A008615 -id:A008615 - cp's OEIS frontend

A090184 Number of partitions of the n-th 3-smooth number into parts 2 and 3.

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 22, 25, 28, 33, 37, 41, 43, 49, 55, 65, 73, 82, 86, 97, 109, 122, 129, 145, 163, 171, 193, 217, 244, 257, 289, 325, 342, 365, 385, 433, 487, 513, 577, 649, 683, 730, 769, 865, 973, 1025, 1094, 1153

Offset: 1

Reinhard Zumkeller, Jan 21 2004

nonn

			n=11: A003586(11) = 2^3 * 3 = 24: 3+3+3+3+3+3+3+3 = 3+3+3+3+3+3+2+2+2 = 3+3+3+3+2+2+2+2+2+2 = 3+3+2+2+2+2+2+2+2+2+2 = 2+2+2+2+2+2+2+2+2+2+2+2: a(11)=5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022328, A022329.
Cf. A003586, A008615.
Cf. A103221, A117222, A117220, A117221.

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Length[IntegerPartitions[#, All, {2, 3}]]& /@ Select[Range[10000], smooth3Q] (* Jean-François Alcover, Oct 13 2021 *)
With[{nn = 6^5}, Map[Floor[#/2] - Floor[#/3] &, Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}] &[nn] + 2]] (* Michael De Vlieger, Oct 13 2021 *)

from sympy import integer_log
def A090184(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return ((m:=bisection(f,n,n)+2)>>1)-m//3 # Chai Wah Wu, Oct 22 2024

a(2^i * 3^j) = floor(2^(i-1) * 3^(j-1) + 1), i*j>0.

a(n) = A103221(A003586(n)).

Offset changed to 1 by Alois P. Heinz, Oct 15 2021

A128115 Mobius inversion of A103221.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 3, 2, 3, 2, 5, 2, 5, 3, 3, 2, 4, 2, 6, 3, 4, 2, 7, 2, 7, 4, 4, 3, 8, 3, 7, 4, 5, 4, 9, 3, 6, 4, 6, 4, 10, 2, 10, 5, 6, 5, 8, 4, 11, 6, 7, 4, 12, 4, 12, 6, 7, 6, 10, 4, 13, 6, 9, 6, 14, 4, 10, 7, 9, 6, 15, 4, 12, 8, 10, 7, 12, 5, 16, 7

Offset: 1

Paulo de Almeida Sachs (sachs6(AT)yahoo.de), Feb 15 2007

nonn

Number of uniform n-grammic crossed antiprisms.
Agrees with Mobius inversion of A008615 for n != 3. - Andrew Baxter, Jun 06 2008
Number of primitive equivalence classes of period 2n billiards on an equilateral triangle. - Andrew Baxter, Jun 06 2008

Andrew M. Baxter and Ron Umble, Periodic Orbits of Billiards on an Equilateral Triangle, Amer. Math. Monthly, 115 (No. 6, 2008), 479-491.

Cf. A055684.

Cf. A008615, A103221.

with(numtheory): A103221:=n->floor((n+2)/2)-floor((n+2)/3): A128115:=n->add(mobius(d)*A103221(n/d), d in divisors(n)): # Andrew Baxter, Jun 06 2008

SUM_{d|n} mu(d) * A103221(n/d), where mu is Mobius function (A008683). - Andrew Baxter, Jun 06 2008

Edited by Andrew Baxter, Jun 06 2008

A075993 Triangle read by rows: T(n,m) is the number of integers k such that floor(n/k) = m, n >= 1, k = 1..n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Sep 28 2002

The sum of numbers in row n is n.

Number of terms > 0 per row: Sum_{k=1..n} A057427(T(n,k)) = A055086(n). - Reinhard Zumkeller, Apr 06 2006

			T(5, 1) = 3 counts k such that floor(5/k) = 1, namely k = 5, 4, 3.
First 10 rows:
  1
  1 1
  2 0 1
  2 1 0 1
  3 1 0 0 1
  3 1 1 0 0 1
  4 1 1 0 0 0 1
  4 2 0 1 0 0 0 1
  5 1 1 1 0 0 0 0 1
  5 2 1 0 1 0 0 0 0 1

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)

Columns 1, 2, 3 are essentially A004526, A008615, A008679.

Cf. A010766.

Table[Floor[n/m] - Floor[n/(m + 1)], {n, 14}, {m, n}] // Flatten (* Michael De Vlieger, Jan 14 2022 *)

T(n, m) = floor(n/m) - floor(n/(m+1)).

A165685 Dimension of the space of Siegel cusp forms of genus 2 and weight n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0, 7, 0, 8, 0, 9, 0, 11, 1, 13, 0, 13, 1, 17, 1, 18, 1, 20, 2, 23, 3, 26, 2, 27, 4, 32, 4, 34, 5, 37, 6, 41, 8, 46, 7, 47, 10, 54, 11, 57, 12, 61, 14, 67, 17, 73, 16, 75, 21, 84, 22, 88, 24, 94, 27, 101, 31, 109, 31, 112

Offset: 1

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 24 2009

nonn

			a(35)=1 as the dimension of the space of Siegel cusp form of genus 2 and weight 35 is 1.

M. Eie, Dimensions of spaces of Siegel cusp forms of degree two and three, AMS, 1984, p. 44-45.

Andy Huchala, Table of n, a(n) for n = 1..20000
Andy Huchala, Proof of generating function.

Cf. A008615, A029143. A165684 gives only the even weights.

N1[k_] := 2^(-7)*3^(-3)*5^(-1) (2 k^3 + 96 k^2 - 52 k - 3231) /; Mod[k, 2] == 0; N1[k_] := 2^(-7)*3^(-3)*5^(-1)*(2 k^3 - 114 k^2 + 2018 k - 9051) /; Mod[k, 2] == 1; N2[k_] := 2^(-5)*3^(-3)*(17 k - 294) /; Mod[k, 12] == 0; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 325) /; Mod[k, 12] == 1; N2[k_] := 2^(-5)*3^(-3)*(-25 k + 254) /; Mod[k, 12] == 2; N2[k_] := 2^(-5)*3^(-3)*(17 k - 261) /; Mod[k, 12] == 3; N2[k_] := 2^(-5)*3^(-3)*(17 k - 86) /; Mod[k, 12] == 4; N2[k_] := 2^(-5)*3^(-3)*(-k + 53) /; Mod[k, 12] == 5; N2[k_] := 2^(-5)*3^(-3)*(-k - 42) /; Mod[k, 12] == 6; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 91) /; Mod[k, 12] == 7; N2[k_] := 2^(-5)*3^(-3)*(-7 k + 2) /; Mod[k, 12] == 8; N2[k_] := 2^(-5)*3^(-3)*(-k - 27) /; Mod[k, 12] == 9;
N2[k_] := 2^(-5)*3^(-3)*(-k + 166) /; Mod[k, 12] == 10; N2[k_] := 2^(-5)*3^(-3)*(17 k - 181) /; Mod[k, 12] == 11; N3[k_] := 2^(-7)*3^(-3)*1131 /; Mod[k, 12] == 0; N3[k_] := 2^(-7)*3^(-3)*229 /; Mod[k, 12] == 1; N3[k_] := 2^(-7)*3^(-3)*(-229) /; Mod[k, 12] == 2; N3[k_] := 2^(-7)*3^(-3)*(-1131) /; Mod[k, 12] == 3; N3[k_] := 2^(-7)*3^(-3)*427 /; Mod[k, 12] == 4; N3[k_] := 2^(-7)*3^(-3)*(-571) /; Mod[k, 12] == 5;
N3[k_] := 2^(-7)*3^(-3)*123 /; Mod[k, 12] == 6; N3[k_] := 2^(-7)*3^(-3)*(-203) /; Mod[k, 12] == 7; N3[k_] := 2^(-7)*3^(-3)*203 /; Mod[k, 12] == 8; N3[k_] := 2^(-7)*3^(-3)*(-123) /; Mod[k, 12] == 9; N3[k_] := 2^(-7)*3^(-3)*571 /; Mod[k, 12] == 10; N3[k_] := 2^(-7)*3^(-3)*(-427) /; Mod[k, 12] == 11; N4[k_] := 5^(-1) /; Mod[k, 5] == 0; N4[k_] := -5^(-1) /; Mod[k, 5] == 3; N4[k_] := 0 /; Mod[k, 5] == 1 || Mod[k, 5] == 2 || Mod[k, 5] == 4;
DimSk[k_] := If[k >= 7, N1[k] + N2[k] + N3[k] + N4[k], 0];
Table[ DimSk[k], {k, 1, 100}]
(* second program: *)
init = {0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 4, 0, 5, 0, 5, 0};
ker = {0, 0, 0, 1, 1, 1, 0, 0, -1, -1, -1, 1, 0, 0, 1, -1, -1, -1, 0, 0, 1, 1, 1, 0, 0, 0, -1};
ans = LinearRecurrence[ker, init, 100];
ans[[3]] = 0 ; ans (* Andy Huchala, Mar 03 2022 *)

R. = PowerSeriesRing(ZZ, 100);
p = x^26 + x^24 - x^21 - x^19 + x^18 - x^17 - x^14 - x^13 + x^10 + x^9 + x^8 + x^7 - x^3;
q = x^27 - x^23 - x^22 - x^21 + x^18 + x^17 + x^16 - x^15 - x^12 + x^11 + x^10 + x^9 - x^6 - x^5 - x^4 + 1;
(x^3 + p/q).list()[1:] # Andy Huchala, Mar 03 2022

G.f.: x^10 (1+x^2-x^5-x^7+x^10-x^15+x^20) / ((-1+x)^4 (1+x)^3 (1+2x^2+2x^4+x^6)^2 (1+x+x^4+x^7+x^8)). - Andy Huchala, Mar 03 2022

a(2n) = A165684(n) and a(2n+35) = A029143(n). - Andy Huchala, Mar 04 2022

a(73) corrected by Andy Huchala, Mar 02 2022

A321203 Irregular triangle T giving the coefficients of x^n = x^{2e2 + 3e3} of (1 + x^2 + x^3)^n, with the pair of nonnegative numbers [e2, e3] listed in row n of A321201, for n >= 2.

Original entry on oeis.org

2, 3, 6, 20, 15, 20, 105, 168, 70, 84, 504, 1260, 252, 1320, 2310, 495, 7920, 924, 12870, 10296, 10010, 45045, 3432, 3003, 100100, 45045, 120120, 240240, 12870, 74256, 680680, 194480, 18564, 1113840, 1225224, 48620, 1058148, 4232592, 831402, 542640, 8817900, 6046560, 184756

Offset: 2

Wolfdieter Lang, Nov 05 2018

The row length is r(n), with r(n) = A008615(n+2) for n >= 2:  [1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, ...].
The row sums give A176806(n).
For n = 0 with the trivial [e2, e3] = [0, 0] solution the multinomial is 1 with the row sum A176806(0) = 1. For n = 1 there is no solution (with row sum set to A176806(1) = 0).
This multinomial array for pairs [e2, e3] with 2*2e2 + 3*e3 = n, with nonnegative numbers e2 and e3, is obtained from the multinomial array n!/(e1!*e2!*e3!) with n = e1 + e2 + e3, giving the coefficient x_1^{e1}* x_2^{e2}*x_3^{e3} of (x_1 + x_2 + x_3)^n. Here, in order to find the coefficients of (1 + x^2 + x^3)^n, one sets x_1 = 1, x_2 = x^2 and x_3 = x^3. Hence n = e1 + e2 + e3, and the power of x^n becomes n = 2*e2 + 3*e3. Therefore, e1 = n - (e2 + e3), and the array gives n!/((n-(e2+e3))!*e2!*e3!).

			The triangle T(n, m), and the row sums begin:
n\m        0        1        2       3  ...  Row sums A176806(n)
2:         2                                         2
3:         3                                         3
4:         6                                         6
5:        20                                        20
6:        15       20                               35
7:       105                                       105
8:       168       70                              238
9:        84      504                              588
10:     1260      252                             1512
11:     1320     2310                             3630
12:      495     7920      924                    9339
13:    12870    10296                            23166
14:    10010    45045     3432                   58487
15:     3003   100100    45045                  148148
16:   120120   240240    12870                  373230
17:    74256   680680   194480                  949416
18:    18564  1113840  1225224   48620         2406248
19:  1058148  4232592   831402                 6122142
20:   542640  8817900  6046560  184756        15591856
...
------------------------------------------------------------------------------
n = 8: (1 + x^2 + x^3)^8 has coefficients 238 of x^n arising from the two [e2, e3] pairs [1, 2] and [4, 0], given in row n = 8 of A321201. The multinomial values are 8!/((8-3)!*1!*2!) = 168 and 8!/((8-4)!*4!*0!) = 70, summing to 238.

Cf. A008615, A130561, A176806, A321201, A319204.

T(n, m) is obtained from the pair(s) [e2, e3] given in row n of A321201 by n!/((n - (e2 +e3))!*e2!*e3!), for n >= 2 and m = 1, 2, ..., A008615(n+2).

A321202 Row sums of the irregular triangle A321201.

Original entry on oeis.org

1, 1, 2, 2, 5, 3, 7, 7, 9, 9, 15, 11, 18, 18, 21, 21, 30, 24, 34, 34, 38, 38, 50, 42, 55, 55, 60, 60, 75, 65, 81, 81, 87, 87, 105, 93, 112, 112, 119, 119, 140, 126, 148, 148, 156, 156, 180, 164, 189, 189, 198, 198, 225, 207, 235, 235, 245, 245, 275, 255, 286, 286, 297

Offset: 2

Wolfdieter Lang, Nov 05 2018

nonn

Total number of parts in the partitions of n into parts of size 2 and 3. - Andrew Howroyd, Nov 10 2018

Andrew Howroyd, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (-1,1,3,2,-2,-3,-1,1,1).

Cf. A008615, A321201.

row[n_] := Reap[Do[If[2 e2 + 3 e3 == n, Sow[{e2, e3}]], {e2, 0, n/2}, {e3, 0, n/3}]][[2, 1]];
a[n_] := row[n] // Flatten // Total;
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Nov 23 2018 *)

Vec((1 + 2*x + 2*x^2)/((1 + x + x^2)^2*(1 + x)^2*(1 - x)^3) + O(x^60)) \\ Andrew Howroyd, Nov 10 2018

a(n) = Sum_{k=1..2*A008615(n+2)} A321201(n, k),  n >= 2.
From Andrew Howroyd, Nov 10 2018: (Start)
G.f.: x^2*(1 + 2*x + 2*x^2)/((1 + x + x^2)^2*(1 + x)^2*(1 - x)^3).
a(n) = Sum_{k=0..floor(n/6)} 2*k + (n-6*k)/2 for even n.
a(n) = Sum_{k=0..floor((n-3)/6)} 2*k + 1 + (n-3-6*k)/2 for odd n.
(End)

Terms a(27) and beyond from Andrew Howroyd, Nov 10 2018

A340985 Irregular triangle read by rows: n-th row gives the order of all undirected (also called weakly connected) components of the Collatz digraph of order n, sorted from largest to smallest.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 3, 2, 3, 3, 3, 3, 1, 7, 1, 7, 1, 1, 8, 1, 1, 8, 2, 1, 9, 2, 1, 9, 2, 1, 1, 9, 4, 1, 9, 4, 1, 1, 10, 4, 1, 1, 10, 5, 1, 1, 10, 6, 1, 1, 10, 6, 1, 1, 1, 12, 6, 1, 1, 12, 6, 1, 1, 1, 12, 7, 1, 1, 1, 12, 7, 2, 1, 1, 13, 7, 2, 1, 1, 13, 7, 2, 1, 1

Offset: 1

Sebastian Karlsson, Feb 01 2021

The Collatz digraph of order n is the directed graph with the vertex set V = {1, 2, ..., n} and the arrow set A = {m -> A014682(m) | m and A014682(m) are elements of V}.
Some notes:
- First column is A340010.
- The sum of the n-th row is n - the n-th row can be seen as a partition of n.
- Assuming the Collatz conjecture to be true, the length of each row for n > 1 is A008615(n+7). If the Collatz conjecture is true, then the (infinite) Collatz digraph is an undirected tree except for the cycle 1 -> 2 -> 1. This means that for the Collatz digraph of order n > 1, there will be one undirected component containing the cycle 1 -> 2 -> 1, and precisely one undirected component for every odd k such that 1 < k <= n and (3*k+1)/2 > n. The cardinality of the set {1} U {k | 1 < k <= n, k is odd and (3*k+1)/2 > n} is 1 + floor((n+1)/2) - floor((n+1)/3) = A008615(n+7).

			+-----------------+  +--------------------------+
|Array begins:    |  | Continues:               |
+-----------------+  +--------------------------+
| 1;              |  | 12, 6, 1, 1, 1;          |
| 2;              |  | 12, 7, 1, 1, 1;          |
| 2,  1;          |  | 12, 7, 2, 1, 1;          |
| 3,  1;          |  | 13, 7, 2, 1, 1;          |
| 3,  2;          |  | 13, 7, 2, 1, 1, 1;       |
| 3,  3;          |  | 21, 2, 1, 1, 1;          |
| 3,  3, 1;       |  | 21, 2, 1, 1, 1, 1;       |
| 7,  1;          |  | 22, 2, 1, 1, 1, 1;       |
| 7,  1, 1;       |  | 22, 2, 2, 1, 1, 1;       |
| 8,  1, 1;       |  | 22, 3, 2, 1, 1, 1;       |
| 8,  2, 1;       |  | 22, 3, 2, 1, 1, 1, 1;    |
| 9,  2, 1;       |  | 24, 3, 2, 1, 1, 1;       |
| 9,  2, 1, 1;    |  | 24, 3, 2, 1, 1, 1, 1;    |
| 9,  4, 1;       |  | 25, 3, 2, 1, 1, 1, 1;    |
| 9,  4, 1, 1;    |  | 25, 4, 2, 1, 1, 1, 1;    |
| 10, 4, 1, 1;    |  | 26, 4, 2, 1, 1, 1, 1;    |
| 10, 5, 1, 1;    |  | 26, 4, 2, 1, 1, 1, 1, 1; |
| 10, 6, 1, 1;    |  | 26, 4, 4, 1, 1, 1, 1;    |
| 10, 6, 1, 1, 1; |  | 26, 4, 4, 1, 1, 1, 1, 1; |
| 12, 6, 1, 1;    |  | 27, 4, 4, 1, 1, 1, 1, 1; |
+-----------------+  +--------------------------+
.
First row is [1] since the Collatz digraph of order 1 is the singleton 1, i.e., there is one weakly connected component which has order 1.
Third row is [2, 1] since the Collatz digraph of order 3 consists of the cycle 1 -> 2 -> 1 and the singleton 3. That gives one weakly connected component of order 2 and one with order 1.
Fifth row is [3, 2] since the Collatz digraph of order 5 consists of the weakly connected components 4 -> 2 -> 1 -> 2 and 3 -> 5. These components have order 3 and 2 respectively.

Sebastian Karlsson, Rows n = 1..400, flattened
Thijs Laarhoven, The 3n+1 conjecture, Eindhoven University of Technology, Bachelor thesis (2009). See also.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A008615, A014682, A088975, A127824, A248573, A340010.

import networkx as nx
def A014682(n):
    return n//2 if n%2 == 0 else (3*n+1)//2
def Row(n): #Returns n-th row
    G = nx.Graph()
    G.add_nodes_from(range(1, n+1))
    G.add_edges_from([(m,A014682(m)) for m in range(1,n+1) if A014682(m) <= n])
    return sorted([len(c) for c in nx.connected_components(G)], reverse=True)

A060550 a(n) is the number of distinct patterns (modulo geometric D_3-operations) with no other than strict 120-degree rotational symmetry which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 6, 2, 6, 12, 6, 12, 28, 12, 28, 56, 28, 56, 120, 56, 120, 240, 120, 240, 496, 240, 496, 992, 496, 992, 2016, 992, 2016, 4032, 2016, 4032, 8128, 4032, 8128, 16256, 8128, 16256, 32640, 16256, 32640, 65280, 32640

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Harry J. Smith, Table of n, a(n) for n = 1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,2,0,0,-4).

Cf. A060547, A060548, A008611, A008615.

a(n) = { 2^(floor(n/3) + (n%3)%2 - 1) - 2^(floor((n + 3)/6) + (n%6==1) - 1) } \\ Harry J. Smith, Jul 07 2009

a(n) = 2^(floor(n/3) + (n mod 3) mod 2 - 1) - 2^(floor((n+3)/6) + d(n)-1), with d(n)=1 if n mod 6=1, otherwise d(n)=0.
a(n) = (A060547(n) - A060548(n))/2.
a(n) = 2^(A008611(n-1) - 1) + 2^(A008615(n+1) - 1), for n >= 1.
G.f.: x^4*(x^2 - x + 1)*(x^2 + x + 1) / ((2*x^3-1)*(2*x^6-1)). - Colin Barker, Aug 29 2013

A060552 a(n) is the number of distinct (modulo geometric D3-operations) nonsymmetric (no reflective nor rotational symmetry) patterns which can be formed by an equilateral triangular arrangement of closely packed black and white cells satisfying the local matching rule of Pascal's triangle modulo 2, where n is the number of cells in each edge of the arrangement. The matching rule is such that any elementary top-down triangle of three neighboring cells in the arrangement contains either one or three white cells.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 14, 35, 70, 154, 310, 650, 1300, 2666, 5332, 10788, 21588, 43428, 86856, 174244, 348488, 697992, 1396040, 2794120, 5588240, 11180680, 22361360, 44730896, 89462032, 178940432, 357880864, 715794960

Offset: 1

André Barbé (Andre.Barbe(AT)esat.kuleuven.ac.be), Apr 03 2001

Harry J. Smith, Table of n, a(n) for n=1..500
A. Barbé, Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2, Discr. Appl. Math. 105(2000), 1-38.
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-4,-4,10,-4,-4,4,8,8,-16).

Cf. A000079, A060547, A060548, A060546, A060552, A008611, A008615, A008619.

a(n) = { (2^(n-1)-2^(floor(n/3)+(n%3)%2-1))/3+2^(floor((n+3)/6)+(n%6==1)-1)-2^floor((n-1)/2) } \\ Harry J. Smith, Jul 07 2009

a(n) = (2^(n-1) - 2^(floor(n/3) + (n mod 3)mod 2 - 1))/3 + 2^(floor((n+3)/6) + d(n) - 1) - 2^floor((n-1)/2), with d(n)=1 if n mod 6=1 else d(n)=0.
a(n) = (A000079(n-1) - A060547(n)/2)/3 + A060548(n)/2 -A060546(n)/2.
a(n) = (A000079(n-1) - 2^(A008611(n-1) - 1))/3 + 2^(A008615(n+1) - 1) - 2^(A008619(n-1) - 1), n >= 1.
From R. J. Mathar, Aug 03 2009: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - 4*a(n-4) - 4*a(n-5) + 10*a(n-6) - 4*a(n-7) - 4*a(n-8) + 4*a(n-9) + 8*a(n-10) + 8*a(n-11) - 16*a(n-12).
G.f.: -x^4*(-1 - x^2 - x^4 + 2*x^3 + 2*x^5 + 2*x^6)/((2*x-1)*(2*x^2-1)*(2*x^3-1)*(2*x^6-1)). (End)

A081753 a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Apr 08 2003

a(2n) = dimension of M(2n), where M(2n) denotes the complex vector space of modular forms of weight 2n for the group : PSL2(Z). dimension of M(2n+1) = 0.
See A103221(n) for the dimension of M(2n). The Apostol reference, p. 119, eq. (9) uses even k. - Wolfdieter Lang, Sep 16 2016
The space of modular forms is generated by E_4 (A004009) and E_6 (A013973) both of even weight. This is why the space of modular forms of odd weight is trivial. - Michael Somos, Dec 11 2018

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + 2*x^12 + ... - _Michael Somos_, Dec 11 2018

Apostol, Tom M., Modular Functions and Dirichlet Series in Number Theory, second edition, Springer, 1990.
Yves Hellegouarch, "Invitation aux mathématiques de Fermat-Wiles", Dunod, 2ème édition, p. 285

Robert Israel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A008615, A097992, A103221.

seq(floor(n/12)+1-charfcn[0](n-2 mod 12), n=0..100); # Robert Israel, Sep 16 2016

Table[If[Mod[n, 12] == 2, Floor[n/12], Floor[n/12] + 1], {n, 0, 120}] (* or *)
CoefficientList[Series[(1 - x^2 + x^3)/(1 - x - x^12 + x^13), {x, 0, 120}], x] (* Michael De Vlieger, Sep 19 2016 *)
a[ n_] := Quotient[n, 12] + Boole[Mod[n, 12] != 2]; (* Michael Somos, Dec 11 2018 *)

a(k) = if(k%12-2, floor(k/12)+1, floor(k/12))

{a(n) = n\12 + (n%12!=2)}; /* Michael Somos, Dec 11 2018 */

a(n) = floor(n/12) if n == 2 (mod 12); a(n) = floor(n/12) + 1 otherwise.
G.f.: (1-x^2+x^3)/(1-x-x^12+x^13). - Robert Israel, Sep 16 2016
a(2*n) = A008615(n+2), a(2*n+1) = A097992(n). - Michael Somos, Dec 11 2018

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