A156574 a(n) = 34*a(n-1) - a(n-2) - 4232 for n > 2; a(1)=1369, a(2)=42025.

1369, 42025, 1423249, 48344209, 1642275625, 55789022809, 1895184495649, 64380483825025, 2187041265550969, 74295022544903689, 2523843725261170225, 85736391636334879729, 2912513471910124736329, 98939721653307906151225

Offset: 1

Author

Klaus Brockhaus, Feb 11 2009

Keywords

• nonn
• easy

Comments

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

Examples

			a(3) = 34*a(2) - a(1) - 4232 = 34*42025 - 1369 - 4232 = 1423249.
		

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

Third trisection of A156572.

Cf. A156164 (decimal expansion of 17+12*sqrt(2)), A156573, A156575.

Programs

Mathematica

LinearRecurrence[{35,-35,1}, {1369, 42025, 1423249}, 30] (* G. C. Greubel, Jan 04 2022 *)

PARI

{m=14; v=concat([1369, 42025], vector(m-2)); for(n=3, m, v[n]=34*v[n-1]-v[n-2]-4232); v}

Sage

def a(n): return -289*bool(n==0) + (529/4) + (3/4)*(209*chebyshev_U(n, 17) - 5457*chebyshev_U(n-1, 17))
[a(n) for n in (1..30)] # G. C. Greubel, Jan 04 2022

Formula

a(n) = (1058 + (627 - 238*sqrt(2))*(17 + 12*sqrt(2))^n + (627 + 238*sqrt(2))*(17 - 12*sqrt(2))^n)/8.

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

a(n) = -289*[n=0] + (529/4) + (3/4)*(209*ChebyshevU(n, 17) - 5457*ChebyshevU(n - 1, 17)). - G. C. Greubel, Jan 04 2022

Extensions

Revised by Klaus Brockhaus, Feb 16 2009

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