A042934 Numerators of continued fraction convergents to sqrt(999).

31, 32, 63, 95, 158, 885, 5468, 6353, 37233, 80819, 441328, 522147, 3574210, 18393197, 21967407, 40360604, 62328011, 102688615, 6429022141, 6531710756, 12960732897, 19492443653, 32453176550

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 205377230, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Crossrefs

Cf. A042935 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042936 (m=1000).

Programs

Mathematica

Numerator[Convergents[Sqrt[999], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

PARI

A42934=contfracpnqn(c=contfrac(sqrt(999)), #c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n) = A42934[n+1]! For more terms, use:
A042934(n)={n<#A42934 || A42934_upto(n+10); A42934[n+1]}
{A42934_upto(N,A=Vec(A42934,N))=for(n=#A42934+1,N, A[n]=205377230*A[n-18]-A[n-36]); A42934=A} \\ M. F. Hasler, Nov 01 2019

Formula

a(n) = 205377230*a(n-18) - a(n-36). - Wesley Ivan Hurt, May 28 2021