A010549 -id:A010549 - cp's OEIS frontend

A248321 Egyptian fraction representation of sqrt(98) (A010549) using a greedy function.

9, 2, 3, 16, 274, 83555, 12961139206, 198730293272988591339, 183217938497357958578283307441647343725436, 36471466160614116433355003352211955756967345223328177891902116459111899483704330391

Offset: 0

Robert G. Wilson v, Oct 05 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[ 98]]

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)

A010169 Continued fraction for sqrt(98).

Original entry on oeis.org

9, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18, 1, 8, 1, 18

Offset: 0

N. J. A. Sloane

			9.89949493661166534161182106... = 9 + 1/(1 + 1/(8 + 1/(1 + 1/(18 + ...)))). - _Harry J. Smith_, Jun 12 2009

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. A010549 (decimal expansion).

ContinuedFraction[Sqrt[98],300] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2011 *)
PadRight[{9},120,{18,1,8,1}] (* Harvey P. Dale, Dec 13 2015 *)

{ allocatemem(932245000); default(realprecision, 24000); x=contfrac(sqrt(98)); for (n=0, 20000, write("b010169.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 12 2009

From Wesley Ivan Hurt, Jun 23 2021: (Start)
a(n) = a(n-4).
a(0) = 9; a(n) = 7 + 6*(-1)^n + 5*cos(n*Pi/2) for n > 0. (End)
From Amiram Eldar, Nov 14 2023: (Start)
Multiplicative with a(2) = 8, a(2^e) = 18 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 7/2^s + 5/2^(2*s-1)). (End)

A041176 Numerators of continued fraction convergents to sqrt(98).

Original entry on oeis.org

9, 10, 89, 99, 1871, 1970, 17631, 19601, 370449, 390050, 3490849, 3880899, 73347031, 77227930, 691170471, 768398401, 14522341689, 15290740090, 136848262409, 152139002499, 2875350307391, 3027489309890

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 (0,0,0,198,0,0,0,-1).

Cf. A010549, A041177.

Numerator[Convergents[Sqrt[98], 30]] (* Vincenzo Librandi, Oct 30 2013 *)

G.f.: (9 + 10*x + 89*x^2 + 99*x^3 + 89*x^4 - 10*x^5 + 9*x^6 - x^7) / ((1 + 14*x^2 - x^4)*(1 - 14*x^2 - x^4)). [Bruno Berselli, Oct 30 2013]

A381689 Decimal expansion of the isoperimetric quotient of a truncated cuboctahedron (great rhombicuboctahedron).

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 06 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.8390038051045342803688923433479361567507803475...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377343 (surface area), A377344 (volume).

Cf. A002193, A002194, A010549, A019679, A381684.

First[RealDigits[Pi/12*(11 + Sqrt[98])^2/(2 + Sqrt[2] + Sqrt[3])^3, 10, 100]]

Equals 36*Pi*A377344^2/(A377343^3).

Equals (Pi/12)*(11 + 7*sqrt(2))^2/((2 + sqrt(2) + sqrt(3))^3) = A019679*(11 + A010549)^2/((2 + A002193 + A002194)^3).

cp's OEIS Frontend

A248321 Egyptian fraction representation of sqrt(98) (A010549) using a greedy function.

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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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A010169 Continued fraction for sqrt(98).

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A041176 Numerators of continued fraction convergents to sqrt(98).

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A381689 Decimal expansion of the isoperimetric quotient of a truncated cuboctahedron (great rhombicuboctahedron).

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