A164547 a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 11.

1, 11, 93, 743, 5849, 45859, 359157, 2811967, 22014001, 172336571, 1349127693, 10561555223, 82680381449, 647257375699, 5067007272357, 39666697336687, 310527849736801, 2430944642644331, 19030472980917693, 148978670884223303

Offset: 0

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

• nonn
• easy

Comments

Binomial transform of A164546.

Fifth binomial transform of A164640.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..144
• Index entries for linear recurrences with constant coefficients, signature (10,-17).

Crossrefs

Cf. A164546, A164640.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(5+2*r)^n+(2-3*r)*(5-2*r)^n)/4: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009

Mathematica

LinearRecurrence[{10,-17},{1,11},30] (* Harvey P. Dale, Jun 04 2012 *)

Sage

[(17)^((n-1)/2)*(sqrt(17)*chebyshev_U(n, 5/sqrt(17)) + chebyshev_U(n-1, 5/sqrt(17))) for n in (0..30)] # G. C. Greubel, Jul 17 2021

Formula

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 11.

a(n) = ((2+3*sqrt(2))*(5+2*sqrt(2))^n + (2-3*sqrt(2))*(5-2*sqrt(2))^n)/4.

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

a(n) = (17)^((n-1)/2)*(sqrt(17)*ChebyshevU(n, 5/sqrt(17)) + ChebyshevU(n-1, 5/sqrt(17))). - G. C. Greubel, Jul 17 2021

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009