A215502 -id:A215502 - cp's OEIS frontend

A215500 a(n) = ((sqrt(5) + 3)^n + (-sqrt(5) -1)^n + (-sqrt(5) + 3)^n + (sqrt(5) - 1)^n) / 2^n.

4, 2, 10, 14, 54, 112, 340, 814, 2254, 5702, 15250, 39404, 104004, 270922, 711490, 1859134, 4873054, 12748472, 33391060, 87394454, 228841254, 599050102, 1568437210, 4106054164, 10750060804, 28143585362, 73681573690, 192899714414, 505019869254, 1322156172352

Offset: 0

Peter Luschny, Aug 13 2012

Inverse binomial transform is (-1)^n * a(n). - Michael Somos, Jun 02 2014

			G.f. = 4 + 2*x + 10*x^2 + 14*x^3 + 54*x^4 + 112*x^5 + 340*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-4,1).

Cf. A051927, A215502, A215503.

Cf. A124697.

I:=[4,2,10,14]; [n le 4 select I[n] else 2*Self(n-1) + 3*Self(n-2) - 4*Self(n-3) + Self(n-4): n in [1..30]]; // G. C. Greubel, Apr 23 2018

A215500 := x -> ((sqrt(5)+3)^x+(-sqrt(5)-1)^x+(-sqrt(5)+3)^x+(sqrt(5)-1)^x)/2^x;
seq(simplify(A215500(i)),i=0..29);

a[n_] := ((Sqrt[5] + 3)^n + (-Sqrt[5] - 1)^n + (-Sqrt[5] + 3)^n + (Sqrt[5] - 1)^n)/2^n; Table[a[n] // Simplify, {n, 0, 29}] (* Jean-François Alcover, Jul 02 2013 *)
LinearRecurrence[{2,3,-4,1}, {4, 2, 10, 14}, 50] (* G. C. Greubel, Apr 23 2018 *)

{a(n) = polsym( (1 + (-1)^(n>0)*x - x^2) * (1 - 3*x + x^2), abs(n))[1 + abs(n)]}; /* Michael Somos, Jun 02 2014 */

def A215500(x) :
    return ((sqrt(5)+3)^x+(-sqrt(5)-1)^x+(-sqrt(5)+3)^x+(sqrt(5)-1)^x)/2^x
[A215500(i).round() for i in (0..29)]

G.f.: 2*(2-x)*(1+x)*(1-2*x)/((1-3*x+x^2)*(1+x-x^2)). - Colin Barker, Aug 19 2012
a(n) = 2*a(n-1)+3*a(n-2)-4*a(n-3)+a(n-4). - Colin Barker, Aug 20 2012
a(-n) = A124697(n) if n>0. - Michael Somos, Jun 02 2014

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).

Original entry on oeis.org

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

Peter Luschny, Aug 13 2012

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).

Cf. A051927, A215500, A215502.

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

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);

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

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

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

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)]

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

New name from Altug Alkan, Apr 27 2018

cp's OEIS Frontend

A215500 a(n) = ((sqrt(5) + 3)^n + (-sqrt(5) -1)^n + (-sqrt(5) + 3)^n + (sqrt(5) - 1)^n) / 2^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

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).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions