A248163 Chebyshev's S polynomials (A049310) evaluated at 34/3 and multiplied by powers of 3 (A000244).

1, 34, 1147, 38692, 1305205, 44028742, 1485230383, 50101574344, 1690086454249, 57012025275370, 1923198081274339, 64875626535849196, 2188462519487403613, 73823845023749080078, 2490314568132082090135, 84006280711277049343888, 2833800713070230938880977, 95593167717986358477858226

Offset: 0

Author

Wolfdieter Lang, Nov 07 2014

Keywords

• nonn
• easy

Comments

This sequence appears in the solution for the curvature sequence of the touching circle and chord example given in A249457. See also the pair A249862(n) and a(n-1), with a(-1) = 0, for which details are given in A249862.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..650
• Index entries for linear recurrences with constant coefficients, signature (34,-9).

Crossrefs

Cf. A000244, A049310, A249457, A249862.

Programs

Magma

I:=[1, 34]; [n le 2 select I[n] else 34*Self(n-1) - 9*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 08 2014

Mathematica

CoefficientList[Series[1/(1-34 x +(3 x)^2), {x,0,40}], x] (* Vincenzo Librandi, Nov 08 2014 *)
Table[3^n*ChebyshevU[n,17/3], {n,0,40}] (* G. C. Greubel, May 31 2025 *)

PARI

a(n) = 3^n*polchebyshev(n, 2, 17/3); \\ Michel Marcus, May 31 2025

SageMath

def A248163(n): return 3^n*chebyshev_U(n,17/3)
print([A248163(n) for n in range(41)]) # G. C. Greubel, May 31 2025

Formula

a(n) = 3^n*S(n, 34/3) with Chebyshev's S polynomial (for S see the coefficient triangle A049310).

O.g.f.: 1/(1 - 34*x + 9*x^2).

a(n) = 34*a(n-1) - 9*a(n-2), a(-1) = 0, a(0) = 1 .

E.g.f.: exp(17*x)*(140*cosh(2*sqrt(70)*x) + 17*sqrt(70)*sinh(2*sqrt(70)*x))/140. - Stefano Spezia, Mar 24 2023

Extensions

a(16)-a(17) from Stefano Spezia, Mar 24 2023