A106528 Values of y in x^2 - 289 = 2*y^2.

6, 20, 34, 56, 126, 204, 330, 736, 1190, 1924, 4290, 6936, 11214, 25004, 40426, 65360, 145734, 235620, 380946, 849400, 1373294, 2220316, 4950666, 8004144, 12940950, 28854596, 46651570, 75425384, 168176910, 271905276, 439611354, 980206864

Offset

1,1

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

a(3n) = 34*A001109(n).

a(3n+1) = A001109(n+2) + 4*A001109(n+1) - 3*A001109(n) + 4*A001109(n-1).

a(3n+2) = 4*A001109(n+2) - 3*A001109(n+1) + 4*A001109(n) + A001109(n-1).

From Colin Barker, Mar 29 2012: (Start)

a(n) = 6*a(n-3) - a(n-6).

G.f.: 2*x*(3 +10*x +17*x^2 +10*x^3 +3*x^4)/(1 -6*x^3 +x^6). (End)

Example

a(9) = 1190, 34*A001109(3) = 34*35 = 1190;
a(10) = 1924, A001109(5) +4*A001109(4) -3*A001109(3) +4*A001109(2) = 1189 + 4*204 - 3*35 + 4*6 = 1924;
a(11) =4290, 4*A001109(5) -3*A001109(4) +4*A001109(3) +A001109(2) = 4*1189 - 3*204 + 4*35 + 6 = 4290; also 2*4290^2 = A106527(11)^2 - 289 = 6067^2 - 289.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {6, 20, 34, 56, 126, 204}, 40] (* G. C. Greubel, Aug 18 2021 *)

Prog

(Magma) I:=[6, 20, 34, 56, 126, 204]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..41]]; // G. C. Greubel, Aug 18 2021
(Sage)
def A106528_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( 2*x*(3 +10*x +17*x^2 +10*x^3 +3*x^4)/(1 -6*x^3 +x^6) ).list()
a=A106528_list(41); a[1:] # G. C. Greubel, Aug 18 2021

Crossrefs

Cf. A001109, A106527.

Sequence in context: A226363 A253906 A031005 * A031068 A031052 A097811

Adjacent sequences: A106525 A106526 A106527 * A106529 A106530 A106531

Keyword

nonn

Author

Andras Erszegi (erszegi.andras(AT)chello.hu), May 09 2005

Status

approved