A161852 Solutions to the simultaneous equations m(n)+1=a(n)^2 and 7*m(n)+1=b(n)^2.

1, 5, 11, 79, 175, 1259, 2789, 20065, 44449, 319781, 708395, 5096431, 11289871, 81223115, 179929541, 1294473409, 2867582785, 20630351429, 45701395019, 328791149455, 728354737519, 5240028039851, 11607974405285, 83511657488161, 184999235747041, 1330946491770725

Offset

1,2

Comments

The equations are equivalent to the Pell equation x(n)^2-7*y(n)^2=9 with x(n)=7*m(n)+4 and y(n)=a(n)*b(n).

x-values in the solution to 7x^2 - 6 = y^2.

Primes in the sequence are 5, 11, 79, 1259, 2789, 44449, 11289871, 20630351429, ...- R. J. Mathar, May 09 2013

Links

Table of n, a(n) for n=1..26.

Index entries for linear recurrences with constant coefficients, signature (0,16,0,-1).

Formula

From Bruno Berselli, Oct 28 2011: (Start)

G.f.: x*(1-x)*(1+6*x+x^2)/(1-16*x^2+x^4).

a(n) = ((7+(-1)^n*t)*(8-3*t)^floor(n/2)+(7-(-1)^n*t)*(8+3*t)^floor(n/2))/14 with t=sqrt(7). (End)

a(n) = 16*a(n-2) - a(n-4) with a(1)=1, a(2)=5, a(3)=11, a(4)=79. - Sture Sjöstedt, Nov 18 2011

Maple

n=0: for a from 1 to 1000000 do b:=sqrt(7*a^2-6):
if (trunc(b)=b) then n:=n+1: m:=a^2-1: x:=7*m+4: y:=a*b:
print(n, a, b, m, x, y): end if: end do:

Prog

(Maxima) makelist(expand(((7+(-1)^n*sqrt(7))*(8-3*sqrt(7))^floor(n/2)+(7-(-1)^n*sqrt(7))*(8+3*sqrt(7))^floor(n/2))/14), n, 1, 26);  \\ Bruno Berselli, Oct 28 2011
(PARI) Vec((1-x)*(1+6*x+x^2)/(1-16*x^2+x^4)+O(x^99)) \\ Charles R Greathouse IV, Oct 28 2011

Crossrefs

Cf. A195878.

Sequence in context: A303549 A192436 A154797 * A319537 A345015 A002359

Adjacent sequences: A161849 A161850 A161851 * A161853 A161854 A161855

Keyword

nonn,easy

Author

Paul Weisenhorn, Jun 20 2009

Extensions

More terms from Bruno Berselli, Oct 28 2011

Status

approved