A010464 -id:A010464 - cp's OEIS frontend

A277645 Beatty sequence for 3+sqrt(6).

5, 10, 16, 21, 27, 32, 38, 43, 49, 54, 59, 65, 70, 76, 81, 87, 92, 98, 103, 108, 114, 119, 125, 130, 136, 141, 147, 152, 158, 163, 168, 174, 179, 185, 190, 196, 201, 207, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 277, 283, 288, 294, 299, 305

Offset: 1

Jason Kimberley, Oct 26 2016

Eggleton et al. show that k is in this sequence if and only if A277515(k) > 3.

			a(4) = 3*4 + 9 because 9^2 = 81 < 6*4^2 = 96 < 100 = 10^2.

R. B. Eggleton, J. S. Kimberley, and J. A. MacDougall, Square-free rank of integers, submitted.

Jason Kimberley, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences.

Cf. A000196, A008585, A010464, A033581, A277515.

Complement of A277644.

Magma
```
[3*n+Isqrt(6*n^2): n in [1..60]];
```

Floor[Range[100]*(3 + Sqrt[6])] (* Paolo Xausa, Jul 11 2024 *)

a(n) = floor(n*(3+sqrt(6))).
a(n) = 3*n + A000196(A033581(n)).
a(n) = A008585(n) + A000196(A033581(n)).

A004564 Expansion of sqrt(6) in base 5.

Original entry on oeis.org

2, 2, 1, 1, 0, 4, 3, 1, 1, 4, 3, 1, 1, 3, 3, 1, 3, 4, 1, 2, 1, 4, 1, 1, 2, 1, 2, 4, 4, 3, 0, 1, 4, 1, 3, 0, 1, 0, 4, 4, 0, 3, 3, 1, 1, 1, 4, 3, 0, 0, 0, 0, 0, 3, 1, 2, 2, 0, 4, 1, 1, 2, 0, 0, 2, 3, 1, 4, 1, 0, 3, 1, 3, 3, 3, 0, 1, 2, 3, 0, 3, 1, 4, 1, 0, 0, 3, 2, 2, 0, 0, 0, 0, 3, 3, 2, 3, 2, 3

Offset: 1

N. J. A. Sloane

			In base 5: sqrt(11) = 2.21104311431133134121411...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A040003, A010464.

d:= 6; m:=5; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

RealDigits[Sqrt[6], 5, 100][[1]] (* Alonso del Arte, Aug 27 2012 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A154236 a(n) = ( (5 + sqrt(6))^n - (5 - sqrt(6))^n )/(2*sqrt(6)).

Original entry on oeis.org

1, 10, 81, 620, 4661, 34830, 259741, 1935640, 14421321, 107436050, 800355401, 5962269060, 44415937981, 330876267670, 2464859855061, 18361949464880, 136787157402641, 1018994534193690, 7590989351286721, 56548997363187100

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Jan 05 2009

First differences are in A164551.

Lim_{n -> infinity} a(n)/a(n-1) = 5 + sqrt(6) = 7.4494897427....

G. C. Greubel, Table of n, a(n) for n = 1..999
Index entries for linear recurrences with constant coefficients, signature (10, -19).

Cf. A010464 (decimal expansion of square root of 6), A164551.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-6); S:=[ ((5+r)^n-(5-r)^n)/(2*r): n in [1..25] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 07 2009

LinearRecurrence[{10, -19}, {1, 10}, 25] (* or *) Table[Simplify[((5 + Sqrt[6])^n -(5-Sqrt[6])^n)/(2*Sqrt[6])], {n, 1, 25}] (* G. C. Greubel, Sep 06 2016 *)

a(n)=([0,1; -19,10]^(n-1)*[1;10])[1,1] \\ Charles R Greathouse IV, Sep 07 2016

[lucas_number1(n,10,19) for n in range(1, 25)] # Zerinvary Lajos, Apr 26 2009

From Philippe Deléham, Jan 06 2009: (Start)
a(n) = 10*a(n-1) - 19*a(n-2) for n > 1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 10*x + 19*x^2). (End)

Extended beyond a(7) by Klaus Brockhaus, Jan 07 2009

Edited by Klaus Brockhaus, Oct 04 2009

A171539 Decimal expansion of sqrt(6/35).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -2 3/2 | 5/2 -1/2>.

			sqrt(6/35) = sqrt(210)/35 = 0.414039335605412530677676118776...

Wikipedia, Table of Clebsch-Gordan coefficients

RealDigits[Sqrt[6/35],10,120][[1]] (* Harvey P. Dale, Jun 09 2023 *)

equals A010464/A010490 = A010494*A171537/10.

A174925 Decimal expansion of (2+sqrt(6))/4.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 02 2010

Continued fraction expansion of (2+sqrt(6))/4 is A010689.

			(2+sqrt(6))/4 = 1.11237243569579452454...

Cf. A010464 (decimal expansion of sqrt(6)), A010689 (repeat 1, 8).

RealDigits[N[(2+Sqrt[6])/4,300]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

A176051 Decimal expansion of (2+sqrt(6))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (2+sqrt(6))/2 is A010694.

a(n) = A115754(n) for n > 1; a(1) = 2.

			(2+sqrt(6))/2 = 2.22474487139158904909...
Note also that (1+sqrt(6))/2 = 1.724744871391589049098642..., the mis-typed golden ratio. - _N. J. A. Sloane_, Jan 19 2025

Cf. A010464 (decimal expansion of sqrt(6)), A115754 (decimal expansion of sqrt(3/2)), A010694 (repeat 2, 4).

A176213 Decimal expansion of 2+sqrt(6).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of 2+sqrt(6) is A010694.

a(n) = A010464(n) = A086180(n) for n > 1, a(1) = 4.

			2+sqrt(6) = 4.44948974278317809819...

Cf. A010464 (decimal expansion of sqrt(6)), A086180 (decimal expansion of 1+sqrt(6)), A010694 (repeat 4, 2).

RealDigits[2+Sqrt[6],10,120][[1]] (* Harvey P. Dale, Aug 19 2018 *)

A004563 Expansion of sqrt(6) in base 4.

Original entry on oeis.org

2, 1, 3, 0, 3, 0, 1, 0, 1, 3, 0, 0, 2, 2, 0, 0, 1, 0, 2, 1, 0, 0, 2, 1, 1, 3, 0, 3, 3, 2, 2, 0, 0, 1, 2, 1, 0, 0, 3, 0, 2, 0, 2, 2, 0, 1, 2, 1, 0, 0, 3, 1, 0, 2, 2, 2, 2, 3, 0, 2, 0, 3, 2, 1, 3, 2, 3, 1, 2, 0, 0, 1, 0, 1, 1, 0, 0, 3, 2, 2, 1, 3, 2, 3, 3, 1, 2, 2, 0, 2, 3, 1, 3, 3, 0, 0, 1, 3, 2

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010464.

d:= 6; m:=4; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

RealDigits[Sqrt[6],4,120][[1]] (* Harvey P. Dale, May 06 2011 *)

A004565 Expansion of sqrt(6) in base 6.

Original entry on oeis.org

2, 2, 4, 1, 0, 3, 1, 2, 2, 0, 5, 5, 2, 1, 4, 5, 3, 2, 5, 0, 0, 4, 3, 2, 0, 4, 0, 4, 1, 1, 0, 5, 5, 2, 0, 5, 3, 2, 1, 0, 4, 3, 2, 0, 1, 5, 5, 0, 3, 1, 1, 1, 5, 3, 4, 1, 3, 3, 1, 2, 2, 3, 1, 5, 2, 4, 0, 1, 5, 4, 3, 0, 3, 0, 4, 4, 0, 4, 0, 2, 5, 3, 2, 5, 2, 5, 4, 3, 5, 1, 1, 0, 4, 5, 5, 3, 5, 5, 3

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010464.

d:= 6; m:=6; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

RealDigits[Sqrt[6], 6, 100][[1]] (* G. C. Greubel, Mar 26 2018 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

A004566 Expansion of sqrt(6) in base 7.

Original entry on oeis.org

2, 3, 1, 0, 1, 1, 4, 0, 0, 6, 3, 0, 0, 5, 2, 3, 4, 1, 1, 2, 1, 6, 2, 5, 6, 6, 1, 6, 3, 2, 0, 5, 4, 5, 0, 1, 5, 1, 6, 6, 6, 6, 4, 2, 6, 1, 1, 3, 2, 6, 6, 0, 5, 4, 6, 3, 1, 3, 6, 6, 2, 4, 0, 4, 4, 1, 0, 1, 0, 2, 3, 2, 2, 3, 4, 5, 5, 6, 1, 2, 5, 5, 0, 1, 6, 4, 6, 2, 4, 2, 0, 0, 4, 5, 0, 6, 2, 3, 6

Offset: 1

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 1..10000
Jason Kimberley, Index of expansions of sqrt(d) in base b

Cf. A010464.

d:= 6; m:=7; Prune(Reverse(IntegerToSequence(Isqrt(d*m^100), m))); // G. C. Greubel, Mar 26 2018

RealDigits[Sqrt[6], 7, 100][[1]] (* G. C. Greubel, Mar 26 2018 *)

Updated by Alois P. Heinz at the suggestion of Kevin Ryde, Feb 19 2012

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A277645 Beatty sequence for 3+sqrt(6).

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A004564 Expansion of sqrt(6) in base 5.

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A154236 a(n) = ( (5 + sqrt(6))^n - (5 - sqrt(6))^n )/(2*sqrt(6)).

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A171539 Decimal expansion of sqrt(6/35).

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A174925 Decimal expansion of (2+sqrt(6))/4.

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A176051 Decimal expansion of (2+sqrt(6))/2.

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A176213 Decimal expansion of 2+sqrt(6).

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A004563 Expansion of sqrt(6) in base 4.

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