A072065 -id:A072065 - cp's OEIS frontend

A008594 Multiples of 12.

0, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 36 ).
The positive terms are the differences of consecutive star numbers (A003154). - Mihir Mathur, Jun 07 2013
A089911(a(n)) = 0. - Reinhard Zumkeller, Jul 05 2013
a(1) = 12 is a primitive abundant number, thus all a(n), n >= 2, are nonprimitive abundant numbers. - Daniel Forgues, Sep 24 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 324.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Wikipedia, Star Number.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A072065 and A121032.

Cf. A003154, A008588, A008592, A008593, A008606, A089911.

a008594 = (* 12)
a008594_list = [0, 12 ..]  -- Reinhard Zumkeller, Dec 12 2012

[12*n: n in [0..50]]; // Vincenzo Librandi, Jun 11 2011

A008594:=n->12*n: seq(A008594(n), n=0..100); # Wesley Ivan Hurt, Sep 24 2016

12*Range[0,200] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
NestList[12+#&,0,60] (* Harvey P. Dale, Feb 02 2022 *)

a(n)=12*n \\ Charles R Greathouse IV, Apr 21 2015

From Vincenzo Librandi, Jun 11 2011: (Start)
a(n) = 12*n.
a(n) = 2*a(n-1) - a(n-2) for n > 1.
G.f.: 12*x/(1-x)^2. (End)
a(n) = A003154(n) - A003154(n-1). - Mihir Mathur, Jun 07 2013
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 12*x*exp(x).
a(n) = 2*A008588(n) = A008606(n)/2. (End)

A017629 a(n) = 12*n + 9.

Original entry on oeis.org

9, 21, 33, 45, 57, 69, 81, 93, 105, 117, 129, 141, 153, 165, 177, 189, 201, 213, 225, 237, 249, 261, 273, 285, 297, 309, 321, 333, 345, 357, 369, 381, 393, 405, 417, 429, 441, 453, 465, 477, 489, 501, 513, 525, 537, 549, 561, 573, 585, 597, 609, 621, 633

Offset: 0

N. J. A. Sloane

Numbers k such that k mod 2 = (k+1) mod 3 = 1 and (k+2) mod 4 != 1. - Klaus Brockhaus, Jun 15 2004
For n > 3, the number of squares on the infinite 3-column chessboard at <= n knight moves from any fixed point. - Ralf Stephan, Sep 15 2004
A016946 is the subsequence of squares (for n = 3*k*(k+1) = A028896(k), then a(n) = (6k+3)^2 = A016946(k)). - Bernard Schott, Apr 05 2021

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

Subsequences include A072065, A016945, A016813, A008585, and A005408.

Cf. A008594, A017533, A017545, A017137, A004767, A001019, A028896, A016946, A089911.

a017629 = (+ 9) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

12*Range[0,200]+9 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
LinearRecurrence[{2,-1},{9,21},60] (* Harvey P. Dale, Apr 14 2019 *)

a(n)=12*n+9 \\ Charles R Greathouse IV, Jul 10 2016

[i+9 for i in range(525) if gcd(i,12) == 12] # Zerinvary Lajos, May 21 2009

a(n) = 6*(4*n+1) - a(n-1) (with a(0)=9). - Vincenzo Librandi, Dec 17 2010
A089911(2*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
G.f.: (9 + 3*x)/(1 - x)^2. - Alejandro J. Becerra Jr., Jul 08 2020
Sum_{n>=0} (-1)^n/a(n) = (Pi + log(3-2*sqrt(2)))/(12*sqrt(2)). - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(3 + 4*x). - Stefano Spezia, Feb 25 2023

A017545 a(n) = 12*n + 2.

Original entry on oeis.org

2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 122, 134, 146, 158, 170, 182, 194, 206, 218, 230, 242, 254, 266, 278, 290, 302, 314, 326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554, 566, 578, 590, 602, 614, 626, 638

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 40 ).

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A017533, A017557.

Subsequence of A072065.

List([0..60], n-> 2*(6*n+1) ); # G. C. Greubel, Sep 18 2019

[12*n+2: n in [0..60]]; // Vincenzo Librandi, Jun 07 2011

A017545:=n->12*n+2: seq(A017545(n), n=0..60); # Wesley Ivan Hurt, Apr 27 2017

12*Range[0,60]+2 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=12*n+2 \\ Charles R Greathouse IV, Jul 10 2016

[2*(6*n+1) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 07 2011
From G. C. Greubel, Sep 18 2019: (Start)
G.f.: 2*(1 + 5*x)/(1-x)^2.
E.g.f.: 2*(1 + 6*x)*exp(x). (End)
Sum_{n>=0} (-1)^n/a(n) = Pi/12 + sqrt(3)*log(2 + sqrt(3))/12. - Amiram Eldar, Dec 12 2021

A017653 a(n) = 12*n + 11.

Original entry on oeis.org

11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, 143, 155, 167, 179, 191, 203, 215, 227, 239, 251, 263, 275, 287, 299, 311, 323, 335, 347, 359, 371, 383, 395, 407, 419, 431, 443, 455, 467, 479, 491, 503, 515, 527, 539, 551, 563, 575, 587, 599, 611, 623, 635

Offset: 0

N. J. A. Sloane

Or, with a different offset, 12*n - 1. In any case, numbers congruent to -1 (mod 12). - Alonso del Arte, May 29 2011

Numbers congruent to 2 (mod 3) and 3 (mod 4). - Bruno Berselli, Jul 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
John Elias, 60*n+55 Triangular Snowflakes.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1000.
Tanya Khovanova, Recursive Sequences.
Leo Tavares, Illustration: Twin Hexagonal Frames.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A003215, A008594, A016969, A017533, A017545, A089911.

Subsequence of A072065.

a017653 = (+ 11) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+11: n in [0..60]]; // Vincenzo Librandi, Jun 08 2011

Array[12*#+11&,100,0] (* Vladimir Joseph Stephan Orlovsky, Dec 14 2009 *)

PARI
```
a(n)=12*n+11
```

a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jun 08 2011
G.f.: (11+x)/(1-x)^2. - Colin Barker, Feb 19 2012
A089911(2*a(n)) = 11. - Reinhard Zumkeller, Jul 05 2013
a(n) = 2*A003215(n+1) - 1 - 2*A003215(n). See Twin Hexagonal Frames illustration. - Leo Tavares, Aug 19 2021
From Elmo R. Oliveira, Apr 12 2025: (Start)
E.g.f.: exp(x)*(11 +  12*x).
a(n) = A016969(2*n+1). (End)

A074229 Numbers n such that Kronecker(6,n)==mu(gcd(6,n)).

Original entry on oeis.org

1, 5, 19, 23, 25, 29, 43, 47, 49, 53, 67, 71, 73, 77, 91, 95, 97, 101, 115, 119, 121, 125, 139, 143, 145, 149, 163, 167, 169, 173, 187, 191, 193, 197, 211, 215, 217, 221, 235, 239, 241, 245, 259, 263, 265, 269, 283, 287, 289, 293, 307, 311, 313, 317, 331, 335

Offset: 1

Jon Perry, Sep 17 2002

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Equals 2 * A072065 + 1.

for (x=1,200, for (y=1,200,if (kronecker(x,y)==moebius(gcd(x,y)),write("km.txt",x,";",y," : ",kronecker(x,y)))))

isok(n) = kronecker(6, n) == moebius(gcd(6, n)); \\ Michel Marcus, Mar 17 2014

Vec(x*(1+4*x+14*x^2+4*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100)) \\ Colin Barker, Dec 14 2015

From Colin Barker, Dec 14 2015: (Start)
a(n) = (3/2+(3*i)/2)*(i^n-i*(-i)^n)-(-1)^n+6*(n+1)-9 where i = sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
G.f.: x*(1+4*x+14*x^2+4*x^3+x^4) / ((1-x)^2*(1+x)*(1+x^2)).
(End)

More terms from Michel Marcus, Mar 17 2014

A301752 Clique covering number of the n-triangular grid graph.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 13, 15, 19, 22, 26, 31, 35, 41, 46, 52, 58, 64, 71, 77, 85, 92, 100, 109, 117, 127, 136, 146, 156, 166, 177, 187, 199, 210, 222, 235, 247, 261, 274, 288, 302, 316, 331, 345, 361, 376, 392, 409, 425, 443, 460, 478, 496, 514, 533, 551, 571

Offset: 1

Eric W. Weisstein, Mar 26 2018

nonn

Maximal cliques are triangles in the n-triangular grid graph. The clique covering number cannot be less than the number of nodes divided by three. Perfect nonoverlapping coverings are possible for n + 1 in A072065. - Andrew Howroyd, Jun 27 2018

Georg Fischer, Table of n, a(n) for n = 1..1000
Stan Wagon, Graph Theory Problems from Hexagonal and Traditional Chess., The College Mathematics Journal, Vol. 45, No. 4, September 2014, pp. 278-287.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Triangular Grid Graph.
Eric Weisstein's World of Mathematics, Triangular Honeycomb King Graph.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-2,2,1,0,-2,1).

Cf. A072065.

Table[(Sqrt[3] (16 + 3 n (3 + n)) - 9 Cos[n Pi/6] + 2 Sqrt[3] Cos[2 n Pi/3] + 9 Cos[5 n Pi/6] + 9 Sin[n Pi/6] - 9 Sin[5 n Pi/6])/(18 Sqrt[3]), {n, 20}] (* Eric W. Weisstein, Apr 18 2019 *)
LinearRecurrence[{2, 0, -1, -2, 2, 1, 0, -2, 1}, {1, 3, 4, 6, 8, 10, 13, 15, 19}, 20] (* Eric W. Weisstein, Apr 18 2019 *)
CoefficientList[Series[(-1 - x + 2 x^2 + x^3 - x^4 - 2 x^5 + 2 x^7 - x^8)/((-1 + x)^3 (1 + x - x^3 + x^5 + x^6)), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 18 2019 *)

a(n) ~ (n+1)*(n+2)/6. - Andrew Howroyd, Jun 27 2018
a(n) = 2*a(n-1) - a(n-3) - 2*a(n-4) + 2*a(n-5) + a(n-6) - 2*a(n-8) + a(n-9). - Eric W. Weisstein, Apr 18 2019
G.f.: x (-1 - x + 2*x^2 + x^3 - x^4 - 2*x^5 + 2*x^7 - x^8)/((-1 + x)^3*(1 + x - x^3 + x^5 + x^6)). - Eric W. Weisstein, Apr 18 2019

a(11)-a(24) from Andrew Howroyd, Jun 27 2018

More terms from Georg Fischer, Jun 04 2019

A334875 Number of tribone tilings of an n-triangle.

Original entry on oeis.org

1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 8, 12, 0, 72, 0, 0, 0, 0, 0, 0, 185328, 0, 4736520, 21617456, 0, 912370744, 0, 0, 0, 0, 0, 0, 3688972842502560, 0, 717591590174000896, 9771553571471569856, 0, 3177501183165726091520, 0, 0, 0, 0, 0, 0

Offset: 0

Jonathan Oswald, May 13 2020

This sequence was requested to be added by the author of the link Code Golf challenge. It is based on the work of J. H. Conway, who proved that n = 12k + 0,2,9,11 if and only if T(n) can be tiled (i.e., exactly covered without overlapping) by tribones.

Code Golf Challenge, Number of Distinct Tribone Tilings, posted by CGCC user Bubbler.
J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), 183-208.
James Propp, Trimer covers in the triangular grid: twenty mostly open problems, arXiv:2206.06472 [math.CO], 2022.

The sequence of nonzero indices is A072065.

Cf. A155219.

# tribone tilings
def h(coords):
  def anyhex(i, j):
    c = [x in coords for x in [(i-1, j), (i, j+1), (i+1, j+1), (i+1, j), (i, j-1), (i-1, j-1)]]
    return any(map(lambda x, y: x and y, c, c[1:] + c[:1]))
  return all(anyhex(*z) for z in coords)
def g(coords):
  if not coords: return 1
  #if not h(coords): return 0
  i, j = min(coords)
  if (i+1, j+1) not in coords: return 0
  cases = 0
  if (i+1, j) in coords: cases += g(coords - {(i, j), (i+1, j), (i+1, j+1)})
  if (i, j+1) in coords: cases += g(coords - {(i, j), (i, j+1), (i+1, j+1)})
  return cases
def f(n):
  coords = {(i, j) for i in range(n) for j in range(i+1)}
  #if n%12 not in [0, 2, 9, 11]: return 0
  print(n, g(coords) if n%12 in [0, 2, 9, 11] else 0)
[f(x) for x in range(21)]

Name clarified by James Propp, Mar 28 2022

cp's OEIS Frontend

A008594 Multiples of 12.

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A017629 a(n) = 12*n + 9.

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A017545 a(n) = 12*n + 2.

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A017653 a(n) = 12*n + 11.

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A074229 Numbers n such that Kronecker(6,n)==mu(gcd(6,n)).

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PARI

PARI

PARI

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A301752 Clique covering number of the n-triangular grid graph.

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A334875 Number of tribone tilings of an n-triangle.