A010472 -id:A010472 - cp's OEIS frontend

A176016 Decimal expansion of (3+sqrt(15))/6.

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(15))/6 is A010687.

			(3+sqrt(15))/6 = 1.14549722436790281419...

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

Cf. A010472 (sqrt(15)), A176020 ((3+sqrt(15))/3), A010687 (repeat 1, 6).

RealDigits[(3+Sqrt[15])/6,10,120][[1]] (* Harvey P. Dale, May 01 2012 *)

Equals 1/2+A140246. - R. J. Mathar, Mar 17 2025

A176020 Decimal expansion of (3+sqrt(15))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (3+sqrt(15))/3 is A010693.

a(n) = A020817(n-1) for n > 1; a(1) = 2.

			(3+sqrt(15))/3 = 2.29099444873580562839...

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

Cf. A010472 (decimal expansion of sqrt(15)), A176016 (decimal expansion of (3+sqrt(15))/6), A010693 (repeat 2, 3), A020817 (decimal expansion of 1/sqrt(60)).

RealDigits[(3+Sqrt[15])/3,10,120][[1]] (* Harvey P. Dale, May 20 2013 *)

A041022 Numerators of continued fraction convergents to sqrt(15).

Original entry on oeis.org

3, 4, 27, 31, 213, 244, 1677, 1921, 13203, 15124, 103947, 119071, 818373, 937444, 6443037, 7380481, 50725923, 58106404, 399364347, 457470751, 3144188853, 3601659604, 24754146477, 28355806081

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 (0,8,0,-1).

Cf. A010472, A041023.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[15],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[15], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := (-((4-Sqrt[15])^n*(3+Sqrt[15]))+(-3+Sqrt[15])*(4+Sqrt[15])^n)/2 // Simplify
a1[n_] := ((4-Sqrt[15])^n+(4+Sqrt[15])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

G.f.: (3+4*x+3*x^2-x^3)/(1-8*x^2+x^4).
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = (-((4-sqrt(15))^n*(3+sqrt(15)))+(-3+sqrt(15))*(4+sqrt(15))^n)/2.
a1(n) = ((4-sqrt(15))^n+(4+sqrt(15))^n)/2. (End)

A176058 Decimal expansion of (3+sqrt(15))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (3+sqrt(15))/2 is A176059.

			(3+sqrt(15))/2 = 3.43649167310370844258...

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

Cf. A010472 (decimal expansion of sqrt(15)), A176059 (repeat 3, 2).

First[RealDigits[(3 + Sqrt[15])/2, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

Equals 1.5 + A088543. [R. J. Mathar, Apr 12 2010]

A177187 Union of A057080 and A001090.

Original entry on oeis.org

1, 1, 9, 8, 71, 63, 559, 496, 4401, 3905, 34649, 30744, 272791, 242047, 2147679, 1905632, 16908641, 15003009, 133121449, 118118440, 1048062951, 929944511, 8251382159, 7321437648, 64962994321, 57641556673, 511452572409, 453811015736, 4026657584951, 3572846569215

Offset: 1

Mark Dols, May 04 2010

Column sums of shifted Pascal-like array:
  1..1..10..10..100..100.1000.1000
  ......-1..-2..-30..-40.-500.-600
  ................1....3...60..100
  .........................-1...-4
  -------------------------------- +
  1..1...9...8...71...63..559..496
Decimal expansion of ratio n/(n+1) is accumulation of Catalan numbers; (5 +/- sqrt(15)).

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Cf. A057080, A001090, A041059, A004189, A000108.

Cf. A010472.

For n odd a(n) = 10*a(n-1) - a(n-2), for n even a(n) = a(n-1) - a(n-2); with a(0) = 0, a(1) = 1.

G.f.: x*(1+x+x^2) / ( 1-8*x^2+x^4 ). - R. J. Mathar, Nov 11 2011

A017949 Powers of sqrt(15) rounded down.

Original entry on oeis.org

1, 3, 15, 58, 225, 871, 3375, 13071, 50625, 196069, 759375, 2941046, 11390625, 44115700, 170859375, 661735513, 2562890625, 9926032708, 38443359375, 148890490631, 576650390625, 2233357359474, 8649755859375

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Cf. A010472 (sqrt(15)).

[Floor(Sqrt(15^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Floor[(Sqrt[15])^Range[0,30]] (* Harvey P. Dale, Jul 03 2020 *)

a(n)=sqrtint(15^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(15^n)). - Vincenzo Librandi, Jun 24 2011

A171535 Decimal expansion of 2*sqrt(2/15).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

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

			sqrt(8/15) = 0.73029674334022148460929304... = 2*0.365148371..

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients

SetDefaultRealField(RealField(100)); Sqrt(8/15); // G. C. Greubel, Oct 02 2018

RealDigits[2*Sqrt[2/15], 10, 100][[1]] (* G. C. Greubel, Oct 02 2018 *)

default(realprecision, 100); sqrt(8/15) \\ G. C. Greubel, Oct 02 2018

Equals A010466/A010472.

A176403 Decimal expansion of (15+4*sqrt(15))/5.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 17 2010

Continued fraction expansion of (15+4*sqrt(15))/5 is A010726.

			(15+4*sqrt(15))/5 = 6.09838667696593350814...

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

Cf. A010472 (decimal expansion of sqrt(15)), A010726 (repeat 6, 10).

A277091 a(n) = ((1 + sqrt(15))^n - (1 - sqrt(15))^n)/sqrt(15).

Original entry on oeis.org

0, 2, 4, 36, 128, 760, 3312, 17264, 80896, 403488, 1939520, 9527872, 46209024, 225808256, 1098542848, 5358401280, 26096402432, 127210422784, 619770479616, 3020486878208, 14717760471040, 71722337236992, 349493321068544, 1703099363454976, 8299105221869568, 40441601532108800

Offset: 0

Ilya Gutkovskiy, Sep 29 2016

Number of zeros in substitution system {0 -> 1111111, 1 -> 1001} at step n from initial string "1" (see example).

			Evolution from initial string "1": 1 -> 1001 -> 1001111111111111111001 -> ...
Therefore, number of zeros at step n:
a(0) = 0;
a(1) = 2;
a(2) = 4, etc.

Ilya Gutkovskiy, Illustration (substitution system {0 -> 1111111, 1 -> 1001}) and similar sequences
Eric Weisstein's World of Mathematics, Substitution System
Index entries for linear recurrences with constant coefficients, signature (2,14).

Cf. A010472, A103435, A274520, A274526.

Mathematica
```
LinearRecurrence[{2, 14}, {0, 2}, 26]
```

concat(0, Vec(2*x/(1-2*x-14*x^2) + O(x^99))) \\ Altug Alkan, Oct 01 2016

O.g.f.: 2*x/(1 - 2*x - 14*x^2).
E.g.f.: 2*sinh(sqrt(15)*x)*exp(x)/sqrt(15).
a(n) = 2*a(n-1) + 14*a(n-2).
Lim_{n->infinity} a(n+1)/a(n) = 1 + sqrt(15) = 1 + A010472.

A176533 Decimal expansion of (15+4*sqrt(15))/3.

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of (15+4*sqrt(15))/3 is A010726 preceded by 10.

			(15+4*sqrt(15))/3 = 10.16397779494322251357...

Cf. A010472 (decimal expansion of sqrt(15)), A010726 (repeat 6, 10).

RealDigits[(15+4*Sqrt[15])/3,10,120][[1]] (* Harvey P. Dale, Nov 10 2024 *)

cp's OEIS Frontend

A176016 Decimal expansion of (3+sqrt(15))/6.

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A176020 Decimal expansion of (3+sqrt(15))/3.

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A041022 Numerators of continued fraction convergents to sqrt(15).

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A176058 Decimal expansion of (3+sqrt(15))/2.

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A177187 Union of A057080 and A001090.

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A017949 Powers of sqrt(15) rounded down.

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A171535 Decimal expansion of 2*sqrt(2/15).

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A176403 Decimal expansion of (15+4*sqrt(15))/5.

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