A006566 -id:A006566 - cp's OEIS frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316

Offset: 1

Henry Bottomley, Apr 02 2001

Triangular numbers not == 0 (mod 3). - Amarnath Murthy, Nov 13 2005
Shallow diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003
Equals the triangular numbers convolved with [1, 7, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
a(n) is congruent to 1 (mod 9) for all n. The sequence of digital roots of the a(n) is A000012(n). The sequence of units' digits of the a(n) is period 20: repeat [1, 0, 8, 5, 1, 6, 0, 3, 5, 6, 6, 5, 3, 0, 6, 1, 5, 8, 0, 1]. - Ant King, Jun 18 2012
Divide each side of any triangle ABC with area (ABC) into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_(2n) on side a, and similarly for sides b and c. If the hexagon with area (Hex(n)) delimited by AA_n, AA_(n+1), BB_n, BB_(n+1), CC_n and CC_(n+1) cevians, we have a(n+1) = (ABC)/(Hex(n)) for n >= 1, (see link with java applet). - Ignacio Larrosa Cañestro, Jan 02 2015; edited by Wolfdieter Lang, Jan 30 2015
For the case n = 1 see the link for Marion's Theorem (actually Marion Walter's Theorem, see the Cugo et al, reference). Also, the generalization considered here has been called there (Ryan) Morgan's Theorem.  - Wolfdieter Lang, Jan 30 2015
Pollock states that every number is the sum of at most 11 terms of this sequence, but note that "1, 10, 28, 35, &c." has a typo (35 should be 55). - Michel Marcus, Nov 04 2017
a(n) is also the number of (nontrivial) paths as well as the Wiener sum index of the (n-1)-alkane graph. - Eric W. Weisstein, Jul 15 2021

T. D. Noe, Table of n, a(n) for n = 1..1000
Ignacio Larrosa Cañestro, Hexágono y estrella determinados por tres pares de cevianas simétricas, (java applet).
Al Cugo et al., Marion's theorem, The Mathematics Teacher 86 (1993) p. 619.
John Elias, Illustration of Initial Terms
F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
Eric Weisstein's World of Mathematics, Alkane Graph
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Marion's Theorem
Eric Weisstein's World of Mathematics, Wiener Sum Index
Index entries for two-way infinite sequences
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A027468, A081266, A190152.

List([1..50],n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019

[(2*n-1)^2+(n-1)*n/2: n in [1..50]]; // Vincenzo Librandi, Nov 18 2015

H := n -> simplify(1/hypergeom([-3*n,3*n+3,1],[3/2,2],3/4)); A060544 := n -> H(n-1); seq(A060544(i),i=1..19); # Peter Luschny, Jan 09 2012

Take[Accumulate[Range[150]], {1, -1, 3}] (* Harvey P. Dale, Mar 11 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* Harvey P. Dale, Mar 11 2013 *)
FoldList[#1 + #2 &, 1, 9 Range @ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(3 n - 1) (3 n - 2)/2, {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[Binomial[3 n - 1, 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[PolygonalNumber[3 n - 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)

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

[(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019

a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.
a(n) = a(n-1) + 9*(n-1) = A060543(n, 3) = A006566(n)/n.
a(n) = A025035(n)/A025035(n-1) = A027468(n-1) + 1 = A000217(3*n-2).
a(1-n) = a(n).
From Paul Barry, Mar 15 2003: (Start)
a(n) = C(n-1, 0) + 9*C(n-1, 1) + 9*C(n-1, 2); binomial transform of (1, 9, 9, 0, 0, 0, ...).
a(n) = 9*A000217(n-1) + 1.
G.f.: x*(1 + 7*x + x^2)/(1-x)^3. (End)
Narayana transform (A001263) of [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).
a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - Peter Luschny, Jan 09 2012
From Ant King, Jun 18 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 9.
a(n) = A000217(n) + 7*A000217(n-1) + A000217(n-2).
Sum_{n>=1} 1/a(n) = 2*Pi/(3*sqrt(3)) = A248897.
(End)
a(n) = (2*n-1)^2 + (n-1)*n/2. - Ivan N. Ianakiev, Nov 18 2015
a(n) = A101321(9,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: (2 + 9*x^2)*exp(x)/2 - 1. - G. C. Greubel, Mar 02 2019
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 11*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)
a(n) = A000567(n) + A005449(n-1) (see illustration in links). - John Elias, Nov 10 2020
a(n) = P(2*n,4)*P(3*n,3)/24 for n>=2, where P(s,k) = ((s - 2)*k^2 - (s - 4)*k)/2 is the k-th s-gonal number. - Lechoslaw Ratajczak, Jul 18 2021

Additional description from Terrel Trotter, Jr., Apr 06 2002

Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015

A006564 Icosahedral numbers: a(n) = n(5n^2 - 5*n + 2)/2.

Original entry on oeis.org

1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, 6384, 7890, 9616, 11577, 13788, 16264, 19020, 22071, 25432, 29118, 33144, 37525, 42276, 47412, 52948, 58899, 65280, 72106, 79392, 87153, 95404, 104160, 113436, 123247, 133608

Offset: 1

N. J. A. Sloane

Schlaefli symbol for this polyhedron: {3,5}.

One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012). - Daniel Forgues, May 14 2010

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 = 1..1000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., Vol. 131, No. 1 (2002), pp. 65-75.
Victor Meally, Letter to N. J. A. Sloane, no date.
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.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000566, A053012, A175578.

Cf. A000292 (tetrahedral numbers), A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers).

a006564 n = n * (5 * n * (n - 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013

[(5*n^3-5*n^2+2*n)/2: n in [1..100]] // Vincenzo Librandi, Nov 21 2010

A006564:=(1+8*z+6*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Table[n (5n^2-5n+2)/2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,12,48,124},40] (* Harvey P. Dale, May 26 2011 *)

a(n)=5*n^2*(n-1)/2+n \\ Charles R Greathouse IV, Oct 07 2015

a(n) = C(n+2,3) + 8*C(n+1,3) + 6*C(n,3).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) with a(0)=1, a(1)=12, a(2)=48, a(3)=124. - Harvey P. Dale, May 26 2011
G.f.: x*(6*x^2 + 8*x + 1)/(x-1)^4. - Harvey P. Dale, May 26 2011
a(n) = A006566(n) - A035006(n). - Peter M. Chema, May 04 2016
E.g.f.: x*(2 + 10*x + 5*x^2)*exp(x)/2. - Ilya Gutkovskiy, May 04 2016
Sum_{n>=1} 1/a(n) = A175578. - Amiram Eldar, Jan 03 2022

A007613 a(n) = (8^n + 2*(-1)^n)/3.

Original entry on oeis.org

1, 2, 22, 170, 1366, 10922, 87382, 699050, 5592406, 44739242, 357913942, 2863311530, 22906492246, 183251937962, 1466015503702, 11728124029610, 93824992236886, 750599937895082, 6004799503160662, 48038396025285290, 384307168202282326, 3074457345618258602

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Also, the cogrowth sequence of C3 X C3 = ; that is, the number of words of length 3n that reduce to the identity. - Sean A. Irvine, Nov 04 2024

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. S. Clark, Proof without words, Math. Mag., 63 (1990), 29.
Index entries for linear recurrences with constant coefficients, signature (7,8).

Cf. A001045, A006566, A078008, A082311, A139459.

[(8^n + 2*(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 14 2011

LinearRecurrence[{7,8}, {1,2}, 41] (* G. C. Greubel, Apr 23 2023 *)

a(n)=(8^n + 2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 06 2011

Vec((5*x-1)/((x+1)*(8*x-1)) + O(x^50)) \\ Colin Barker, Sep 29 2014

[(8^n -4*(n%2) +2)/3 for n in range(41)] # G. C. Greubel, Apr 23 2023

a(n) = A078008(3*n). - Paul Barry, Nov 29 2003
From Paul Barry, Mar 24 2004: (Start)
a(n) = (A082311(n) + (-1)^n)/2.
a(n) = (A001045(3*n+1) + (-1)^n)/2. (End)
a(n) = Sum_{k=0..n} binomial(3*n, 3*k). - Paul Barry, Jan 13 2005
a(n) = 8*a(n-1) + 6*(-1)^n. - Paul Curtz, Nov 19 2007
From Colin Barker, Sep 29 2014: (Start)
a(n) = 7*a(n-1) + 8*a(n-2).
G.f.: (1-5*x) / ((1+x)*(1-8*x)). (End)
E.g.f.: (1/3)*(exp(8*x) + 2*exp(-x)). - G. C. Greubel, Apr 23 2023

More terms from Colin Barker, Sep 29 2014

A060539 Table by antidiagonals of number of ways of choosing k items from n*k.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 20, 15, 4, 1, 70, 84, 28, 5, 1, 252, 495, 220, 45, 6, 1, 924, 3003, 1820, 455, 66, 7, 1, 3432, 18564, 15504, 4845, 816, 91, 8, 1, 12870, 116280, 134596, 53130, 10626, 1330, 120, 9, 1, 48620, 735471, 1184040, 593775, 142506, 20475, 2024, 153, 10

Offset: 1

Henry Bottomley, Apr 02 2001

			Square array A(n,k) begins:
  1,  1,    1,     1,      1,       1,        1, ...
  2,  6,   20,    70,    252,     924,     3432, ...
  3, 15,   84,   495,   3003,   18564,   116280, ...
  4, 28,  220,  1820,  15504,  134596,  1184040, ...
  5, 45,  455,  4845,  53130,  593775,  6724520, ...
  6, 66,  816, 10626, 142506, 1947792, 26978328, ...
  7, 91, 1330, 20475, 324632, 5245786, 85900584, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened (first 20 antidiagonals from Harry J. Smith)
H. J. Brothers, Pascal's Prism: Supplementary Material.

Rows include A000012, A000984, A005809, A005810, A001449, A004355, A004368.
Columns include A000027, A000384, A006566, A060541.
Main diagonal is A014062.
Cf. A295772.

A:= (n, k)-> binomial(n*k, k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);  # Alois P. Heinz, Jul 28 2023

{ i=0; for (m=1, 20, for (n=1, m, k=m - n + 1; write("b060539.txt", i++, " ", binomial(n*k, k))); ) } \\ Harry J. Smith, Jul 06 2009

A(n,k) = binomial(n*k,k) = A007318(n*k,k) = A060538(n,k)/A060538(n-1,k).

A035006 Number of possible rook moves on an n X n chessboard.

Original entry on oeis.org

0, 8, 36, 96, 200, 360, 588, 896, 1296, 1800, 2420, 3168, 4056, 5096, 6300, 7680, 9248, 11016, 12996, 15200, 17640, 20328, 23276, 26496, 30000, 33800, 37908, 42336, 47096, 52200, 57660, 63488, 69696, 76296, 83300, 90720, 98568, 106856

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

Obviously A035005(n) = A002492(n-1) + a(n) since Queen = Bishop + Rook. - Johannes W. Meijer, Feb 04 2010

X values of solutions of the equation: (X-Y)^3-2*X*Y=0. Y values are b(n)=2*n*(n-1)^2 (see A181617). - Mohamed Bouhamida, Jul 06 2023

			On a 3 X 3-board, rook has 9*4 moves, so a(3)=36.

E. Bonsdorff, K. Fabel and O. Riihimaa, Schach und Zahl (Chess and numbers), Walter Rau Verlag, Dusseldorf, 1966.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Alexander M. Haupt, Bijective enumeration of rook walks, arXiv:2007.01018 [math.CO], 2020.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
Richard P. Stanley, Bijective Proof Problems, Problem 540 p. 63, (2015).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A033586 (King), A035005 (Queen), A035008 (Knight), A002492 (Bishop) and A049450 (Pawn).

Cf. A006002, A006564, A006566.

[(n-1)*2*n^2: n in [1..40]]; // Vincenzo Librandi, Jun 16 2011

Table[(n-1) 2 n^2,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,8,36,96},40] (* Harvey P. Dale, May 12 2012 *)

a(n) = (n-1)*2*n^2.
a(n) = Sum_{j=1..n} ((n+j-1)^2 - (n-j+1)^2). - Zerinvary Lajos, Sep 13 2006
1/a(n+1) = Integral_{x=1/(n+1)..1/n} x*h(x) = Integral_{x=1/(n+1)..1/n} x*(1/x - floor(1/x)) = 1/((2*(n^2+2*n+1))*n) and Sum_{n>=1} 1/((2*(n^2+2*n+1))*n) = 1-Zeta(2)/2 where h(x) is the Gauss (continued fraction) map h(x)={x^-1} and {x} is the fractional part of x. - Stephen Crowley, Jul 24 2009
a(n) = 4 * A006002(n-1). - Johannes W. Meijer, Feb 04 2010
G.f.: 4*x^2*(2+x)/(1-x)^4. - Colin Barker, Mar 11 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=0, a(2)=8, a(3)=36, a(4)=96. - Harvey P. Dale, May 12 2012
a(n) = A006566(n) - A006564(n). - Peter M. Chema, Feb 10 2016
E.g.f.: 2*exp(x)*x^2*(2 + x). - Stefano Spezia, May 10 2022
From Amiram Eldar, May 14 2022: (Start)
Sum_{n>=2} 1/a(n) = 1 - Pi^2/12.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/24 + log(2) - 1. (End)

A005904 Centered dodecahedral numbers.

Original entry on oeis.org

1, 33, 155, 427, 909, 1661, 2743, 4215, 6137, 8569, 11571, 15203, 19525, 24597, 30479, 37231, 44913, 53585, 63307, 74139, 86141, 99373, 113895, 129767, 147049, 165801, 186083, 207955, 231477, 256709, 283711, 312543, 343265, 375937, 410619, 447371, 486253, 527325

Offset: 0

N. J. A. Sloane

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

Amiram Eldar, Table of n, a(n) for n = 0..10000
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.
Boon K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558; alternative link.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A001620, A006566.

A005904:=(z+1)*(z**2+28*z+1)/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation.]

a[n_] := (2*n + 1) * (5*n^2 + 5*n + 1); Array[a, 30, 0] (* Amiram Eldar, Sep 12 2022 *)

a(n) = (2*n+1)*(5*n^2+5*n+1).
Sum_{n>=0} 1/a(n) = -psi((5+sqrt(5))/10) - psi((5-sqrt(5))/10) - 2*gamma - 4*log(2), where psi is the digamma function and gamma is Euler's constant (A001620). - Amiram Eldar, Sep 12 2022
E.g.f.: exp(x)*(1 + 32*x + 45*x^2 + 10*x^3). - Stefano Spezia, Jun 06 2025

A228888 a(n) = binomial(3*n + 2, 3).

Original entry on oeis.org

10, 56, 165, 364, 680, 1140, 1771, 2600, 3654, 4960, 6545, 8436, 10660, 13244, 16215, 19600, 23426, 27720, 32509, 37820, 43680, 50116, 57155, 64824, 73150, 82160, 91881, 102340, 113564, 125580, 138415, 152096, 166650, 182104, 198485, 215820, 234136, 253460

Offset: 1

Peter Bala, Sep 09 2013

			From _Bruno Berselli_, Jun 26 2018: (Start)
Including 0, row sums of the triangle:
| 0|   .................................................................. 0
| 1|   2   3   4   ..................................................... 10
| 5|   6   7   8   9  10  11   ......................................... 56
|12|  13  14  15  16  17  18  19  20  21   ............................ 165
|22|  23  24  25  26  27  28  29  30  31  32  33  34   ................ 364
|35|  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50   .... 680
...
in the first column of which we have the pentagonal numbers (A000326).
(End)

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 A228887 (binomial(3*n + 1,3)).
Cf. A228889.
Similar sequences are listed in A316224.

[Binomial(3*n + 2, 3): n in [1..40]]; // Vincenzo Librandi, Sep 09 2013

Maple
```
seq(binomial(3*n+2,3), n = 1..38);
```

Table[(Binomial[3 n + 2, 3]), {n, 1, 40}] (* Vincenzo Librandi, Sep 09 2013 *)

a(n) = binomial(3*n + 2, 3) = 1/6*(3*n)*(3*n + 1)*(3*n + 2).
a(-n) = - A006566(n).
a(n) = 1/6*A228889(n).
G.f.: (10*x + 16*x^2 + x^3)/(1 - x)^4 = 10*x + 56*x^2 + 165*x^3 + ....
Sum {n >= 1} 1/a(n) = 9/2 - 3/2*log(3) - 1/2*sqrt(3)*Pi.
Sum {n >= 1} (-1)^n/a(n) = 9/2 - 4*log(2) - 1/3*sqrt(3)*Pi.

A054776 a(n) = 3n(3n-1)(3*n-2).

Original entry on oeis.org

0, 6, 120, 504, 1320, 2730, 4896, 7980, 12144, 17550, 24360, 32736, 42840, 54834, 68880, 85140, 103776, 124950, 148824, 175560, 205320, 238266, 274560, 314364, 357840, 405150, 456456, 511920, 571704, 635970, 704880, 778596, 857280, 941094

Offset: 0

Henry Bottomley, May 19 2000

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 46.
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 268.

Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A006566, A007531, A097321.

A054776:=n->3*n*(3*n-1)*(3*n-2): seq(A054776(n), n=0..50); # Wesley Ivan Hurt, Apr 14 2017

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

a(n) = A007531(3n-2) = 6*A006566(n).
Sum_{n>=1} 1/a(n) = Pi*sqrt(3)/12 - log(3)/4 = 0.178796768891527... [Jolley eq. 250]. - Benoit Cloitre, Apr 05 2002
G.f.: 6*x*(1+16*x+10*x^2)/(1-x)^4.
E.g.f.: 3*exp(x)*x*(2 + 18x + 9x^2). - Indranil Ghosh, Apr 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - Pi/(6*sqrt(3)). - Amiram Eldar, Mar 08 2022

A124388 a(n) = 27*n + 18.

Original entry on oeis.org

18, 45, 72, 99, 126, 153, 180, 207, 234, 261, 288, 315, 342, 369, 396, 423, 450, 477, 504, 531, 558, 585, 612, 639, 666, 693, 720, 747, 774, 801, 828, 855, 882, 909, 936, 963, 990, 1017, 1044, 1071, 1098, 1125, 1152, 1179, 1206, 1233, 1260, 1287, 1314, 1341, 1368

Offset: 0

Reinhard Zumkeller, Oct 30 2006

Second differences of dodecahedral numbers (A006566).

Also, first differences of dodecahedral gnomic numbers (A093485); a(n+1) - a(n) = 27.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006566, A016789, A093485.

[27*n+18: n in [0..50]]; // Vincenzo Librandi, Sep 29 2011

Range[18, 7000, 27] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)

for(n=0, 1e2, print1(27*n+18, ", ")) \\ Felix Fröhlich, Jul 07 2014

a(n) = 9*A016789(n).
G.f.: 9*(2 + x)/(x-1)^2. - R. J. Mathar, Jul 02 2011
From Elmo R. Oliveira, Dec 08 2024: (Start)
E.g.f.: 9*exp(x)*(2 + 3*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
Sum_{n>=0} (-1)^n/a(n) = (Pi/sqrt(3) - log(2))/27. - Amiram Eldar, May 11 2025

A060541 a(n) = binomial(4*n, 4).

Original entry on oeis.org

1, 70, 495, 1820, 4845, 10626, 20475, 35960, 58905, 91390, 135751, 194580, 270725, 367290, 487635, 635376, 814385, 1028790, 1282975, 1581580, 1929501, 2331890, 2794155, 3321960, 3921225, 4598126, 5359095, 6210820, 7160245, 8214570, 9381251, 10668000, 12082785

Offset: 1

Henry Bottomley, Apr 02 2001

Harry J. Smith, Table of n, a(n) for n=1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000027, A000332, A000384, A006566, A015219, A060539.

[Binomial(4 n, 4): n in [1..40]]; // Vincenzo Librandi, Jan 20 2015

Table[Binomial[4n, 4], {n, 100}] (* Wesley Ivan Hurt, Sep 27 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,70,495,1820,4845},40] (* Harvey P. Dale, Jan 13 2015 *)

a(n) = n*(2*n - 1)*(4*n - 1)*(4*n - 3)/3; \\ Harry J. Smith, Jul 06 2009

a(n) = n*(2n-1)*(4n-1)*(4n-3)/3.
a(n) = n * A015219(n-1) = A000332(4n) = A060539(n, 4).
G.f.: x*(1+65*x+155*x^2+35*x^3) / (1-x)^5. - R. J. Mathar, Oct 03 2011
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*log(2) - Pi.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*sqrt(2)*log(sqrt(2)-1) - log(2) + (2*sqrt(2) - 3/2)*Pi. (End)

Offset changed from 0 to 1 by Harry J. Smith, Jul 06 2009

cp's OEIS Frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

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A006564 Icosahedral numbers: a(n) = n*(5*n^2 - 5*n + 2)/2.

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A007613 a(n) = (8^n + 2*(-1)^n)/3.

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A060539 Table by antidiagonals of number of ways of choosing k items from n*k.

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A035006 Number of possible rook moves on an n X n chessboard.

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A005904 Centered dodecahedral numbers.

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A006564 Icosahedral numbers: a(n) = n(5n^2 - 5*n + 2)/2.

A054776 a(n) = 3n(3n-1)(3*n-2).