A042860 -id:A042860 - cp's OEIS frontend

A040930 Continued fraction for sqrt(962).

31, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62, 62

Offset: 0

N. J. A. Sloane

			31 + 1/(62 + 1/(62 + 1/(62 + 1/(62 + ...)))) = sqrt(962).

Cf. A042860/A042861 (convergents).

Continued fraction for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040000 (contfrac(sqrt(2)) = (1,2,2,...)), A040002, A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870 (contfrac(sqrt(901)) = (30,60,60,...)).

with(numtheory): Digits := 300: convert(evalf(sqrt(962)),confrac);

PadRight[{31},100,62] (* Harvey P. Dale, Sep 18 2012 *)

G.f.: 31*(1+x)/(1-x). - Colin Barker, Aug 11 2012
From Elmo R. Oliveira, Feb 16 2024: (Start)
a(n) = 62 for n >= 1.
E.g.f.: 62*exp(x) - 31.
a(n) = 31*A040000(n). (End)

A042861 Denominators of continued fraction convergents to sqrt(962).

Original entry on oeis.org

1, 62, 3845, 238452, 14787869, 917086330, 56874140329, 3527113786728, 218737928917465, 13565278706669558, 841266017742430061, 52172058378737333340, 3235508885499457097141, 200653722959345077356082, 12443766332364894253174225, 771714166329582788774158032

Offset: 0

N. J. A. Sloane

From Michael A. Allen, Jan 22 2024: (Start)
Also called the 62-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 62 kinds of squares available. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (62,1).

Cf. A042860, A040930.

Row n=62 of A073133, A172236 and A352361 and column k=62 of A157103.

Denominator[Convergents[Sqrt[962],20]] (* Harvey P. Dale, Jun 15 2013 *)

a(n) = F(n, 62), the n-th Fibonacci polynomial evaluated at x=62. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 62*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=62.
G.f.: 1/(1 - 62*x - x^2). (End)

Additional term from Colin Barker, Dec 25 2013

cp's OEIS Frontend

A040930 Continued fraction for sqrt(962).

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