A215503 a(n) = (u+1)^n + (-s-1)^n + (t+1)^n + (-1)^n + (-t+1)^n + (s-1)^n + (-u+1)^n where s = sqrt(2), t = sqrt(2-s), u = sqrt(2+s).

7, 1, 19, 13, 111, 121, 763, 1093, 5575, 9697, 42099, 84173, 324591, 717081, 2538331, 6023173, 20049671, 50079553, 159514963, 413387789, 1275778031, 3394968121, 10242581819, 27780675397, 82461727687, 226743641121, 665232392883, 1847286687181, 5374409263215

Offset: 0

Author

Peter Luschny, Aug 13 2012

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wikipedia, Nested radical
• Index entries for linear recurrences with constant coefficients, signature (1,9,-5,-17,1,5,-1).

Crossrefs

Cf. A051927, A215500, A215502.

Programs

Magma

I:=[7,1,19,13,111,121,763]; [n le 7 select I[n] else Self(n-1) + 9*Self(n-2) -5*Self(n-3) -17*Self(n-4) + Self(n-5) + 5*Self(n-6) - Self(n-7): n in [1..30]]; // G. C. Greubel, Apr 23 2018

Maple

A215503 := n -> (sqrt(2+sqrt(2))+1)^n+(-sqrt(2)-1)^n+(sqrt(2-sqrt(2))+1)^n+(-1)^n+(-sqrt(2-sqrt(2))+1)^n+(sqrt(2)-1)^n+(-sqrt(2+sqrt(2))+1)^n;
seq(simplify(A215503(i)),i=0..28);

Mathematica

LinearRecurrence[{1,9,-5,-17,1,5,-1}, {7,1,19,13,111,121,763}, 50] (* G. C. Greubel, Apr 23 2018 *)

PARI

x='x+O('x^30); Vec((7-6*x-45*x^2+20*x^3+51*x^4 -2*x^5-5*x^6)/( (1+x)*(1+2*x-x^2)*(1 -4*x +2*x^2 +4*x^3-x^4))) \\ G. C. Greubel, Apr 23 2018

PARI

polsym(polrecip(((1+x)*(1+2*x-x^2)*(1-4*x+2*x^2+4*x^3-x^4))),33) \\ Joerg Arndt, Apr 29 2018

Sage

def A215503(n) :
    return (sqrt(2+sqrt(2))+1)^n+(-sqrt(2)-1)^n+(sqrt(2-sqrt(2))+1)^n+(-1)^n+(-sqrt(2-sqrt(2))+1)^n+(sqrt(2)-1)^n+(-sqrt(2+sqrt(2))+1)^n
[A215503(i).round() for i in (0..28)]

Formula

a(n) = (sqrt(2 + sqrt(2)) + 1)^n + (-sqrt(2) - 1)^n + (sqrt(2 - sqrt(2)) + 1)^n + (-1)^n + (-sqrt(2 - sqrt(2)) + 1)^n + (sqrt(2) - 1)^n + (-sqrt(2 + sqrt(2)) + 1)^n. (Initial name of sequence).

a(n) = a(n-1) + 9*a(n-2) - 5*a(n-3) - 17*a(n-4) + a(n-5) + 5*a(n-6) - a(n-7).

G.f.: (7-6*x-45*x^2+20*x^3+51*x^4-2*x^5-5*x^6)/((1+x)*(1+2*x-x^2)*(1 -4*x +2*x^2+4*x^3-x^4)). - Colin Barker, Aug 20 2012

Extensions

New name from Altug Alkan, Apr 27 2018