A156575 a(n) = 34*a(n-1)-a(n-2)-4232 for n > 2; a(1)=289, a(2)=4225.

289, 4225, 139129, 4721929, 160402225, 5448949489, 185103876169, 6288082836025, 213609712544449, 7256442143671009, 246505423172265625, 8373927945713356009, 284467044731081834449, 9663505592911069011025

Offset

1,1

Comments

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

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).

Formula

a(n) = (1058 + (4947 - 3478*sqrt(2))*(17 + 12*sqrt(2))^n + (4947 + 3478*sqrt(2))*(17 - 12*sqrt(2))^n)/8.

G.f.: x*(289 -5890*x +1369*x^2)/((1-x)*(1-34*x+x^2)).

a(1)=289, a(2)=4225, a(3)=139129, a(n) = 35*a(n-1) - 35*a(n-2) + a(n-3). - Harvey P. Dale, Dec 15 2011

a(n) = -1369*[n=0] + (529/4) + (3/4)*(1649*ChebyshevU(n, 17) - 55857*ChebyshevU(n-1, 17)). - G. C. Greubel, Jan 04 2022

Example

a(4) = 34*a(3) -a(2) -4232 = 34*139129 -4225 -4232 = 4721929.

Mathematica

RecurrenceTable[{a[1]==289, a[2]==4225, a[n]==34a[n-1]-a[n-2]-4232}, a, {n, 20}] (* or *) LinearRecurrence[{35, -35, 1}, {289, 4225, 139129}, 20] (* Harvey P. Dale, Dec 15 2011 *)

Prog

(PARI) {m=14; v=concat([289, 4225], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}
(Sage)
def a(n): return -1369*bool(n==0) + (529/4) + (3/4)*(1649*chebyshev_U(n, 17) - 55857*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

Crossrefs

First trisection of A156572.

Cf. A156164 (decimal expansion of 17+12*sqrt(2)), A156573, A156574.

Sequence in context: A112077 A152934 A332737 * A296404 A156161 A114762

Adjacent sequences: A156572 A156573 A156574 * A156576 A156577 A156578

Keyword

nonn,easy

Author

Klaus Brockhaus, Feb 11 2009

Extensions

Revised by Klaus Brockhaus, Feb 16 2009

G.f. corrected by Klaus Brockhaus, Sep 22 2009

Status

approved