A010472 -id:A010472 - cp's OEIS frontend

A175578 Decimal expansion of the sum over the inverse icosahedral numbers.

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

Offset: 1

R. J. Mathar, Jul 15 2010

Defined by sum_{n>=1} 1/A006564(n) = 1/1 + 1/12 +1/48 + 1/124 +...

Equals gamma + Pi*sqrt(5/3)*tanh(Pi*sqrt(15)/10)/2 + Re psi( 1/2+i*sqrt(15)/10 ), where psi is the digamma function, i the imaginary unit, Pi = A000796, sqrt(15)=A010472, gamma=A001620.

			1.12356596689925188757393..

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

Cf. A006564 (icosahedral numbers).

Cf. sums of inverses: A152623 (tetrahedral numbers), A002117 (cubes), A175577 (octahedral numbers), A295421 (dodecahedral numbers).

Digits := 120 : gamma+ Psi(1/2+sqrt(15)*I/10)+sqrt(15)/6*Pi*tanh(Pi*sqrt(15)/10) ; evalf(Re(%)) ;

N[EulerGamma + PolyGamma[1/2 + (I*Sqrt[15])/10] + (1/2)*Tanh[(Pi*Sqrt[15])/10]*Pi*Sqrt[5/3] // Re, 105] // RealDigits // First (* Jean-François Alcover, Feb 05 2013 *)

Euler+Pi*sqrt(5/3)*tanh(Pi*sqrt(15)/10)/2+real(psi(1/2+ I*sqrt(15)/10)) \\ Charles R Greathouse IV, Jul 19 2013

A040011 Continued fraction for sqrt(15).

Original entry on oeis.org

3, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6

Offset: 0

N. J. A. Sloane

Decimal expansion of 313/990. - R. J. Mathar, Aug 22 2025

			3.872983346207416885179265399... = 3 + 1/(1 + 1/(6 + 1/(1 + 1/(6 + ...)))). - _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. A010472 (decimal expansion). A010687.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[15],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{3},120,{6,1}] (* Harvey P. Dale, Apr 14 2020 *)

{ allocatemem(932245000); default(realprecision, 19000); x=contfrac(sqrt(15)); for (n=0, 20000, write("b040011.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2^e) = 6, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 5/2^s). (End)
G.f.: (3 + x + 3*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A041023 Denominators of continued fraction convergents to sqrt(15).

Original entry on oeis.org

1, 1, 7, 8, 55, 63, 433, 496, 3409, 3905, 26839, 30744, 211303, 242047, 1663585, 1905632, 13097377, 15003009, 103115431, 118118440, 811826071, 929944511, 6391493137, 7321437648, 50320119025

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 6 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,8,0,-1).

Cf. A010472, A041022, A002530.

Denominator[NestList[(6/(6+#))&,0,60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
a0[n_] := (-((-5+Sqrt[15])*(4+Sqrt[15])^n)+(4-Sqrt[15])^n*(5+Sqrt[15]))/10 // Simplify
a1[n_] := (-(4-Sqrt[15])^n+(4+Sqrt[15])^n)/(2*Sqrt[15]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
Convergents[Sqrt[15],30]//Denominator (* Harvey P. Dale, Aug 13 2016 *)

G.f.: (1+x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(6) + sqrt(10) )/2 and beta = ( sqrt(6) - sqrt(10) )/2 be the roots of the equation x^2 - sqrt(6)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = product {k = 1..floor((n-1)/2)} ( 6 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 6*a(2*n) + a(2*n - 1). (End)
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = (-((-5+sqrt(15))*(4+sqrt(15))^n)+(4-sqrt(15))^n*(5+sqrt(15)))/10.
a1(n) = (-(4-sqrt(15))^n+(4+sqrt(15))^n)/(2*sqrt(15)). (End)

A176104 Decimal expansion of sqrt(285).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of sqrt(285) is A040268.

			sqrt(285) = 16.88194301613413218311...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000
Index entries for algebraic numbers, degree 2.

Cf. A010472 (decimal expansion of sqrt(15)), A010475 (decimal expansion of sqrt(19)), A040268.

A377996 Decimal expansion of the dihedral angle, in radians, between triangular and square faces in a (small) rhombicosidodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Nov 15 2024

Also the dihedral angle, in radians, between square and hexagonal faces in a truncated icosidodecahedron (great rhombicosidodecahedron).

			2.7767288254763100567352450533728425364989164423512...

Eric Weisstein's World of Mathematics, Great Rhombicosidodecahedron.
Eric Weisstein's World of Mathematics, Small Rhombicosidodecahedron.

Cf. A002194, A010472, A377995.

First[RealDigits[ArcCos[-(Sqrt[3] + Sqrt[15])/6], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["Rhombicosidodecahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(sqrt(3) + sqrt(15))/6) = arccos(-(A002194 + A010472)/6).

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi),10,120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PARI
```
(5*sqrt(3)+sqrt(15))/(6*Pi)
```

Equals (5*A002194 + A010472)/(6*A000796).
Equals (5*A002194 + A010472)*A049541/6.
Equals (10*A010527 + A010472)*A049541/6.
Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).
Equals (5 + A002163)*A049541*A020760/2.

A194398 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(15) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 41, 42, 43, 44, 45, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 93

Offset: 1

Clark Kimberling, Aug 24 2011

nonn

See A194368.

Cf. A010472, A194368, A194399, A194400.

r = Sqrt[15]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]       (* A194398 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]       (* A194399 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t3, 1]]       (* A194400 *)

A194399 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(15) and < > denotes fractional part.

Original entry on oeis.org

6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48, 54, 56, 62, 70, 78, 86, 94, 102, 110, 118, 314, 322, 330, 338, 346, 354, 362, 370, 376, 378, 384, 386, 392, 394, 400, 402, 408, 410, 416, 418, 424, 426, 432, 434, 438, 442, 446, 450, 454, 458, 462, 466, 470

Offset: 1

Clark Kimberling, Aug 24 2011

nonn

Every term is even; see A194368.

Cf. A010472, A194368, A194398, A194400.

r = Sqrt[15]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]       (* A194398 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]       (* A194399 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t3, 1]]       (* A194400 *)

A194400 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) > 0, where r=sqrt(15) and < > denotes fractional part.

Original entry on oeis.org

7, 15, 23, 31, 39, 47, 55, 377, 385, 393, 401, 409, 417, 425, 433, 439, 440, 441, 447, 448, 449, 455, 456, 457, 463, 464, 465, 471, 472, 473, 479, 480, 481, 487, 488, 489, 495, 503, 511, 519, 527, 535, 543, 551

Offset: 1

Clark Kimberling, Aug 24 2011

nonn

See A194368.

Cf. A010472, A194368, A194398, A194399.

r = Sqrt[15]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]       (* A194398 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]       (* A194399 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t3, 1]]       (* A194400 *)

A020772 Decimal expansion of 1/sqrt(15).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(15) = 0.258198889747161125678617693318826640722194780352772721772504917740898872796... [Vladimir Joseph Stephan Orlovsky, May 30 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[N[1/Sqrt[15],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals 1/A010472 = A020760 * A020762. - R. J. Mathar, Nov 19 2024

cp's OEIS Frontend

A175578 Decimal expansion of the sum over the inverse icosahedral numbers.

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A040011 Continued fraction for sqrt(15).

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A041023 Denominators of continued fraction convergents to sqrt(15).

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A176104 Decimal expansion of sqrt(285).

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A377996 Decimal expansion of the dihedral angle, in radians, between triangular and square faces in a (small) rhombicosidodecahedron.

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Mathematica

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A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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A194398 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(15) and < > denotes fractional part.

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A194399 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(15) and < > denotes fractional part.

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A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).