A041269 -id:A041269 - cp's OEIS frontend

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

Bruno Berselli, Apr 05 2018

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)

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).

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.

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

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

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

A041268 Numerators of continued fraction convergents to sqrt(147).

Original entry on oeis.org

12, 97, 2340, 18817, 453948, 3650401, 88063572, 708158977, 17083879020, 137379191137, 3314184466308, 26650854921601, 642934702584732, 5170128475599457, 124726018116971700, 1002978273411373057, 24196204579989925068, 194572614913330773601

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 194, 0, -1).

Cf. A041269.

Numerator[Convergents[Sqrt[147], 30]] (* Vincenzo Librandi, Oct 31 2013 *)

G.f.: -(x^3-12*x^2-97*x-12) / ((x^2-14*x+1)*(x^2+14*x+1)). - Colin Barker, Nov 06 2013

More terms from Colin Barker, Nov 06 2013

cp's OEIS Frontend

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

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