A023038 a(n) = 12*a(n-1) - a(n-2).

1, 6, 71, 846, 10081, 120126, 1431431, 17057046, 203253121, 2421980406, 28860511751, 343904160606, 4097989415521, 48831968825646, 581885636492231, 6933795669081126, 82623662392481281, 984550153040694246, 11731978174095849671, 139799187936109501806

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

From Wolfdieter Lang, Nov 08 2002: (Start)

Chebyshev's polynomials T(n,x) evaluated at x=6.

a(n+1) give all (nontrivial, integer) solutions of Pell equation a(n+1)^2 - 35*b(n)^2 = +1 with b(n)=A004191(n), n>=0. (End)

a(35+70k)-1 and a(35+70k)+1 are consecutive odd powerful numbers. The first pair is 23101441813552306872262673994181386126 +- 1. See A076445. - T. D. Noe, May 04 2006

Numbers n such that 35*(n^2-1) is a square. - Vincenzo Librandi, Nov 19 2010

Except for the first term, positive values of x (or y) satisfying x^2 - 12xy + y^2 + 35 = 0. - Colin Barker, Feb 09 2014

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Tanya Khovanova, Recursive Sequences
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (12,-1).

Crossrefs

Cf. A087800.

Programs

Magma

I:=[1, 6]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 19 2017

Maple

A023038:=n->round(((6+sqrt(35))^n + (6-sqrt(35))^n)/2); seq(A023038(n), n=0..30); # Wesley Ivan Hurt, Feb 03 2014

Mathematica

Table[Round[((6 + Sqrt[35])^n + (6 - Sqrt[35])^n)/2], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 03 2014 *)
nn = 20; CoefficientList[Series[(1 - 6*x)/(1 - 12*x + x^2), {x, 0, nn}], x] (* T. D. Noe, Feb 05 2014 *)

PARI

x='x+O('x^30); Vec((1-6*x)/(1-12*x+x^2)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = T(n, 6) = (S(n, 12)-S(n-2, 12))/2 with S(n, x) := U(n, x/2) and T(n), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 12)=A004191(n).

a(n) = ((6+sqrt(35))^n + (6-sqrt(35))^n)/2.

G.f.: (1-6*x)/(1-12*x+x^2).

a(n)*a(n+3) - a(n+1)*a(n+2) = 420. - Ralf Stephan, Jun 06 2005