A020793 -id:A020793 - cp's OEIS frontend

A008588 Nonnegative multiples of 6.

0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348

Offset: 0

N. J. A. Sloane

For n > 3, the number of squares on the infinite 3-column half-strip chessboard at <= n knight moves from any fixed point on the short edge.
Second differences of A000578. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A001018(n). - Reinhard Zumkeller, Feb 24 2009
These numbers can be written as the sum of four cubes (i.e., 6*n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3). - Arkadiusz Wesolowski, Aug 09 2013
A122841(a(n)) > 0 for n > 0. - Reinhard Zumkeller, Nov 10 2013
Surface area of a cube with side sqrt(n). - Wesley Ivan Hurt, Aug 24 2014
a(n) is representable as a sum of three but not two consecutive nonnegative integers, e.g., 6 = 1 + 2 + 3, 12 = 3 + 4 + 5, 18 = 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016 (Corrected by David A. Corneth, Aug 12 2016)
Numbers with three consecutive divisors: for some k, each of k, k+1, and k+2 divide n. - Charles R Greathouse IV, May 16 2016
Numbers k for which {phi(k),phi(2k),phi(3k)} is an arithmetic progression. - Ivan Neretin, Aug 12 2016

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 81.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 318
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Essentially the same as A008458.
Cf. A016921, A016933, A016945, A016957, A016969, A138591.
Cf. A044102 (subsequence).

a008588 = (* 6)
a008588_list = [0, 6 ..]  -- Reinhard Zumkeller, Nov 10 2013

[6*n: n in [0..60] ]; // Vincenzo Librandi, Jul 16 2011

Maple
```
[ seq(6*n,n=0..45) ];
```

Range[0, 500, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
NestList[6+#&,0,60] (* Harvey P. Dale, Nov 17 2024 *)

makelist(6*n,n,0,30); /* Martin Ettl, Nov 12 2012 */

a(n)=6*n \\ Charles R Greathouse IV, Feb 08 2012

[6*n for n in range(59)] # Stefano Spezia, Mar 09 2025

From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 6*n = 2*a(n-1) - a(n-2).
G.f.: 6*x/(1-x)^2. (End)
a(n) = Sum_{k>=0} A030308(n,k)*6*2^k. - Philippe Deléham, Oct 24 2011
a(n) = Sum_{k=2n-1..2n+1} k. - Wesley Ivan Hurt, Nov 22 2015
From Ilya Gutkovskiy, Aug 12 2016: (Start)
E.g.f.: 6*x*exp(x).
Convolution of A010722 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/6 = A002162*A020793. (End)
a(n) = 6 * A001477(n). - David A. Corneth, Aug 12 2016

A364895 Decimal expansion of the 4-volume of the unit regular pentachoron (5-cell).

Original entry on oeis.org

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

Offset: 0

Jianing Song, Aug 12 2023

Decimal expansion of sqrt(5)/96.

In general, the n-volume of the unit regular n-simplex is sqrt(n+1)/(n!*2^(n/2)).

			Equals 0.02329237476562280933...

Wikipedia, 5-cell
Polytope Wiki, Pentachoron

Decimal expansion of 4-volumes: this sequence (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), A364897 (600-cell).

Decimal expansion of the n-volume of the unit regular n-simplex: A120011 (n=2), A020829 (n=3), this sequence (n=4).

First[RealDigits[Sqrt[5]/96, 10, 100, -1]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
sqrt(5)/96
```

A177735 a(0)=1, a(n)=A002445(n)/6 for n>=1.

Original entry on oeis.org

1, 1, 5, 7, 5, 11, 455, 1, 85, 133, 55, 23, 455, 1, 145, 2387, 85, 1, 319865, 1, 2255, 301, 115, 47, 7735, 11, 265, 133, 145, 59, 9464455, 1, 85, 10787, 5, 781, 23350145, 1, 5, 553, 38335, 83, 567385, 1, 10235, 45353, 235, 1, 750295, 1, 5555, 721, 265, 107

Offset: 0

Paul Curtz, May 12 2010

nonn

For n>=1: denominators of the Bernoulli numbers (A002445) divided by 6.
All entries are odd.
a(n)= A002445(n) / A020793(n).
5 divides a(2*n) for n>=1.
These numbers also equal to the lengths of the repeating patterns for the excluded integer values of c/6, when both p^n + c and p^n - c are prime, for an infinite number of primes p>2, and a given integer n>0, arising from the union of one or more prime-based modulo cycles, determined by the divisors of n. See A005097 for details and connection to the von Staudt-Clausen Theorem below. - Richard R. Forberg, Jul 19 2016

Harvey P. Dale, Table of n, a(n) for n = 0..1000
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304062, 1993.
Eric Weisstein's World of Mathematics, von Staudt-Clausen Theorem

Cf. A002445, A165949.

A002445 := proc(n) bernoulli(2*n) ; denom(%) ; end proc:
A177735 := proc(n) if n = 0 then 1; else A002445(n)/6 ; end if; end proc:
seq(A177735(n),n=0..60) ; # R. J. Mathar, Aug 15 2010

Join[{1},Denominator[BernoulliB[Range[2,120,2]]]/6] (* Harvey P. Dale, Oct 19 2012 *)
result = {}; Do[prod = 1; Do[If[PrimeQ[2*Divisors[n][[i]] + 1], prod *= (2*Divisors[n][[i]] + 1)], {i, 2, Length[Divisors[n]]}];
AppendTo[result, prod] , {n, 1, 100}]  ; result (* Richard R. Forberg, Jul 19 2016 *)

a(n)=
{
    my(bd=1);
    forprime (p=5, 2*n+1, if( (2*n)%(p-1)==0, bd*=p ) );
    bd;
}
/* Joerg Arndt, May 06 2012 */

a(n)=if(n<2, return(1)); my(s=1); fordiv(n,d, if(isprime(2*d+1) && d>1, s *= 2*d+1)); s \\ Charles R Greathouse IV, Jul 20 2016

def A177735(n):
    if n == 0: return 1
    M = map(lambda i: i+1, divisors(2*n))
    return mul(filter(lambda s: is_prime(s), M))//6
print([A177735(n) for n in (0..53)]) # Peter Luschny, Feb 20 2016

a(n) = denominator(BernoulliB(2*n, 1/2))/(3*2^(2*n)). - Jean-François Alcover, Apr 16 2013

A simple direct calculation of the denominators, for n>=1, is based on the von Staudt-Clausen Theorem: Product{d|n}(2d+1), for d>1 and 2d+1 prime. See in the Mathematica section below. - Richard R. Forberg, Jul 19 2016

A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 04 2021

The moments of inertia of the five Platonic solids were apparently first calculated by the Canadian physicist John Satterly (1879-1963) in 1957.
The moment of inertia of a solid regular dodecahedron with a uniform mass density distribution, mass M, and edge length L is I = c*M*L^2, where c is this constant.
The corresponding values of c for the other Platonic solids are:
Tetrahedron: 1/20 (= A020761/10).
Octahedron: 1/10 (= A000007).
Cube: 1/6 (= A020793).
Icosahedron: (3 + sqrt(5))/20 (= A104457/10).

			0.60735550374163932719985924360173257727394705341616...

P. K. Aravind, Gravitational collapse and moment of inertia of regular polyhedral configurations, American Journal of Physics, Vol. 59, No. 7 (1991), pp. 647-652.
Frédéric Perrier, Moments of inertia of Archimedean solids, 2015.
John Satterly, Moments of Inertia about Selected Axes of Regular Polygons, Right Pyramids on Regular Polygonal Bases, and of the Platonic and Some Archimedian Polyhedra, American Journal of Physics, Vol. 25, No. 7 (1957), pp. 489-490.
John Satterly, The Moments of Inertia of Some Polyhedra, The Mathematical Gazette, Vol. 42, No. 339 (1958), pp. 11-13.
Wikipedia, List of moments of inertia.
Wikipedia, Moment of inertia.
Wikipedia, Regular dodecahedron.
Index entries for sequences related to moment of inertia.

Cf. A000007, A001622, A020761, A020793, A104457.

Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798.

RealDigits[(95 + 39*Sqrt[5])/300, 10, 100][[1]]

Equals (95 + 39*sqrt(5))/300.

Equals (28 + 39*phi)/150, where phi is the golden ratio (A001622).

A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

Original entry on oeis.org

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

Offset: 3

Jianing Song, Aug 12 2023

Decimal expansion of (1575+705*sqrt(5))/4.

			Equals 787.85698103433793399211...

Polytope Wiki, Hecatonicosachoron
Wikipedia, 120-cell

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), this sequence (120-cell), A364897 (600-cell).

Cf. A102769 (decimal expansion of the volume of the unit regular dodecahedron).

First[RealDigits[(1575 + 705*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(1575+705*sqrt(5))/4
```

A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

Original entry on oeis.org

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

Offset: 2

Jianing Song, Aug 12 2023

Decimal expansion of (50+25*sqrt(5))/4.

			Equals 26.47542485937368560255...

Wikipedia, 600-cell
Polytope Wiki, Hexacosichoron

Decimal expansion of 4-volumes: A364895 (5-cell), A000007 = 1 (8-cell or tesseract), A020793 = 1/6 (16-cell), A000038 = 2 (24-cell), A364896 (120-cell), this sequence (600-cell).

Cf. A102208 (decimal expansion of the volume of the unit regular icosahedron).

First[RealDigits[(50 + 25*Sqrt[5])/4, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

PARI
```
(50+25*sqrt(5))/4
```

A021019 Decimal expansion of 1/15.

Original entry on oeis.org

0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

N. J. A. Sloane

			0.06666666666666666666666666666666666666666666666666666...

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

Cf. A010722, A020793.

From Elmo R. Oliveira, Aug 05 2024: (Start)
G.f.: 6*x/(1-x).
E.g.f.: 6*(exp(x) - 1).
a(n) = 6 for n >= 1. (End)

A234255 Decimal expansion of -B(12) = 691/2730, 13th Bernoulli number without sign.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Dec 22 2013

Essentially of period 6: repeat [5, 3, 1, 1, 3, 5] = A110551(n+3).
691*3663 = 2531133. See A021277.
Seventh part of the constant c=0.6323809537553113569215686274509803711... .
B(24) - B(12) = -86580. See A002882.

			0.2531135531135531135531135531135531135531135...

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. (A027641 or A164555)/A027642, A000367/A002445, A020793, A021046, A234355, A234356, A112828. B(24).

Cf. A002882, A110551, A021277.

[0,2] cat &cat [[5, 3, 1, 1, 3, 5]^^30]; // Wesley Ivan Hurt, Jun 28 2016

A234255:=n->[5, 3, 1, 1, 3, 5][(n mod 6)+1]: 0,2,seq(A234255(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

Join[{0},RealDigits[-BernoulliB[12],10,120][[1]]] (* or *) PadRight[{0,2}, 120, {3,5,5,3,1,1}] (* Harvey P. Dale, Dec 30 2013 *)

default(realprecision, 120);
-bernfrac(12) + 0. \\ Rick L. Shepherd, Jan 15 2014

From Chai Wah Wu, Jun 04 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) for n > 5.
G.f.: x^2*(2 + x - 3*x^2 + 3*x^3)/((1 - x)*(1 - x + x^2)). (End)
From Wesley Ivan Hurt, Jun 28 2016: (Start)
a(n) = a(n-6) for n>8.
a(n) = (9 - 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/3 for n>2. (End)

Offset corrected by and more terms from Rick L. Shepherd, Jan 15 2014

A271427 a(n) = 7^n - a(n-1) for n>0, a(0)=0.

Original entry on oeis.org

0, 7, 42, 301, 2100, 14707, 102942, 720601, 5044200, 35309407, 247165842, 1730160901, 12111126300, 84777884107, 593445188742, 4154116321201, 29078814248400, 203551699738807, 1424861898171642, 9974033287201501, 69818233010410500, 488727631072873507, 3421093417510114542

Offset: 0

Ilya Gutkovskiy, Apr 13 2016

In general, the ordinary generating function for the recurrence b(n) = k^n - b(n-1), where n>0 and b(0)=0, is k*x/((1 + x)*(1 - k*x)). This recurrence gives the closed form b(n) = k*(k^n - (-1)^n)/(k + 1).

			a(2) = 7^2 - a(2-1) = 49 - 7 = 42.
a(4) = 7^4 - a(4-1) = 2401 - 301 = 2100.

Index entries for linear recurrences with constant coefficients, signature (6,7).

Cf. A000420, A015552.

Cf. similar sequences with the recurrence b(n) = k^n - b(n-1): A125122 (k=1), A078008 (k=2), A054878 (k=3), A109499 (k=4), A109500 (k=5), A109501 (k=6), this sequence (k=7), A093134 (k=8), A001099 (k=n).

LinearRecurrence[{6, 7}, {0, 7}, 30]
Table[7 (7^n - (-1)^n)/8, {n, 0, 30}]

vector(50, n, n--; 7*(7^n-(-1)^n)/8) \\ Altug Alkan, Apr 13 2016

for n in range(0,10**2):print((int)((7*(7**n-(-1)**n))/8))
# Soumil Mandal, Apr 14 2016

O.g.f.: 7*x/(1 - 6*x - 7*x^2).
E.g.f.: (7/8)*(exp(7*x) - exp(-x)).
a(n) = 6*a(n-1) + 7*a(n-2).
a(n) = 7*(7^n - (-1)^n)/8.
a(n) = 7*A015552(n).
Sum_{n>0} 1/(a(n) + a(n-1)) = 1/6 = A020793.
Limit_{n->oo} a(n-1)/a(n) = 1/7 = A020806.

A274981 Decimal expansion of gamma(2) = 7/5.

Original entry on oeis.org

1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Natan Arie Consigli, Aug 31 2016

gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002.

Kshitij Education, Molar specific heat.
Wikipedia, Poisson's constant.

Cf. A020793 = gamma(1).

7/5 = (7/2 R)/(5/2 R) = Cp(2)/Cv(2) = A272003/A272002, with R = A081822 (or A070064).

cp's OEIS Frontend

A008588 Nonnegative multiples of 6.

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A364895 Decimal expansion of the 4-volume of the unit regular pentachoron (5-cell).

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A177735 a(0)=1, a(n)=A002445(n)/6 for n>=1.

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A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

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A364896 Decimal expansion of the 4-volume of the unit regular 120-cell.

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A364897 Decimal expansion of the 4-volume of the unit regular 600-cell.

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Mathematica

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A021019 Decimal expansion of 1/15.

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