A010470 -id:A010470 - cp's OEIS frontend

A248242 Egyptian fraction representation of sqrt(13) (A010470) using a greedy function.

3, 2, 10, 181, 37860, 2063394882, 20133724366323386460, 895769948382354175062611801976979893814, 1095684829796116398764171865109547325653507924058299202087102696023776712107256

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter >
0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 13]]

A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

Original entry on oeis.org

2, 3, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8, 8, 3, 6, 7, 5, 7

Offset: 1

Alonso del Arte, Mar 17 2012

The number x given by the infinitely nested radical for n = 3 is such that x^2 = x + 3, bearing some similarity to the golden ratio phi with its property that phi^2 = phi + 1. Also, 3/x = x - 1.
The mentioned polynomial x^2 - x - 3 has the present number as positive root, and the negative one is -A223139. - Wolfdieter Lang, Aug 29 2022
It is the spectral radius of the bull-graph (see Seeger and Sossa, 2023). - Stefano Spezia, Sep 19 2023
c^n = A006130(n) + A006130(n-1) * d, where c = (1 + sqrt(13))/2 and d = (-1 + sqrt(13))/2. - Gary W. Adamson, Nov 25 2023
c^n = A052533(n) + A006130(n-1)*c, with A006130(-1) = 0. This is also valid for powers of 1/c = A356033, with A052533 and A006130 given there in terms of S-Chebyshev polynomials (A049310), used for negative indices. - Wolfdieter Lang, Nov 26 2023

			2.30277563773...

Alberto Seeger and David Sossa, Infinite families of connected graphs with equal spectral radius, Australas. J. Combin. 87 (2) (2023), 258-276. See p. 274.
Index entries for algebraic numbers, degree 2.

Cf. A157532 (continued fraction), A001622, A010470, A085550, A098316, A188943, A223139.

Cf. A006130, A049310, A052533, A209927.

Digits:=140:
evalf((sqrt(13)+1)/2);  # Alois P. Heinz, Sep 19 2023

RealDigits[(1 + Sqrt[13])/2, 10, 130][[1]]
RealDigits[ Fold[ Sqrt[#1 + #2] &, 0, Table[3, {n, 168}]], 10, 111][[1]] (* Robert G. Wilson v, Oct 02 2018 *)

(sqrt(13)+1)/2 \\ Altug Alkan, Oct 03 2018

Closed form: (sqrt(13) + 1)/2 = A098316-1 = A085550+2 = 3*(A188943-1).

A041018 Numerators of continued fraction convergents to sqrt(13).

Original entry on oeis.org

3, 4, 7, 11, 18, 119, 137, 256, 393, 649, 4287, 4936, 9223, 14159, 23382, 154451, 177833, 332284, 510117, 842401, 5564523, 6406924, 11971447, 18378371, 30349818, 200477279, 230827097, 431304376, 662131473

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 (0,0,0,0,36,0,0,0,0,1).

Cf. A010122 (continued fraction for sqrt(13)).

Cf. A010470, A041019 (denominators), A041046, A041090, A041150, A041226, A041318, A041426 and A041550.

a[0]:=3: a[-1]:=1: b(0):=6: b(1):=1; b(2):=1: b(3):=1: b(4):=1:
for n from 1 to 100 do  k:=n mod 5:
   a[n]:=b(k)*a[n-1]+a[n-2]:
   printf("%12d", a[n]):
end do: # Paul Weisenhorn, Aug 17 2018

Numerator[Convergents[Sqrt[13], 30]] (* Vincenzo Librandi, Oct 27 2013 *)
CoefficientList[Series[(3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10),{x,0,50}],x] (* Stefano Spezia, Aug 31 2018 *)

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A006497(3*n+1),
a(5*n+1) = (A006497(3*n+2)-A006497(3*n+1))/2,
a(5*n+2) = (A006497(3*n+2)+A006497(3*n+1))/2,
a(5*n+3) = A006497(3*n+2),
a(5*n+4) = A006497(3*n+3)/2.
(End)
G.f.: (3 + 4*x + 7*x^2 + 11*x^3 + 18*x^4 + 11*x^5 - 7*x^6 + 4*x^7 - 3*x^8 + x^9)/(1 - 36*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013
a(n) = A010122(n)*a(n-1)+a(n-2) with a(0)=3, a(-1)=1. - Paul Weisenhorn, Aug 19 2018

A010122 Continued fraction for sqrt(13).

Original entry on oeis.org

3, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6

Offset: 0

N. J. A. Sloane

Eventual period is (1, 1, 1, 1, 6). - Zak Seidov, Mar 05 2011

			3.605551275463989293119221267... = 3 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 02 2009

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 96 at p. 264.
Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 428.

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 (0,0,0,0,1).

Cf. A010470 (decimal expansion).

ContinuedFraction[Sqrt[13],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)

{ allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(13)); for (n=0, 20000, write("b010122.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(5^e) = 6, and a(p^e) = 1 for p != 5.
Dirichlet g.f.: zeta(s) * (1 + 1/5^(s-1)). (End)
G.f.: (3 + x + x^2 + x^3 + x^4 + 3*x^5)/(1 - x^5). - Stefano Spezia, Aug 17 2024

A295330 Decimal expansion of sqrt(13)/2.

Original entry on oeis.org

1, 8, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8

Offset: 1

Wolfdieter Lang, Nov 20 2017

In a regular hexagon inscribed in a circle of radius R the largest distance between any vertex and a midpoint of a side, after division of R, is sqrt(13)/2. The two smaller distance ratios are sqrt(7)/2 = A242703 and 1/2.
The regular period 6 continued fraction of sqrt(13)/2 is [1; 1, 4, 14, 4, 1, 2], and the convergents are given in A295331/A295332.
Essentially the same as A223139, A209927, A098316 and A085550. - R. J. Mathar, Nov 23 2017

			1.8027756377319946465596106337352479731256482869226231063552265281135834741465...

Index entries for algebraic numbers, degree 2.

Cf. A010470, A242703, A295331/A295332.

RealDigits[Sqrt@13/2, 10, 111][[1]] (* Robert G. Wilson v, Nov 20 2017 *)

sqrt(13)/2 \\ Michel Marcus, Nov 21 2017

A085550 Decimal expansion of (sqrt(13)-3)/2.

Original entry on oeis.org

3, 0, 2, 7, 7, 5, 6, 3, 7, 7, 3, 1, 9, 9, 4, 6, 4, 6, 5, 5, 9, 6, 1, 0, 6, 3, 3, 7, 3, 5, 2, 4, 7, 9, 7, 3, 1, 2, 5, 6, 4, 8, 2, 8, 6, 9, 2, 2, 6, 2, 3, 1, 0, 6, 3, 5, 5, 2, 2, 6, 5, 2, 8, 1, 1, 3, 5, 8, 3, 4, 7, 4, 1, 4, 6, 5, 0, 5, 2, 2, 2, 6, 0, 2, 3, 0, 9, 5, 4, 1, 0, 0, 9, 2, 4, 5, 3, 5, 8, 8, 3, 6, 7, 5, 7

Offset: 0

Cino Hilliard, Jul 04 2003

			0.30277563773199464655961...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Cino Hilliard, + N = Reciprocal Recursion [broken link]
Index entries for algebraic numbers, degree 2.

Cf. A010470 (sqrt(13)).

First[RealDigits[(Sqrt[13] - 3)/2, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

(sqrt(13)-3)/2 \\ Michel Marcus, Jan 16 2020

Equals Sum_{k>=1} (-1)^(k-1)/(A006190(k)*A006190(k+1)). - Vladimir Shevelev, Feb 23 2013

Tan(arctan(c) + arctan(c^3)) = 1/3. - Gary W. Adamson, Apr 02 2024

A171983 Beatty sequence for sqrt(13).

Original entry on oeis.org

3, 7, 10, 14, 18, 21, 25, 28, 32, 36, 39, 43, 46, 50, 54, 57, 61, 64, 68, 72, 75, 79, 82, 86, 90, 93, 97, 100, 104, 108, 111, 115, 118, 122, 126, 129, 133, 137, 140, 144, 147, 151, 155, 158, 162, 165, 169, 173, 176, 180, 183, 187, 191, 194, 198, 201, 205

Offset: 1

Vincenzo Librandi, Jan 21 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A010470.

[Floor(n*Sqrt(13)): n in [1..80]]; // Vincenzo Librandi, Aug 01 2013

f[n_]: = Floor[n Sqrt[13]]; Array[f, 80, 1] (* Vincenzo Librandi, Aug 01 2013 *)

a(n) = floor(n*sqrt(13)). - Jon E. Schoenfield, Jun 18 2010

A176019 Decimal expansion of (3+sqrt(13))/6.

Original entry on oeis.org

1, 1, 0, 0, 9, 2, 5, 2, 1, 2, 5, 7, 7, 3, 3, 1, 5, 4, 8, 8, 5, 3, 2, 0, 3, 5, 4, 4, 5, 7, 8, 4, 1, 5, 9, 9, 1, 0, 4, 1, 8, 8, 2, 7, 6, 2, 3, 0, 7, 5, 4, 1, 0, 3, 5, 4, 5, 1, 7, 4, 2, 1, 7, 6, 0, 3, 7, 8, 6, 1, 1, 5, 8, 0, 4, 8, 8, 3, 5, 0, 7, 4, 2, 0, 0, 7, 6, 9, 8, 4, 7, 0, 0, 3, 0, 8, 1, 7, 8, 6, 2, 7, 8, 9, 1

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(13))/6 is A010690.

Minimal polynomial: 9*x^2 - 9*x - 1. - Amiram Eldar, Dec 03 2024

			1.10092521257733154885320354457841599104188276230754...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010470 (decimal expansion of sqrt(13)), A010690 (repeat 1, 9), A092936.

RealDigits[(3 + Sqrt[13])/6, 10, 120][[1]] (* Amiram Eldar, Dec 03 2024 *)

Equals Product_{k>=2} (1 + (-1)^k/A092936(k)). - Amiram Eldar, Dec 03 2024

A176322 Decimal expansion of sqrt(1365).

Original entry on oeis.org

3, 6, 9, 4, 5, 9, 0, 6, 4, 0, 3, 8, 2, 2, 3, 3, 1, 9, 9, 1, 6, 3, 5, 6, 1, 1, 5, 9, 9, 7, 9, 7, 1, 4, 2, 0, 5, 6, 5, 1, 5, 9, 7, 4, 2, 2, 3, 1, 7, 1, 5, 5, 8, 7, 4, 6, 3, 9, 0, 6, 4, 1, 8, 8, 8, 4, 9, 2, 6, 6, 0, 6, 6, 7, 2, 5, 1, 5, 9, 4, 7, 0, 2, 0, 9, 0, 3, 0, 4, 0, 6, 0, 5, 4, 4, 6, 5, 2, 9, 0, 3, 8, 1, 2, 5

Offset: 2

Klaus Brockhaus, Apr 15 2010

Continued fraction expansion of sqrt(1365) is (repeat 1, 17, 2, 17, 1, 72) preceded by 36.

			36.94590640382233199163...

G. C. Greubel, Table of n, a(n) for n = 2..1001

Cf. A002194 (sqrt(3)), A002163 (sqrt(5)), A010465 (sqrt(7)), A010470 (sqrt(13)).

Cf. A176321 ((35+sqrt(1365))/14).

SetDefaultRealField(RealField(120)); Sqrt(1365); // G. C. Greubel, Nov 26 2019

evalf( sqrt(1365), 120); # G. C. Greubel, Nov 26 2019

RealDigits[Sqrt[1365],10,120][[1]] (* Harvey P. Dale, Oct 09 2017 *)

default(realprecision, 120); sqrt(1365) \\ G. C. Greubel, Nov 26 2019

numerical_approx(sqrt(1365), digits=120) # G. C. Greubel, Nov 26 2019

Equals sqrt(3)*sqrt(5)*sqrt(7)*sqrt(13).

A194392 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(13) and < > denotes fractional part.

Original entry on oeis.org

1, 3, 29, 31, 33, 34, 35, 36, 37, 39, 41, 67, 69, 71, 72, 73, 74, 75, 77, 79, 105, 107, 143, 145, 181, 183, 209, 211, 213, 214, 215, 216, 217, 219, 221, 247, 249, 251, 252, 253, 254, 255, 257, 259, 285, 287, 323, 325, 361, 363, 389, 391, 393, 394, 395

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010470, A194368, A194393, A194394.

r = Sqrt[13]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 400}];
Flatten[Position[t1, 1]]       (* A194392 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]       (* A194393 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]       (* A194394 *)

cp's OEIS Frontend

A248242 Egyptian fraction representation of sqrt(13) (A010470) using a greedy function.

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A209927 Decimal expansion of sqrt(3 + sqrt(3 + sqrt(3 + sqrt(3 + ... )))).

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Maple

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A041018 Numerators of continued fraction convergents to sqrt(13).

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A010122 Continued fraction for sqrt(13).

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A295330 Decimal expansion of sqrt(13)/2.

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A085550 Decimal expansion of (sqrt(13)-3)/2.

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A171983 Beatty sequence for sqrt(13).

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Magma

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A176019 Decimal expansion of (3+sqrt(13))/6.

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