A228888 -id:A228888 - cp's OEIS frontend

A006566 Dodecahedral numbers: a(n) = n(3n - 1)(3n - 2)/2.

0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711, 45760, 52394, 59640, 67525, 76076, 85320, 95284, 105995, 117480, 129766, 142880, 156849, 171700, 187460, 204156

Offset: 0

N. J. A. Sloane

Schlaefli symbol for this polyhedron: {5,3}.
A093485 = first differences; A124388 = second differences; third differences = 27. - Reinhard Zumkeller, Oct 30 2006
One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010
From Peter Bala, Sep 09 2013: (Start)
a(n) = binomial(3*n,3). Two related sequences are binomial(3*n+1,3) (A228887) and binomial(3*n+2,3) (A228888). The o.g.f.'s for these three sequences are rational functions whose numerator polynomials are obtained from the fourth row [1, 4, 10, 16, 19, 16, 10, 4, 1] of the triangle of trinomial coefficients A027907 by taking every third term:
Sum_{n >= 1} binomial(3*n,3)*x^n = (x + 16*x^2 + 10*x^3)/(1-x)^4;
Sum_{n >= 1} binomial(3*n+1,3)*x^n = (4*x + 19*x^2 + 4*x^3)/(1-x)^4;
Sum_{n >= 1} binomial(3*n+2,3)*x^n = (10*x + 16*x^2 + x^3)/(1-x)^4. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Victor Meally, Letter to N. J. A. Sloane, no date.
Ed Pegg Jr, Dodecahedral 2024.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A027907, A035006, A060544, A093485, A124388, A228887, A228888, A295421.

Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006564 (icosahedral numbers).

a006566 n = n * (3 * n - 1) * (3 * n - 2) `div` 2
a006566_list = scanl (+) 0 a093485_list  -- Reinhard Zumkeller, Jun 16 2013

[n*(3*n-1)*(3*n-2)/2: n in [0..40]]; // Vincenzo Librandi, Dec 11 2015

A006566:=(1+16*z+10*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n(3n-1)(3n-2)/2,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,20,84},40] (* Harvey P. Dale, Jul 24 2013 *)
CoefficientList[Series[x (1 + 16 x + 10 x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 11 2015 *)

PARI
```
a(n)=n*(3*n-1)*(3*n-2)/2
```

G.f.: x(1 + 16x + 10x^2)/(1 - x)^4.
a(n) = A000292(3n-3) = A054776(n)/6 = n*A060544(n).
a(n) = C(n+2,3) + 16 C(n+1,3) + 10 C(n,3).
a(0)=0, a(1)=1, a(2)=20, a(3)=84, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jul 24 2013
a(n) = binomial(3*n,3). a(-n) = - A228888(n). Sum_{n>=1} 1/a(n) = 1/2*( sqrt(3)*Pi - 3*log(3) ). Sum_{n>=1} (-1)^n/a(n) = 1/3*sqrt(3)*Pi - 4*log(2). - Peter Bala, Sep 09 2013
a(n) = A006564(n) + A035006(n). - Peter M. Chema, May 04 2016
E.g.f.: x*(2 + 18*x + 9*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016
From Amiram Eldar, Jan 09 2024: (Start)
Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi - 3*log(3))/2 (A295421).
Sum_{n>=1} (-1)^(n+1)/a(n) = (12*log(2) - sqrt(3)*Pi)/3. (End)

More terms from Henry Bottomley, Nov 23 2001

A228887 a(n) = binomial(3*n + 1,3).

Original entry on oeis.org

4, 35, 120, 286, 560, 969, 1540, 2300, 3276, 4495, 5984, 7770, 9880, 12341, 15180, 18424, 22100, 26235, 30856, 35990, 41664, 47905, 54740, 62196, 70300, 79079, 88560, 98770, 109736, 121485, 134044, 147440, 161700, 176851, 192920, 209934, 227920, 246905

Offset: 1

Peter Bala, Sep 09 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006566 (binomial(3*n,3)) and A228888 (binomial(3*n + 2,3)).

[Binomial(3*n+1,3): n in [1..40]]; // Vincenzo Librandi, Sep 10 2013

Maple
```
seq(binomial(3*n+1,3), n = 1..38);
```

Table[(Binomial[3n + 1, 3]), {n, 40}] (* Vincenzo Librandi, Sep 10 2013 *)
LinearRecurrence[{4,-6,4,-1},{4,35,120,286},40] (* Harvey P. Dale, Jan 11 2015 *)

a(n) = -a(-n) = binomial(3*n + 1,3) = 1/6*(3*n + 1)*(3*n)*(3*n - 1).
G.f.: x*(4 + 19*x + 4*x^2)/(1 - x)^4 = 4*x + 35*x^2 + 120*x^3 + ....
Sum_{n>=1} 1/a(n) = 3*log(3) - 3.
Sum_{n>=1} (-1)^n/a(n) = 4*log(2) - 3.
E.g.f.: exp(x)*x*(8 + 27*x + 9*x^2)/2. - Stefano Spezia, Sep 20 2024

A316224 a(n) = n(2n + 1)(4n + 1).

Original entry on oeis.org

0, 15, 90, 273, 612, 1155, 1950, 3045, 4488, 6327, 8610, 11385, 14700, 18603, 23142, 28365, 34320, 41055, 48618, 57057, 66420, 76755, 88110, 100533, 114072, 128775, 144690, 161865, 180348, 200187, 221430, 244125, 268320, 294063, 321402, 350385, 381060, 413475, 447678, 483717

Offset: 0

Bruno Berselli, Jun 27 2018

Sums of the consecutive integers from A000384(n) to A000384(n+1)-1. This is the case s=6 of the formula n*(n*(s-2) + 1)*(n*(s-2) + 2)/2 related to s-gonal numbers.

The inverse binomial transform is 0, 15, 60, 48, 0, ... (0 continued).

			Row sums of the triangle:
|  0 |  ................................................................. 0
|  1 |  2  3  4  5  .................................................... 15
|  6 |  7  8  9 10 11 12 13 14  ........................................ 90
| 15 | 16 17 18 19 20 21 22 23 24 25 26 27  ........................... 273
| 28 | 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44  ............... 612
| 45 | 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65  .. 1155
...
where:
. first column is A000384,
. second column is A130883 (without 1),
. third column is A033816,
. diagonal is A014106,
. 0, 2, 8, 18, 32, 50, ... are in A001105.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First bisection of A059270 and subsequence of A034828, A047866, A109900, A290168.

Sums of the consecutive integers from P(s,n) to P(s,n+1)-1, where P(s,k) is the k-th s-gonal number: A027480 (s=3), A055112 (s=4), A228888 (s=5).

GAP
```
List([0..40], n -> n*(2*n+1)*(4*n+1));
```

[n*(2*n+1)*(4*n+1) for n in 0:40] |> println

Magma
```
[n*(2*n+1)*(4*n+1): n in [0..40]];
```

seq(n*(2*n+1)*(4*n+1),n=0..40); # Muniru A Asiru, Jun 27 2018

Table[n (2 n + 1) (4 n + 1), {n, 0, 40}]

Maxima
```
makelist(n*(2*n+1)*(4*n+1), n, 0, 40);
```
PARI
```
vector(40, n, n--; n*(2*n+1)*(4*n+1))
```
Python
```
[n*(2*n+1)*(4*n+1) for n in range(40)]
```
Sage
```
[n*(2*n+1)*(4*n+1) for n in (0..40)]
```

O.g.f.: 3*x*(5 + 10*x + x^2)/(1 - x)^4.
E.g.f.: x*(15 + 30*x + 8*x^2)*exp(x).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) =  3*A258582(n).
a(n) = -3*A100157(-n).
Sum_{n>0} 1/a(n) = 2*(3 - log(4)) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) + 2*sqrt(2)*log(1+sqrt(2)) + (sqrt(2)-1/2)*Pi - 6. - Amiram Eldar, Sep 17 2022

A228889 a(n) = 3n(3n + 1)(3*n + 2).

Original entry on oeis.org

60, 336, 990, 2184, 4080, 6840, 10626, 15600, 21924, 29760, 39270, 50616, 63960, 79464, 97290, 117600, 140556, 166320, 195054, 226920, 262080, 300696, 342930, 388944, 438900, 492960, 551286, 614040, 681384, 753480, 830490, 912576, 999900, 1092624, 1190910

Offset: 1

Peter Bala, Sep 09 2013

Related sequences are A054776 and A097321.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A054776, A097321, A228888.

[3*n*(3*n+1)*(3*n+2): n in [1..40]]; // Vincenzo Librandi, Sep 10 2013

Maple
```
seq(3*n*(3*n+1)*(3*n+2), n = 1..35);
```

CoefficientList[Series[6 (10 + 16 x + x^2)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 10 2013 *)
Table[Times@@(3n+{0,1,2}),{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{60,336,990,2184},40] (* Harvey P. Dale, Dec 20 2023 *)

a(n) = 3*n*(3*n + 1)*(3*n + 2) = 6*binomial(3*n + 2,3) = 6*A228888(n).
a(-n) = - A054776(n).
G.f.: 6*x*(10 + 16*x + x^2)/(1 - x)^4 = 60*x + 336*x^2 + 990*x^3 + ....
Sum {n >= 1} 1/a(n) = 3/4 - log(3)/4 - 1/12*sqrt(3)*Pi;
Sum {n >= 1} (-1)^n/a(n) = 3/4 - 2/3*log(2) - 1/18*sqrt(3)*Pi.

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).

Original entry on oeis.org

1, 10, 1, 28, 56, 1, 55, 462, 165, 1, 91, 2002, 3003, 364, 1, 136, 6188, 24310, 12376, 680, 1, 190, 15504, 125970, 167960, 38760, 1140, 1, 253, 33649, 490314, 1352078, 817190, 100947, 1771, 1, 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1

Offset: 1

Peter Luschny, Mar 31 2023

			Table T(n, k) starts:
  [1]   1;
  [2]  10,     1;
  [3]  28,    56,       1;
  [4]  55,   462,     165,       1;
  [5]  91,  2002,    3003,     364,       1;
  [6] 136,  6188,   24310,   12376,     680,       1;
  [7] 190, 15504,  125970,  167960,   38760,    1140,      1;
  [8] 253, 33649,  490314, 1352078,  817190,  100947,   1771,    1;
  [9] 325, 65780, 1562275, 7726160, 9657700, 3124550, 230230, 2600, 1.

Cf. A082365 (row sums), A228888 (subdiagonal), A060544 (column 1), A066802 (central column).

T := (n, k) -> binomial(3*n - 1, 3*k - 1):
seq(print(seq(T(n, k), k = 1..n)), n = 1..8);

cp's OEIS Frontend

A006566 Dodecahedral numbers: a(n) = n*(3*n - 1)*(3*n - 2)/2.

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A228887 a(n) = binomial(3*n + 1,3).

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A316224 a(n) = n*(2*n + 1)*(4*n + 1).

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Julia

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A228889 a(n) = 3*n*(3*n + 1)*(3*n + 2).

Views

Author

Keywords

Comments

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Crossrefs

Programs

Magma

Maple

Mathematica

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A361949 Triangle read by rows. T(n, k) = binomial(3*n - 1, 3*k - 1).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A006566 Dodecahedral numbers: a(n) = n(3n - 1)(3n - 2)/2.

A316224 a(n) = n(2n + 1)(4n + 1).

A228889 a(n) = 3n(3n + 1)(3*n + 2).

A361949 Triangle read by rows. T(n, k) = binomial(3n - 1, 3k - 1).