A072065 -id:A072065

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A008594

Multiples of 12: a(n) = 12*n.

+10
51

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

a(n) = 12*n. a(n) = 2*a(n-1)-a(n-2) for n>1. G.f.: 12*x/(1-x)^2. - Vincenzo Librandi, Jun 11 2011

a(n) = A003154(n)- A003154(n-1). - Mihir Mathur, Jun 07 2013

MAPLE

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

MATHEMATICA

12*Range[0, 200] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

NestList[12+#&, 0, 60] (* Harvey P. Dale, Feb 02 2022 *)

PROG

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

(Haskell)

a008594 = (* 12)

a008594_list = [0, 12 ..] -- Reinhard Zumkeller, Dec 12 2012

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

CROSSREFS

Subsequence of A072065 and A121032.

Cf. A008592, A008593, A003154.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017629

a(n) = 12*n + 9.

+10
20

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

Table of n, a(n) for n=0..52.

Tanya Khovanova, Recursive Sequences.

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

FORMULA

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

MATHEMATICA

12*Range[0, 200]+9 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

LinearRecurrence[{2, -1}, {9, 21}, 60] (* Harvey P. Dale, Apr 14 2019 *)

PROG

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

(Haskell)

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

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

CROSSREFS

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

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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017545

a(n) = 12*n + 2.

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

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).

FORMULA

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

MAPLE

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

MATHEMATICA

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

PROG

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

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

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

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

CROSSREFS

Cf. A008594, A017533, A017557.

Subsequence of A072065.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A017653

a(n) = 12*n + 11.

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

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

LINKS

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).

FORMULA

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

MATHEMATICA

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

PROG

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

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

(Haskell)

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

CROSSREFS

Cf. A003215, A008594, A017533, A017545.

Subsequence of A072065.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A074229

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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).

FORMULA

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)

PROG

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

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

(PARI) 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

CROSSREFS

Equals 2 * A072065 + 1.

KEYWORD

nonn,easy

AUTHOR

Jon Perry, Sep 17 2002

EXTENSIONS

More terms from Michel Marcus, Mar 17 2014

STATUS

approved

A301752

Clique covering number of the n-triangular grid graph.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Georg Fischer, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Clique Covering Number

Eric Weisstein's World of Mathematics, Triangular Grid Graph

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-2,2,1,0,-2,1).

FORMULA

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A072065.

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Mar 26 2018

EXTENSIONS

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

More terms from Georg Fischer, Jun 04 2019

STATUS

approved

A334875

Number of tribone tilings of an n-triangle.

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..44.

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.

PROG

(Python) # 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)]

CROSSREFS

The sequence of nonzero indices is A072065.

Cf. A155219.

KEYWORD

nonn,hard

AUTHOR

Jonathan Oswald, May 13 2020

EXTENSIONS

Name clarified by James Propp, Mar 28 2022

STATUS

approved

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