A295330 -id:A295330 - cp's OEIS frontend

A295331 Numerators of continued fraction convergents to sqrt(13)/2 = A295330.

1, 2, 9, 128, 521, 649, 1819, 2468, 11691, 166142, 676259, 842401, 2361061, 3203462, 15174909, 215652188, 877783661, 1093435849, 3064655359, 4158091208, 19697020191, 279916373882, 1139362515719, 1419278889601, 3977920294921, 5397199184522, 25566717033009, 363331237646648, 1478891667619601, 1842222905266249, 5163337478152099, 7005560383418348, 33185579011825491

Offset: 0

Wolfdieter Lang, Nov 20 2017

The denominators are given in A295332.

The continued fraction expansion of sqrt(13)/2 is [1, repeat(1, 4, 14, 4, 1, 2)] = [b(0), repeat(b(1), b(2), b(3), b(4), b(5), b(6))].

			The convergents a(n)/A295332 begin: 1, 2, 9/5, 128/71, 521/289, 649/360, 1819/1009, 2468/1369, 11691/6485, 166142/92159, 676259/375121, 842401/467280, 2361061/1309681, 3203462/1776961, 15174909/8417525, 215652188/119622311, 877783661/486906769, 1093435849/606529080, ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).

Cf. A294972/A294973 (sqrt(7/2)), A295330, A295332.

Numerator[Convergents[Sqrt[13]/2,40]] (* or *) LinearRecurrence[ {0,0,0,0,0,1298,0,0,0,0,0,-1},{1,2,9,128,521,649,1819,2468,11691,166142,676259,842401},40] (* Harvey P. Dale, Jan 23 2019 *)

G.f.: (1 + 2*x + 9*x^2 + 128*x^3 + 521*x^4 + 649*x^5 + 521*x^6 - 128*x^7 + 9*x^8 + 2*x^9 + x^10 - x^11) / ((1 - 3*x - x^2)*(1 + 3*x - x^2)*(1 + 3*x + 10*x^2 - 3*x^3 + x^4)*(1 - 3*x + 10*x^2 + 3*x^3 + x^4)). For the derivation see the period 4 example for the denominators of the convergents of sqrt(7)/2 given in A294972. Here the period is 6 and the input for the recurrence a(n) = b(n)*a(n-1) + a(n-2) is a(-1) = 1 = a(0) (a(-2) = 1 - b(0) = 0) with the b(n) modulo 6 given above. Note that a(6*k) = 2*a(6*k-1) + a(6*k-2) is valid only for k >= 1 because a(0) = 1 is not equal to 2*a(-1) + a(-2) = 2. The denominator in the unfactorized form is 1 - 1298*x^6 + x^12.

a(n) = 1298*a(n-6) - a(n-12), n >= 12, with inputs a(0)..a(11).

A295332 Denominators of the continued fraction convergents to sqrt(13)/2 = A295330.

Original entry on oeis.org

1, 1, 5, 71, 289, 360, 1009, 1369, 6485, 92159, 375121, 467280, 1309681, 1776961, 8417525, 119622311, 486906769, 606529080, 1699964929, 2306494009, 10925940965, 155269667519, 632004611041, 787274278560, 2206553168161, 2993827446721, 14181862955045, 201539908817351, 820341498224449, 1021881407041800, 2864104312308049, 3885985719349849, 18408047189707445

Offset: 0

Wolfdieter Lang, Nov 20 2017

The numerators are given in A295331.

The continued fraction expansion of sqrt(13)/2 is [1, repeat(1, 4, 14, 4, 1, 2)].

			See A295331 for the first convergents.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1298, 0, 0, 0, 0, 0, -1).

Cf. A295330, A295331.

G.f.: (1 + x + 5*x^2 + 71*x^3 + 289*x^4 + 360*x^5 - 289*x^6 + 71*x^7 - 5*x^8 + x^9 - x^10) / ((1 - 3*x - x^2)*(1 + 3*x - x^2)*(1 + 3*x + 10*x^2 - 3*x^3 + x^4)*(1 - 3*x + 10*x^2 + 3*x^3 + x^4)). See A295331 for a hint for the derivation. Here the a(n) recurrence is the same as there but the inputs are a(0) = 1, a(-1) = 0, (a(-2) = 1). The unfactorized denominator is 1 - 1298*x^6 + x^12.

a(n) = 1298*a(n-6) - a(n-12), n >= 12, with inputs a(0)..a(11).

A010470 Decimal expansion of square root of 13.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 3 followed by {1, 1, 1, 1, 6} repeated. - Harry J. Smith, Jun 02 2009
The convergents to sqrt(13) are given in A041018/A041019. - Wolfdieter Lang, Nov 23 2017
The fundamental algebraic (integer) number in the field Q(sqrt(13)) is (1 + sqrt(13))/2  = A209927. - Wolfdieter Lang, Nov 21 2023

			3.605551275463989293119221267470495946251296573845246212710453056227166...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.31.4, p. 201.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A010122 (continued fraction), A041018/A041019 (convergents), A248242 (Egyptian fraction), A171983 (Beatty sequence).

Cf. A020770 (reciprocal), A209927, A295330, A344069.

RealDigits[N[Sqrt[13],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(13); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010470.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009

A020797 Decimal expansion of 1/sqrt(40).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

With offset 1, decimal expansion of sqrt(5/2). - Eric Desbiaux, May 01 2008
sqrt(5/2) appears as a coordinate in a degree-5 integration formula on 13 points in the unit sphere [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011
With offset 2, decimal expansion of sqrt(250). - Michel Marcus, Nov 04 2013
From Wolfdieter Lang, Nov 21 2017: (Start)
The regular continued fraction of 1/sqrt(40) = 1/(2*sqrt(10)) is [0; 6, 3, repeat(12, 3)], and the convergents are given by A(n-1)/B(n-1), n >= 0, with A(-1) = 0, A(n-1) = A041067(n) and B(-1) = 1, B(n-1) = A041066(n).
The regular continued fraction of sqrt(5/2) = sqrt(10)/2 is [1; repeat(1, 1, 2)], and the convergents are given in A295333/A295334.
sqrt(10)/2 is one of the catheti of the rectangular triangle with hypotenuse sqrt(13)/2 = A295330 and the other cathetus sqrt(3)/2 = A010527. This can be constructed from a regular hexagon inscribed in a circle with a radius of 1 unit. If the vertex V_0 has coordinates (x, y) = (1, 0) and the midpoint M_4 = (0, -sqrt(3)/2) then the point L = (sqrt(10)/2, 0) is obtained as intersection of the x-axis and a circle around M_4 with radius taken from the distance between M_4 and V_1 = (1/2, sqrt(3)/2) which is sqrt(13)/2. (End)

			1/sqrt(40) = 0.15811388300841896659994467722163592668597775696626084134287...
sqrt(5/2) = 1.5811388300841896659994467722163592668597775696626084134287...
sqrt(250) = 15.811388300841896659994467722163592668597775696626084134287...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
A. H. Stroud, D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2.

Cf. A010467 (sqrt(10)), A010527, A010494 (sqrt(40)), A041067/A041066, A295330, A295333/A295334.

RealDigits[N[1/Sqrt[40],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

1/sqrt(40) \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(40*x^2-1)[2] \\ Charles R Greathouse IV, Feb 04 2025

Equals Re(sqrt(5*i)/10) = Im(sqrt(5*i)/10). - Karl V. Keller, Jr., Sep 01 2020

Equals A010467/20. - R. J. Mathar, Feb 23 2021

A041613 Denominators of continued fraction convergents to sqrt(325).

Original entry on oeis.org

1, 36, 1297, 46728, 1683505, 60652908, 2185188193, 78727427856, 2836372591009, 102188140704180, 3681609437941489, 132640127906597784, 4778726214075461713, 172166783834623219452, 6202782944260511361985, 223472352777213032250912

Offset: 0

N. J. A. Sloane

From Michael A. Allen, Jul 13 2023: (Start)
Also called the 36-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 36 kinds of squares available. (End)
a(2*n) and b(2*n) = A041612(2*n) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = -1. a(2*n+1) and b(2*n+1) = A041612(2*n+1) give all (positive integer) solutions to the Pell equation b^2 - 13*a^2 = 1. - Robert FERREOL, Oct 09 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (36,1).

Cf. A041612 (numerators), A040306 (continued fraction), A295330.

Row n=36 of A073133, A172236 and A352361 and column k=36 of A157103.

with (combinat):seq(fibonacci(3*n,3)/10, n=1..15); # Zerinvary Lajos, Apr 20 2008

a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*36,{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[325], 30]] (* Vincenzo Librandi Dec 21 2013 *)

a(n) = F(n, 36), the n-th Fibonacci polynomial evaluated at x=36. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 36*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=36.
G.f.: 1/(1 - 36*x - x^2). (End)
a(n) = ((18 + 5*sqrt(13))^(n+1) - (18 - 5*sqrt(13))^(n+1)) / (2*sqrt(13)). - Robert FERREOL, Oct 09 2024

More terms from Colin Barker, Nov 20 2013

A041612 Numerators of continued fraction convergents to sqrt(325).

Original entry on oeis.org

18, 649, 23382, 842401, 30349818, 1093435849, 39394040382, 1419278889601, 51133434066018, 1842222905266249, 66371158023650982, 2391203911756701601, 86149711981264908618, 3103780835237293411849, 111822259780523827735182, 4028705132934095091878401

Offset: 0

N. J. A. Sloane

a(2*n) and b(2*n) = A041613(2*n) give all (positive integer) solutions to the Pell equation a^2 - 13*b^2 = -1. a(2*n+1) and b(2*n+1) = A041613(2*n+1) give all (positive integer) solutions to the Pell equation a^2 - 13*b^2 = 1. - Robert FERREOL, Oct 09 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (36,1).

Cf. A040306 (continued fraction), A041613 (denominators), A295330.

Numerator[Convergents[Sqrt[325], 30]] (* Vincenzo Librandi, Nov 04 2013 *)

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 36*a(n-1) + a(n-2), n > 1; a(0)=18, a(1)=649.
G.f.: (18+x)/(1-36*x-x^2). (End)
a(n) = ((18 + 5*sqrt(13))^(n+1) + (18 - 5*sqrt(13))^(n+1))/2. - Robert FERREOL, Oct 09 2024

Additional term from Colin Barker, Nov 09 2013

A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 27 2017

The regular continued fraction of sqrt(14)/2 is [1, repeat(1, 6, 1, 2)].
The convergents are given in A295336/A295337.
sqrt(14)/2 appears in a regular hexagon inscribed in a circle of radius 1 unit in the following way. Draw a straight line through two opposed midpoints of a side (halving the hexagon). The length between one of the midpoints, say M, and one of the two vertices nearest to the opposed midpoint is sqrt(13)/2 = A295330 units. A circle through M with this length ratio sqrt(13)/2 intersects the line below the hexagon at a point, say P. Then the length ratio between P and one of the two vertices nearest to M is sqrt(14)/2 (from a right triangle (1/2, sqrt(13)/2, sqrt(14)/2)).

			1.87082869338697069279187436615827465087800990388936347315187273366001757815...

Cf. A010471, A295330, A295336/A295337.

First[RealDigits[Sqrt[14]/2, 10, 100]] (* Paolo Xausa, May 23 2025 *)

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

6, 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, 9

Offset: 0

Amiram Eldar, Feb 24 2025

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

			0.60092521257733154885320354457841599104188276230754...

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
Index entries for algebraic numbers, degree 2.

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Mathematica
```
RealDigits[Sqrt[13] / 6, 10, 120][[1]]
```

list(len) = digits(floor(10^len*quadgen(52)/6));

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.

cp's OEIS Frontend

A295331 Numerators of continued fraction convergents to sqrt(13)/2 = A295330.

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A295332 Denominators of the continued fraction convergents to sqrt(13)/2 = A295330.

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A010470 Decimal expansion of square root of 13.

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A020797 Decimal expansion of 1/sqrt(40).

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A041613 Denominators of continued fraction convergents to sqrt(325).

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A041612 Numerators of continued fraction convergents to sqrt(325).

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A294969 Decimal expansion of sqrt(14)/2 = sqrt(7/2) = A010471/2.

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A381485 Decimal expansion of sqrt(13)/6.

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