0,1

a(n) gives the general (positive integer) solution of the Pell equation a^2 - 165*b^2 =+4 with companion sequence b(n)=A078362(n-1), n>=1.

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

a(n)=13*a(n-1)-a(n-2), n >= 1; a(-1)=13, a(0)=2.

a(n) = S(n, 13) - S(n-2, 13) = 2*T(n, 13/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 13)=A078362(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, case. See A049310 and A053120.

G.f.: (2-13*x)/(1-13*x+x^2).

a(n) = ap^n + am^n, with ap := (13+sqrt(165))/2 and am := (13-sqrt(165))/2.

a[0] = 2; a[1] = 13; a[n_] := 13a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 16}] (from Robert G. Wilson v Jan 30 2004)

(PARI) a(n)=if(n<0, 0, 2*subst(poltchebi(n), x, 13/2))

(PARI) a(n)=if(n<0, 0, polsym(1-13*x+x^2, n)[n+1])

sage: [lucas_number2(n, 13, 1) for n in xrange(0, 20)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

a(n)=sqrt(4 + 165*A078362(n-1)^2), n>=1, (Pell equation d=165, +4).

Cf. A077428, A078355 (Pell +4 equations).

Sequence in context: A098638 A090643 A132521 this_sequence A143851 A088316 A006905

Adjacent sequences: A078360 A078361 A078362 this_sequence A078364 A078365 A078366

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 29 2002