A216073 The list of the a(n)-values in the common solutions to k+1=b^2 and 6*k+1=a^2.

1, 7, 17, 71, 169, 703, 1673, 6959, 16561, 68887, 163937, 681911, 1622809, 6750223, 16064153, 66820319, 159018721, 661452967, 1574123057, 6547709351, 15582211849, 64815640543, 154247995433, 641608696079, 1526897742481, 6351271320247, 15114729429377, 62871104506391

Offset: 1

Author

Paul Weisenhorn, Sep 01 2012

Keywords

• nonn
• easy

Comments

The equations are equivalent to the Pell equation a^2 - 6*b^2 = -5 with the 2 fundamental solutions (1;1) and (7;3) and the solution (5;2) for the unit form.

The associated b(n) are in A080806.

A181442(n) = (A080806(n) + 1)/2.

A180483(n) = (a(n) + 5)/2.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Crossrefs

Cf. A080806, A180483, A181442.

Programs

Maple

a(1)=1: a(2)=7: a(3)=17: a(4)=71:
for n from 5 to 20 do
  a(n)=10*a(n-2)-a(n-4):
  printf("%9d%20d\n",n,a(n)):
end do:

Mathematica

LinearRecurrence[{0,10,0,-1}, {1,7,17,71}, 50] (* G. C. Greubel, Feb 22 2017 *)

PARI

a(n) = if(n<1, 0, if(n<5, [1,7,17, 71][n], 10*a(n-2)-a(n-4) ) );
/* Joerg Arndt, Sep 03 2012 */

SageMath

def b(n): return (1/2)*(1+(-1)^n)*chebyshev_U(n//2, 5)
def A216073(n): return b(n) +7*b(n-1) +7*b(n-2) +b(n-3)
[A216073(n) for n in (0..50)] # G. C. Greubel, Apr 28 2022

Formula

a(n) = 10*a(n-2) - a(n-4).

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).

G.f.: x*(1+7*x+7*x^2+x^3)/(1-10*x^2+x^4).

a(2*n+1) = ((1+r)*(5+2*r)^n + (1-r)*(5-2*r)^n)/2 where r=sqrt(6) and 0<=n.

a(2*n+2) = ((7+3*r)*(5+2*r)^n + (7-3*r)*(5-2*r)^n)/2 where r=sqrt(6) and 0<=n.

a(n) = -((5-2*r)^(1/4)*((2*r+5)^((-1)^n/4+n/2)*(-1)^n - r*(2*r+5)^((-1)^n/4+n/2)) + (2*r+5)^(1/4)*((5-2*r)^((-1)^n/4+n/2)*(-1)^n + (5-2*r)^((-1)^n/4+n/2)*r))/(2*(5-2*r)^(1/4)*(2*r+5)^(1/4)) with r=sqrt(6) and 1<=n. - Alexander R. Povolotsky, Sep 01 2012

a(n) = b(n) +7*b(n-1) +7*b(n-2) +b(n-3), where b(n) = (1/2)*(1 +(-1)^n)* ChebyshevU(n/2, 5). - G. C. Greubel, Apr 28 2022