A156573 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=529, a(2)=13225.

529, 13225, 444889, 15108769, 513249025, 17435353849, 592288777609, 20120383080625, 683500735959409, 23218904639535049, 788759257008228025, 26794595833640213569, 910227499086759029089, 30920940373116166771225

Offset: 1

Author

Klaus Brockhaus, Feb 11 2009

Keywords

• nonn
• easy

Examples

			a(3) = 34*a(2) - a(1) - 4232 = 34*13225 - 529 - 4232 = 444889.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..600
• Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Crossrefs

Second trisection of A156572.

Cf. A008844, A156164 (decimal expansion of 17+12*sqrt(2)), A156574, A156575.

Programs

Mathematica

LinearRecurrence[{35,-35,1}, {529,13225,444889}, 30] (* G. C. Greubel, Jan 04 2022 *)

PARI

{m=14; v=concat([529, 13225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}

Sage

def a(n): return -529*bool(n==0) + (529/4) + (3/4)*(chebyshev_U(n, 17) - 33*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

Formula

a(n) = 529*(2 + (3 - 2*sqrt(2))*(17 + 12*sqrt(2))^n + (3 + 2*sqrt(2))*(17 - 12*sqrt(2))^n)/8.

a(n) = 529*A008844(n).

G.f.: 529*x*(1 -10*x +x^2)/((1-x)*(1-34*x+x^2)). [corrected by Klaus Brockhaus, Sep 22 2009]

Limit_{n -> infinity} a(n)/a(n-1) = 17+12*sqrt(2).

a(n) = -529*[n=0] + (529/4) + (1587/4)*(ChebyshevU(n, 17) - 33*ChebyshevU(n-1, 17)). - G. C. Greubel, Jan 04 2022

Extensions

Revised by Klaus Brockhaus, Feb 16 2009