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A159690 Positive numbers y such that y^2 is of the form x^2+(x+881)^2 with integer x. 4
841, 881, 925, 4121, 4405, 4709, 23885, 25549, 27329, 139189, 148889, 159265, 811249, 867785, 928261, 4728305, 5057821, 5410301, 27558581, 29479141, 31533545, 160623181, 171817025, 183790969, 936180505, 1001423009, 1071212269 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

(-41,a(1)) and (A130014(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

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

lim_{n -> infinity} a(n)/a(n-1) = (883+42*sqrt(2))/881 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 1.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..3501

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

FORMULA

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=841, a(2)=881, a(3)=925, a(4)=4121, a(5)=4405, a(6)=4709.

G.f.: (1-x)*(841+1722*x+2647*x^2+1722*x^3+841*x^4) / (1-6*x^3+x^6).

a(3*k-1) = 881*A001653(k) for k >= 1.

EXAMPLE

(-41, a(1)) = (-41, 841) is a solution: (-41)^2+(-41+881)^2 = 1681+705600 = 707281 = 841^2.

(A130014(1), a(2)) = (0, 881) is a solution: 0^2+(0+881)^2 = 776161 = 881^2.

(A130014(3), a(4)) = (2440, 4121) is a solution: 2440^2+(2440+881)^2 = 5953600+11029041 = 16982641 = 4121^2.

MATHEMATICA

CoefficientList[Series[(1 - x)*(841 + 1722*x + 2647*x^2 + 1722*x^3 + 841*x^4)/(1 - 6*x^3 + x^6), {x, 0, 50}], x] (* or *) LinearRecurrence[{0, 0, 6, 0, 0, -1}, {841, 881, 925, 4121, 4405, 4709}, 30] (* G. C. Greubel, Jun 02 2018 *)

PROG

(PARI) {forstep(n=-44, 10000000, [3, 1], if(issquare(2*n^2+1762*n+776161, &k), print1(k, ", ")))}

(Magma) I:=[841, 881, 925, 4121, 4405, 4709]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Jun 02 2018

CROSSREFS

Cf. A130014, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

Sequence in context: A331652 A269894 A252780 * A210470 A108324 A133496

Adjacent sequences: A159687 A159688 A159689 * A159691 A159692 A159693

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 21 2009

STATUS

approved

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Last modified February 1 14:40 EST 2023. Contains 359993 sequences. (Running on oeis4.)