A041053 -id:A041053 - cp's OEIS frontend

A125652 Numbers m such that m^2=A125650(k) for some k (belonging A125651).

1, 3, 9, 105, 306, 3567, 10395, 121173, 353124, 4116315, 11995821, 139833537, 407504790, 4750223943, 13843167039, 161367780525, 470260174536, 5481754313907, 15975002767185, 186218278892313, 542679833909754

Offset: 1

Alexander Adamchuk, Nov 29 2006, corrected Dec 14 2006

nonn

Indices k such that a(n)^2=A125650(k) are listed in A125651.

3 divides a(n) for n>1. For n>1 a(n) = 3*A041053(2n-3), where A041053(n) = {1, 1, 2, 3, 32, 35, 67, 102, 1087, 1189, 2276, 3465, ...} Denominators of continued fraction convergents to sqrt(32). - Alexander Adamchuk, Jan 19 2007

			a(2)=3 because A125650(3)=9=3^2; a(3)=9 because A125650(24)=81=9^2.

Cf. A125650, A125651, A106328.

Cf. A041053.

a(2k)=A106328(2k); for k>0, a(2k+1)=A106328(2k+1)/2.
a(n) = sqrt(A125650(A125651(n))).
a(n) = 3*A041053(2n-3) for n>1. - Alexander Adamchuk, Jan 19 2007

Edited by Max Alekseyev, Jan 11 2007

A010130 Continued fraction for sqrt(32).

Original entry on oeis.org

5, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10, 1, 1, 1, 10

Offset: 0

N. J. A. Sloane

			5.65685424949238019520675489... = 5 + 1/(1 + 1/(1 + 1/(1 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 04 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010487 (decimal expansion).

Cf. A041052/A041053 (convergents), A248259 (Egyptian fraction).

ContinuedFraction[Sqrt[32],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{5},100,{10,1,1,1}] (* Harvey P. Dale, Aug 20 2014 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(32)); for (n=0, 20000, write("b010130.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 1, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 9/4^s). (End)
From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: (5 + x + x^2 + x^3 + 5*x^4)/((1 - x)*(1 + x + x^2 + x^3)).
a(n) = a(n-4), n > 4. (End)

A041052 Numerators of continued fraction convergents to sqrt(32).

Original entry on oeis.org

5, 6, 11, 17, 181, 198, 379, 577, 6149, 6726, 12875, 19601, 208885, 228486, 437371, 665857, 7095941, 7761798, 14857739, 22619537, 241053109, 263672646, 504725755, 768398401, 8188709765, 8957108166, 17145817931, 26102926097, 278175078901, 304278004998

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 (0,0,0,34,0,0,0,-1).

Cf. A010487, A041053.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[32], n]]], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[32], 30]] (* Vincenzo Librandi, Oct 23 2013 *)

G.f.: (5+6*x+11*x^2+17*x^3+11*x^4-6*x^5+5*x^6-x^7)/(1-34*x^4+x^8). - Colin Barker, Jan 03 2012

cp's OEIS Frontend

A125652 Numbers m such that m^2=A125650(k) for some k (belonging A125651).

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A010130 Continued fraction for sqrt(32).

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A041052 Numerators of continued fraction convergents to sqrt(32).

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