A077241 Combined Diophantine Chebyshev sequences A054488 and A077413.

1, 2, 8, 13, 47, 76, 274, 443, 1597, 2582, 9308, 15049, 54251, 87712, 316198, 511223, 1842937, 2979626, 10741424, 17366533, 62605607, 101219572, 364892218, 589950899, 2126747701, 3438485822, 12395593988, 20040964033, 72246816227, 116807298376

Offset

0,2

Comments

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n)= A077242(n).

The number a > 0 belongs to the sequence A077241, if a^2 belongs to the sequence A034856. - Alzhekeyev Ascar M, Apr 27 2012

Numbers k such that k^2 + 2 is a triangular number (see A214838). - Alex Ratushnyak, Mar 07 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for sequences related to Chebyshev polynomials.

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

Formula

a(2k) = A054488(k) and a(2k+1)= A077413(k) for k>=0.

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

a(n) = (-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8. [Bruno Berselli, Mar 10 2013]

Example

8*a(2)^2 + 17 = 8*8^2+17 = 529 = 23^2 = A077242(2)^2.

Mathematica

LinearRecurrence[{0, 6, 0, -1}, {1, 2, 8, 13}, 30] (* Bruno Berselli, Mar 10 2013 *)
CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)

Prog

(Maxima) makelist(expand((-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8), n, 0, 30); /* Bruno Berselli, Mar 10 2013 */
(Magma) I:=[1, 2, 8, 13]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

Crossrefs

Sequence in context: A095825 A106359 A257036 * A228469 A066567 A349818

Adjacent sequences: A077238 A077239 A077240 * A077242 A077243 A077244

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 08 2002

Status

approved