= 1.

EXAMPLE

(-51, a(1)) = (-51, 149) is a solution: (-51)^2+(-51+191)^2 = 2601+19600 = 22201 = 149^2.

(A161486(1), a(2)) = (0, 191) is a solution: 0^2+(0+191)^2 = 36481 = 191^2.

(A161486(3), a(4)) = (336, 625) is a solution: 336^2+(336+191)^2 = 112896+277729 = 390625 = 625^2.

PROG

(PARI) {forstep(n=-52, 100000000, [1, 3], if(issquare(2*n^2+382*n+36481, &k), print1(k, ", ")))}

CROSSREFS

Cf. A161486, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).

Sequence in context: A307472 A209619 A031929 * A121947 A141946 A128390

Adjacent sequences:  A161484 A161485 A161486 * A161488 A161489 A161490

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Jun 13 2009

STATUS

approved

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Last modified June 24 07:00 EDT 2021. Contains 345416 sequences. (Running on oeis4.)