A010474 -id:A010474 - cp's OEIS frontend

A248246 Egyptian fraction representation of sqrt(18) (A010474) using a greedy function.

4, 5, 24, 1027, 3219387, 102715635003972, 28595657331015533671660837004, 1215572475769570408109978391934299568566509985905302163092, 2006120697781748129559395265597556700767017998650179835542888817906954377068504244660639847221485156172682330027607

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[ 18]]

A378393 Decimal expansion of the midradius of a deltoidal icositetrahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 30 2024

The deltoidal icositetrahedron is the dual polyhedron of the (small) rhombicuboctahedron.

			1.5606601717798212866012665431572735589272539065327...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

Cf. A378390 (surface area), A378391 (volume), A378392 (inradius), A378394 (dihedral angle).
Cf. A285871 (midradius of a (small) rhombicuboctahedron with unit edge).
Cf. A010474, A093577.

First[RealDigits[(2 + Sqrt[18])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalIcositetrahedron", "Midradius"], 10, 100]]

Equals (2 + 3*sqrt(2))/4 = (2 + A010474)/4.

A294020 Total number of elements after n-th stage of a hybrid cellular automaton formed by D-toothpicks and toothpicks (see Comments lines for precise definition).

Original entry on oeis.org

0, 1, 5, 9, 15, 23, 27, 41, 65, 81, 103, 111, 115, 129, 153, 169, 191, 199, 203, 217, 241, 257, 279, 287, 291, 305, 329, 345, 367, 375, 379, 393, 417, 433, 455, 463, 467, 481, 505, 521, 543, 551, 555, 569, 593, 609, 631, 639, 643, 657, 681, 697, 719, 727, 731, 745, 769, 785, 807, 815, 819, 833, 857, 873, 895, 903

Offset: 0

Omar E. Pol, Oct 21 2017

The sequence arises from a "hybrid" cellular automaton, which consist of two successive generations using the rules of the D-toothpick sequence A194270 followed by one generation using toothpicks of length 2.
The rules are the same as the rules of A290220 except that here the first element is only one D-toothpick, not two. The result is that here the structure has only two arms instead of four arms as in A290220. On the other hand the structure of each arm is more complex than the structure of the arms of A290220.
The behavior is similar to A289840 and A290220 in the sense that these three cellular automata have the property of self-limiting their growth only in some directions of the square grid. For example: if here the first D-toothpick is placed in the NE-SW direction then the structure grows only in two opposite directions: NW and SE.
On the infinite square grid we start at stage 0 with no toothpicks, so a(0) = 0.
For the next stages we use the following rules:
1) If n is a number of the form 3*k + 1 (cf. A016777), for example: 1, 4, 7, 10, 13, ..., the elements added to the structure at n-th stage must be D-toothpicks of length sqrt(2) connected by their endpoints, in the same way as in the odd-indexed stages of A194270.
2) If n is a number of the form 3*k + 2 (cf. A016789), for example: 2, 5, 8, 11, 14, ..., the elements added to the structure at n-th stage must be toothpicks of length 1 connected by their endpoints, in the same way as in the even-indexed stages of A194270.
3) If n is a positive multiple of 3 (cf. A008585), for example: 3, 6, 9, 12, 15, ..., the elements added to the structure at n-th stage must be toothpicks of length 2. These toothpicks are connected to the structure by their midpoints.
The minimum width of the structure is 3*2^(1/2) = sqrt(18), see A010474.
The maximum width of the structure is 7*2^(1/2) = sqrt(98), see A010549.
The structure contains seven distinct polygons.
a(n) is the total number of elements in the structure after n generations.
A294021 (the first differences) gives the number of elements added at n-th stage.

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A010474, A010549, A139250, A194270, A289840, A290220, A294021.

concat(0, Vec(x*(1 + 4*x + 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 + 13*x^6 + 20*x^7 + 12*x^8 + 16*x^9) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Nov 12 2017

From Colin Barker, Nov 11 2017: (Start)
G.f.: x*(1 + 4*x + 4*x^2 + 6*x^3 + 8*x^4 + 4*x^5 + 13*x^6 + 20*x^7 + 12*x^8 + 16*x^9) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>10.
(End)

A041027 Denominators of continued fraction convergents to sqrt(18).

Original entry on oeis.org

1, 4, 33, 136, 1121, 4620, 38081, 156944, 1293633, 5331476, 43945441, 181113240, 1492851361, 6152518684, 50713000833, 209004522016, 1722749176961, 7100001229860, 58522759015841, 241191037293224

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A010474, A041026.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[18], n]]], {n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+12*Sqrt[2])^n)/6 // Simplify
a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(6*Sqrt[2]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,34,0,-1},{1,4,33,136},20] (* Harvey P. Dale, Jan 05 2019 *)

G.f.: (1+4*x-x^2)/(1-34*x^2+x^4). - Colin Barker, Jan 02 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.
a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(6*sqrt(2)). (End)

A157121 Decimal expansion of 11+3*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}, b = A157119.

lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}, b = A157120.

			11+3*sqrt(2) = 15.24264068711928514640...

Muniru A Asiru, Table of n, a(n) for n = 2..2000
Index entries for algebraic numbers, degree 2.

Cf. A157119, A157120, A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

evalf[120](11+3*sqrt(2)); # Muniru A Asiru, Feb 12 2019

RealDigits[11+3*Sqrt[2],10,120][[1]] (* Harvey P. Dale, Sep 12 2012 *)

Equals 7+A083729 = 11+A010474. [R. J. Mathar, Feb 27 2009]

A380734 Decimal expansion of the medium/short edge length ratio of a disdyakis dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 31 2025

			1.33770871866841824565822846556337733622336049131...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Disdyakis Dodecahedron.
Wikipedia, Disdyakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A380735 (long/short edge length ratio).

Cf. A010474, A378393, A378712, A378713, A378714, A378715, A380736, A380737, A380738.

First[RealDigits[3/14*(2 + 3*Sqrt[2]), 10, 100]]

(2 + 3*sqrt(2))*3/14 \\ Charles R Greathouse IV, Feb 05 2025

Equals (3/14)*(2 + 3*sqrt(2)) = (3/14)*(2 + A010474).

A040013 Continued fraction for sqrt(18).

Original entry on oeis.org

4, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4

Offset: 0

N. J. A. Sloane

Decimal expansion of 74/165. - R. J. Mathar, Aug 25 2025

			4.242640687119285146405066172... = 4 + 1/(4 + 1/(8 + 1/(4 + 1/(8 + ...)))). - _Harry J. Smith_, Jun 03 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 (0,1).

Cf. A010474 Decimal expansion. - Harry J. Smith, Jun 03 2009

ContinuedFraction[Sqrt[18],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,4}] (* Harvey P. Dale, Dec 31 2022 *)

{ allocatemem(932245000); default(realprecision, 31000); x=contfrac(sqrt(18)); for (n=0, 20000, write("b040013.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

G.f.: 4*(1 + x + x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A041026 Numerators of continued fraction convergents to sqrt(18).

Original entry on oeis.org

4, 17, 140, 577, 4756, 19601, 161564, 665857, 5488420, 22619537, 186444716, 768398401, 6333631924, 26102926097, 215157040700, 886731088897, 7309005751876, 30122754096401, 248291038523084

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).

Cf. A010474, A041027.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[18],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[18],20]] (* or *) LinearRecurrence[{0,34,0,-1},{4,17,140,577},20] (* Harvey P. Dale, Jun 12 2014 *)
a0[n_] := ((-4-3*Sqrt[2])/(17+12*Sqrt[2])^n+(-4+3*Sqrt[2])*(17+12*Sqrt[2])^n)/2 // Simplify
a1[n_] := (1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

G.f.: (4+17*x+4*x^2-x^3)/(1-34*x^2+x^4). - Colin Barker, Jan 02 2012
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((-4-3*sqrt(2))/(17+12*sqrt(2))^n+(-4+3*sqrt(2))*(17+12*sqrt(2))^n)/2.
a1(n) = (1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/2. (End)

A083729 Decimal expansion of sqrt(2)/(sqrt(2)-1)^2 = 3*sqrt(2)+4.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 15 2003

			8.242640687119285146405...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A010474, A072449.

SetDefaultRealField(RealField(100)); 4 + 3*Sqrt(2); // G. C. Greubel, Sep 11 2018

RealDigits[4 + 3*Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Sep 11 2018 *)

default(realprecision, 100); 4 + 3*sqrt(2) \\ G. C. Greubel, Sep 11 2018

Equals A010474 plus 4. - R. J. Mathar, Sep 17 2008

Equals Sum_{i>=1} i/sqrt(2)^i. - Christoph B. Kassir, Aug 30 2021

A377203 Decimal expansion of Integral_{x=0..oo} exp(-x)*erf(sqrt(x))^3 dx, where erf is the error function.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Oct 19 2024

			0.458941071620926611833296582700838961085659074877...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Erf.
Wikipedia, Error function.

Cf. A000796, A010474, A188615, A377202.

First[RealDigits[Sqrt[18]*ArcCsc[3]/Pi, 10, 100]]

Equals 3*sqrt(2)*arccot(2*sqrt(2))/Pi = A010474*A188615/A000796 (cf. eq. 38 in Weisstein link).

cp's OEIS Frontend

A248246 Egyptian fraction representation of sqrt(18) (A010474) using a greedy function.

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A378393 Decimal expansion of the midradius of a deltoidal icositetrahedron with unit shorter edge length.

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A294020 Total number of elements after n-th stage of a hybrid cellular automaton formed by D-toothpicks and toothpicks (see Comments lines for precise definition).

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A041027 Denominators of continued fraction convergents to sqrt(18).

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A157121 Decimal expansion of 11+3*sqrt(2).

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A380734 Decimal expansion of the medium/short edge length ratio of a disdyakis dodecahedron.

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A040013 Continued fraction for sqrt(18).

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A041026 Numerators of continued fraction convergents to sqrt(18).

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