cp's OEIS Frontend

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

%I A216073 #26 Apr 28 2022 03:25:02
%S A216073 1,7,17,71,169,703,1673,6959,16561,68887,163937,681911,1622809,
%T A216073 6750223,16064153,66820319,159018721,661452967,1574123057,6547709351,
%U A216073 15582211849,64815640543,154247995433,641608696079,1526897742481,6351271320247,15114729429377,62871104506391
%N A216073 The list of the a(n)-values in the common solutions to  k+1=b^2 and 6*k+1=a^2.
%C A216073 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.
%C A216073 The associated b(n) are in A080806.
%C A216073 A181442(n) = (A080806(n) + 1)/2.
%C A216073 A180483(n) = (a(n) + 5)/2.
%H A216073 G. C. Greubel, <a href="/A216073/b216073.txt">Table of n, a(n) for n = 1..1000</a>
%H A216073 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).
%F A216073 a(n) = 10*a(n-2) - a(n-4).
%F A216073 a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - a(n-4) + a(n-5).
%F A216073 G.f.: x*(1+7*x+7*x^2+x^3)/(1-10*x^2+x^4).
%F A216073 a(2*n+1) = ((1+r)*(5+2*r)^n + (1-r)*(5-2*r)^n)/2  where r=sqrt(6) and 0<=n.
%F A216073 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.
%F A216073 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
%F A216073 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
%p A216073 a(1)=1: a(2)=7: a(3)=17: a(4)=71:
%p A216073 for n from 5 to 20 do
%p A216073   a(n)=10*a(n-2)-a(n-4):
%p A216073   printf("%9d%20d\n",n,a(n)):
%p A216073 end do:
%t A216073 LinearRecurrence[{0,10,0,-1}, {1,7,17,71}, 50] (* _G. C. Greubel_, Feb 22 2017 *)
%o A216073 (PARI)
%o A216073 a(n) = if(n<1, 0, if(n<5, [1,7,17, 71][n], 10*a(n-2)-a(n-4) ) );
%o A216073 /* _Joerg Arndt_, Sep 03 2012 */
%o A216073 (SageMath)
%o A216073 def b(n): return (1/2)*(1+(-1)^n)*chebyshev_U(n//2, 5)
%o A216073 def A216073(n): return b(n) +7*b(n-1) +7*b(n-2) +b(n-3)
%o A216073 [A216073(n) for n in (0..50)] # _G. C. Greubel_, Apr 28 2022
%Y A216073 Cf. A080806, A180483, A181442.
%K A216073 nonn,easy
%O A216073 1,2
%A A216073 _Paul Weisenhorn_, Sep 01 2012