A233593 -id:A233593 - cp's OEIS frontend

A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

2, 3, 30, 34, 111, 235, 3775, 5052, 7352, 9091, 34991, 35530, 53424, 57290, 66023, 1409179, 1519111, 1725990, 1812396, 4370835, 4507156, 4655396, 44257080, 234755198, 261519946, 264374278, 273487975

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on Blazys' expansions, see A233582.

Sqrt(7) is the first square root of a natural number with an a-periodic Blazys' expansion (see A233592 and A233593).

Stanislav Sykora, Table of n, a(n) for n = 1..1000

Cf. Blazys' expansions: A233582 (Pi), A233583 (e), A233584 (sqrt(e)), A233585 (1/gamma), A233585 (2*gamma) and Blazys' continued fractions: A233588, A233589, A233590, A233591.

Cf. A010465, A010121.

BlazysExpansion[n_, mx_] := Block[{k = 1, x = n, lmt = mx + 1, s, lst = {}}, While[k < lmt, s = Floor[x]; x = 1/(x/s - 1); AppendTo[lst, s]; k++]; lst]; BlazysExpansion[Sqrt@7, 32] (* Robert G. Wilson v, May 22 2014 *)

bx(x, nmax)={local(c, v, k); \\ Blazys expansion function
v = vector(nmax); c = x; for(k=1, nmax, v[k] = floor(c); c = v[k]/(c-v[k]); ); return (v); }
bx(sqrt(7), 1000) \\ Execution; use very high real precision

sqrt(7) = 2+2/(3+3/(30+30/(34+34/(111+...)))).

A233592 Positive integers k such that the continued fraction expansion sqrt(k) = c(1) + c(1)/(c(2) + c(2)/(c(3) + c(3)/...)) is periodic.

Original entry on oeis.org

2, 3, 5, 6, 8, 10, 11, 12, 15, 17, 18, 20, 24, 26, 27, 30, 35, 37, 38, 39, 40, 42, 44, 45, 48, 50, 51, 56, 63, 65, 66, 68, 72, 80, 82, 83, 84, 87, 90, 99, 101, 102, 104, 105, 108, 110, 120, 122, 123, 132, 143, 145

Offset: 1

Stanislav Sykora, Jan 06 2014

nonn

For more details on this type of expansion, see A233582.
The cases with aperiodic expansions are listed in A233593.
All the listed cases become periodic after just two leading terms (it is a conjecture that this behavior is general); the validity of their expansions was explicitly tested.

			Blazys's expansion of sqrt(2) is {1, 2, 4, 4, 4, 4, 4, ...}, i.e., it has a periodic termination. Consequently, 2 is a term of this sequence.

Stanislav Sykora, Table of n, a(n) for n = 1..200
Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013.
Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki.

Cf. A233593.

Cf. Blazys's expansions: A233582, A233584, A233585, A233586, A233587.

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cp's OEIS Frontend

A233587 Coefficients of the generalized continued fraction expansion sqrt(7) = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).

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