A041203 -id:A041203 - cp's OEIS frontend

A041202 Numerators of continued fraction convergents to sqrt(112).

10, 11, 21, 53, 74, 127, 2614, 2741, 5355, 13451, 18806, 32257, 663946, 696203, 1360149, 3416501, 4776650, 8193151, 168639670, 176832821, 345472491, 867777803, 1213250294, 2081028097, 42833812234

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 (0,0,0,0,0,254,0,0,0,0,0,-1).

Cf. A041203.

Numerator[Convergents[Sqrt[112], 30]] (* Vincenzo Librandi, Oct 26 2013 *)

A041202(Nmax=99)=Vec(O(x^Nmax)+(10+11*x+21*x^2+53*x^3+74*x^4+127*x^5+74*x^6-53*x^7+21*x^8-11*x^9+10*x^10-x^11)/(1-254*x^6+x^12)) \\ - M. F. Hasler, Oct 26 2013

a(n) = 254*a(n-6)-a(n-12). G.f.: -(x^11-10*x^10+11*x^9-21*x^8+53*x^7-74*x^6-127*x^5-74*x^4-53*x^3-21*x^2-11*x-10)/((x^6-16*x^3+1)*(x^6+16*x^3+1)). [Colin Barker, Jul 19 2012]

A145542 Numerators in continued fraction expansion of sqrt(3/5).

Original entry on oeis.org

1, 3, 7, 24, 55, 189, 433, 1488, 3409, 11715, 26839, 92232, 211303, 726141, 1663585, 5716896, 13097377, 45009027, 103115431, 354355320, 811826071, 2789833533, 6391493137, 21964312944, 50320119025, 172924670019, 396169459063, 1361433047208, 3119035553479

Offset: 1

Gary W. Adamson, Oct 12 2008

a(n)/A145543(n) tends to sqrt(3/5).

A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Related to the Lehmer sequence U_n(sqrt(R),Q) with parameters R = 6 and Q = -1. See A041023. - Peter Bala, Jun 06 2014

			[a(7), a(8)] = [433, 1488] X^4 * [1, 0] = [433, 1488].
a(5) = 55 = 2*a(4) + a(3) = 2*24 + 7.
G.f. = x + 3*x^2 + 7*x^3 + 24*x^4 + 55*x^5 + 189*x^6 + 433*x^7 + 1488*x^8 + ...

Peter Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Eric Weisstein's World of Mathematics, Lehmer Number

Cf. A001090, A041203, A070997, A145543.

Numerator[Convergents[Sqrt[3/5], 30]] (* gives terms with 0 prepended *) (* Wesley Ivan Hurt, Jun 15 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * (1 + 3*x - x^2) / (1 - 8*x^2 + x^4) + x * O(x^n), n))}; /* Michael Somos, Nov 14 2015 */

Numerators in continued fraction expansion of sqrt(3/5); i.e., of [1, 3, 2, 3, 2, 3, 2, 3, 2, ...].
[a(2*n - 1), a(2*n)] = X^n * [1,0], where X is the 2 X 2 matrix [1,2; 3,7].
Empirical G.f.: x*(1+3*x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 04 2012
From Peter Bala, Jun 06 2014: (Start)
a(2*n + 1) = Product_{k=1..n} (6 + 4*cos^2(k*Pi/(2*n+1))).
a(2*n) = 3*Product_{k=1..n-1} (6 + 4*cos^2(k*Pi/(2*n))).
a(2*n + 1) = A070997(n); a(2*n) = 3*A001090(n). (End)

More terms from Wesley Ivan Hurt, Jun 15 2014

cp's OEIS Frontend

A041202 Numerators of continued fraction convergents to sqrt(112).

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