A302332 a(0)=1, a(1)=193; for n>1, a(n) = 194*a(n-1) - a(n-2).

1, 193, 37441, 7263361, 1409054593, 273349327681, 53028360515521, 10287228590683393, 1995669318232062721, 387149560508429484481, 75105019069317087926593, 14569986549887006628274561, 2826502285659009968797338241, 548326873431298046940055344193, 106372586943386162096401939435201

Offset: 0

Author

Bruno Berselli, Apr 05 2018

Keywords

• nonn
• easy
• changed

Comments

Let G and H be sequences of the form G(i) = 4*G(i-1) - G(i-2) and H(j) = 14*H(j-1) - H(j-2) for arbitrary integers i, j and without regard to initial values of G or H, then a(n) = (G(i) + G(i+8*n+4))/(14*G(i+4*n+2)) = (H(j) + H(j+4*n+2))/(14*H(j+2*n+1)) with the exception of G(i+4*n+2) or H(j+2*n+1) != 0. - Klaus Purath, Aug 31 2020

From Klaus Purath, Aug 20 2025: (Start)

Solutions to the Pell equation (7*a(n))^2 - 3*(4*b(n))^2 = 1. The corresponding b(n) are given by A084232.

For any two consecutive terms (x,y), x^2 - 194*x*y + y^2 + 192 = 0. By analogy to this, for three consecutive terms (x, y, z), y^2 - x*z + 192 = 0. (End)

Links

• Colin Barker, Table of n, a(n) for n = 0..400
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (194,-1).

Crossrefs

Seventh row of the array A188646.

First bisection of A041269, A042127.

Similar sequences of the type cosh((2*n+1)*arccosh(k))/k are listed in A302329.

Cf. A084232.

Programs

Mathematica

LinearRecurrence[{194, -1}, {1, 193}, 20]

PARI

x='x+O('x^99); Vec((1-x)/(1-194*x+x^2)) \\ Altug Alkan, Apr 06 2018

Formula

G.f.: (1 - x)/(1 - 194*x + x^2).

a(n) = a(-1-n).

a(n) = cosh((2*n + 1)*arccosh(7))/7.

a(n) = ((7 + 4*sqrt(3))^(2*n + 1) + 1/(7 + 4*sqrt(3))^(2*n + 1))/14.

a(n) = (a(n-1)^2 + 192)/a(n-2). - Klaus Purath, Aug 31 2020

a(n) = (1/7)*T(2*n+1, 7), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Jul 08 2022