A005232 -id:A005232 - cp's OEIS frontend

A008733 Molien series for 3-dimensional group [2+, n] = 2*(n/2).

1, 0, 2, 1, 4, 2, 6, 4, 9, 6, 12, 9, 16, 12, 20, 16, 25, 20, 30, 25, 36, 30, 42, 36, 49, 42, 56, 49, 64, 56, 72, 64, 81, 72, 90, 81, 100, 90, 110, 100, 121, 110, 132, 121, 144, 132, 156, 144, 169, 156, 182, 169, 196, 182, 210, 196, 225, 210, 240, 225, 256

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

List([0..70], n-> Int((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16)); # G. C. Greubel, Jul 30 2019

[Floor((n^2+5*n+13+3*(n+1)*(-1)^n)/16): n in [0..70]]; // Vincenzo Librandi, Aug 24 2013

CoefficientList[Series[(1+x^3)/((1-x^2)^2*(1-x^4)), {x,0,70}], x] (* Vincenzo Librandi, Aug 24 2013 *)
LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,0,2,1,4,2,6},70] (* Harvey P. Dale, Nov 23 2015 *)

a(n)=((n^2+5*n+13+3*(n+1)*(-1)^n))\16 \\ Charles R Greathouse IV, Jun 11 2015

[floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16) for n in (0..70)] # G. C. Greubel, Jul 30 2019

From R. J. Mathar, Nov 04 2008: (Start)
a(n) = A005232(n) - A005232(n-1).
G.f.: (1-x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)). (End)
a(n) = floor((n^2 + 5*n + 13 + 3*(n+1)*(-1)^n)/16). - Tani Akinari, Aug 23 2013
a(n) = Sum_{i=1..floor((n+4)/2)} floor((i-(n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
a(n) = (2*n^2+10*n+13+3*(2*n+5)*(-1)^n+4*(-1)^((6*n-1+(-1)^n)/4))/32. - Luce ETIENNE, Jun 09 2015

A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 6, 16, 81, 299, 1358, 5567, 23350, 91998, 351058, 1269907, 4394634, 14495236, 45779246, 138567568, 403282017, 1130773069, 3062535192, 8028046724, 20411824364, 50429813556, 121280243676, 284360432241, 650972702410

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 148.

Column k=5 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/3840*(1/(1 - x^1)^32 + 231/(1 - x^2)^16 + 20/(1 - x^1)^16/(1 - x^2)^8 + 520/(1 - x^4)^8 + 60/(1 - x^1)^8/(1 - x^2)^12 + 80/(1 - x^1)^8/(1 - x^3)^8 + 720/(1 - x^2)^4/(1 - x^6)^4 + 160/(1 - x^1)^4/(1 - x^2)^2/(1 - x^3)^4/(1 - x^6)^2 + 320/(1 - x^4)^2/(1 - x^12)^2 + 240/(1 - x^1)^4/(1 - x^2)^2/(1 - x^4)^6 + 480/(1 - x^8)^4 + 240/(1 - x^2)^4/(1 - x^4)^6 + 384/(1 - x^1)^2/(1 - x^5)^6 + 384/(1 - x^2)^1/(1 - x^10)^3).

A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

Original entry on oeis.org

1, 1, 7, 23, 153, 849, 6128, 43534, 319119, 2255466, 15307395, 98349144, 597543497, 3430839916, 18653684881, 96273409815, 473010823993, 2218614773950, 9961651259869, 42927432229913, 177963663264430

Offset: 0

Vladeta Jovovic, Aug 05 2000

nonn

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans.Computers, 22 (1973), 1048-1051.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, order 312.

Column k=6 of A363349.

Cf. A005232, A006380, A006381, A006382, A002727, A006148, A052264.

G.f.: 1/46080*(1/(1 - x^1)^64 + 1053/(1 - x^2)^32 + 30/(1 - x^1)^32/(1 - x^2)^16 + 4920/(1 - x^4)^16 + 180/(1 - x^1)^16/(1 - x^2)^24 + 120/(1 - x^1)^8/(1 - x^2)^28 + 160/(1 - x^1)^16/(1 - x^3)^16 + 5280/(1 - x^2)^8/(1 - x^6)^8 + 960/(1 - x^1)^8/(1 - x^2)^4/(1 - x^3)^8/(1 - x^6)^4 + 3840/(1 - x^4)^4/(1 - x^12)^4 + 640/(1 - x^1)^4/(1 - x^3)^20 + 1920/(1 - x^2)^2/(1 - x^6)^10 + 720/(1 - x^1)^8/(1 - x^2)^4/(1 - x^4)^12 + 5760/(1 - x^8)^8 + 2160/(1 - x^2)^8/(1 - x^4)^12 + 1440/(1 - x^1)^4/(1 - x^2)^6/(1 - x^4)^12 + 2304/(1 - x^1)^4/(1 - x^5)^12 + 6912/(1 - x^2)^2/(1 - x^10)^6 + 3840/(1 - x^1)^2/(1 - x^2)^1/(1 - x^3)^2/(1 - x^6)^9 + 3840/(1 - x^4)^1/(1 - x^12)^5).

A275416 Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 3, 4, 3, 1, 1, 3, 8, 5, 3, 1, 1, 4, 10, 10, 5, 3, 1, 1, 4, 16, 15, 11, 5, 3, 1, 1, 5, 20, 27, 17, 11, 5, 3, 1, 1, 5, 29, 38, 32, 18, 11, 5, 3, 1, 1, 6, 35, 60, 49, 34, 18, 11, 5, 3, 1, 1, 6, 47, 84, 83, 54, 35, 18, 11, 5, 3, 1, 1, 7, 56, 122, 123

Offset: 1

R. J. Mathar, Jul 27 2016

By considering the partitions of n into k parts we set a cap on the odd numbers of each part and count the multisets (ordered k-tuples) of odd numbers where each number is not larger than the cap of its part.

Multiset transformation of A110654 or A065033.

			T(6,2) = 3+2+3 = 8 counts {1,1} {1,3}, and {3,3} from taking two odd numbers <= 3; it counts {1,1} and {1,3} from taking an odd number <= 2 and an odd number <= 4; and it counts {1,1}, {1,3} and {1,5} from taking an odd number <= 1 and an odd number <= 5.
T(6,3) = 1+2+2 = 5 counts {1,1,1} from taking three odd numbers <= 2; it counts {1,1,1} and {1,1,3} from taking an odd number <= 1 and an odd number <= 2 and an odd number <= 3; and it counts {1,1,1} and {1,1,3} from taking two odd numbers <= 1 and an odd number <= 4.
  1
  1   1
  2   1   1
  2   3   1   1
  3   4   3   1   1
  3   8   5   3   1   1
  4  10  10   5   3   1   1
  4  16  15  11   5   3   1   1
  5  20  27  17  11   5   3   1   1
  5  29  38  32  18  11   5   3   1   1
  6  35  60  49  34  18  11   5   3   1   1
  6  47  84  83  54  35  18  11   5   3   1   1
  7  56 122 123  94  56  35  18  11   5   3   1   1
  7  72 164 192 146  99  57  35  18  11   5   3   1   1

Alois P. Heinz, Rows n = 1..200, flattened
Index entries for triangles generated by the Multiset Transformation

Cf. A110654 (column 1), A003293 (row sums?), A089353 (equivalent Multiset transformation of A000027), A005232 (2nd column?), A097513 (3rd column?).

T(2n,n) gives A269628.

b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j)*
       binomial(ceil(i/2)+j-1, j), j=0..min(n/i, p)))))
    end:
T:= (n, k)-> b(n$2, k):
seq(seq(T(n, k), k=1..n), n=1..16);  # Alois P. Heinz, Apr 13 2017

b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[Ceiling[i/2] + j - 1, j], {j, 0, Min[n/i, p]}]]]];
T[n_, k_] := b[n, n, k];
Table[T[n, k], {n, 1, 16}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,1) = A110654(n).
T(n,k) = Sum_{c_i*N_i=n,i=1..k} binomial(T(N_i,1)+c_i-1,c_i) for 1 < k <= n.
G.f.: Product_{j>=1} (1-y*x^j)^(-ceiling(j/2)). - Alois P. Heinz, Apr 13 2017

A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

Original entry on oeis.org

0, 1, 1, 3, 4, 8, 10, 15, 19, 26, 31, 39, 46, 56, 64, 75, 85, 98, 109, 123, 136, 152, 166, 183, 199, 218

Offset: 3

Kival Ngaokrajang, May 25 2013

From the drawings as shown in links, it can be separated the distinct quadrilaterals into 3 types:
Q1: Quadrilaterals which have at least one side equal to n-gon sides length.
Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than n-gon sides length.
Q3: Quadrilaterals which have no parallel sides and all sides are longer than n-gon side length.
Q1(n) = A004652(n-3); Q2(n) = A001917(n-6), Q2(3) = 0, Q2(4) = 0; Q3(n) = A005232(n-10), Q3(3) = 0, Q3(4) = 0, Q3(5) = 0, Q3(6) = 0, Q3(7) = 0, Q3(8) = 0, Q3(9) = 0.
a(n) = Q1(n) + Q2(n). The total distinct quadrilaterals is Q1 + Q2 + Q3. Also the total distinct quadrilaterals = A005232(n-4), for n>=4. Also a(n) = A005232(n-4) - A005232(n-10), for n>=10.

			For a pentagon, there are 5 quadrilaterals which are the same size and shape. Therefore a(5) = 1.

Kival Ngaokrajang, The distinct quadrilaterals for n = 4..9
Kival Ngaokrajang, The distinct quadrilaterals for n = 10
Kival Ngaokrajang, Q1 & Q2 for n = 23
Kival Ngaokrajang, Q3 for n = 23

Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n-3) is number of distinct triangles in an n-gon.

Empirical g.f.: -x^4*(x^2-x+1)^2*(x^2+x+1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Oct 31 2013

A141782 Number of connected graphs with one cycle of length m = n-4 and n nodes.

Original entry on oeis.org

18, 28, 32, 45, 52, 69, 79, 100, 114, 140, 158, 189, 212, 249, 277, 320, 354, 404, 444, 501, 548, 613, 667, 740, 802, 884, 954, 1045, 1124, 1225, 1313, 1424, 1522, 1644, 1752, 1885, 2004, 2149, 2279, 2436, 2578, 2748, 2902, 3085, 3252

Offset: 7

Washington Bomfim, Jul 31 2008

nonn

We have unicyclic graphs of order n = m+4 with a cycle of length m. Only 4 nodes of those graphs belong to the rooted trees attached to the cycle, so the orders of those trees can be only 1,2,3,4, or 5. The set of graphs can be divided in five subsets S_1, S_2, S_3, S_4 and S_5, such that
S_1 has trees of orders [5,1,1,...,1],
S_2 has trees of orders [4,2,1,...,1],
S_3 has trees of orders [3,3,1,...,1],
S_4 has trees of orders [3,2,2,1,...,1] and
S_5 has trees of orders [2,2,2,2,1,...,1].
|S_1| = 9 since there are 9 rooted trees with 5 points.
|S_2| = 4floor(m/2).
|S_3| = 3floor(m/2). We consider the 3 2-combinations (with repetition) of the 2 distinct rooted trees of order 3.
|S_4| = 2floor((m-1)^2/4) since floor((m-1)^2/4) is the number of bracelets with m beads, 2 of which are red, 1 of which is blue.
With x=m-4, |S_5| = <(x^3 +9x^2 +(32-9(x mod 2))x)/48 +0.6>. The value of |S_5| is equal to the number of m-bead bracelets with 4 red beads.
This sequence is the fifth column of table T of A058879.

			E.g. a(9)=32. Click the link to see an illustration of the 32 unicyclic graphs of order 9 with a pentagon.

Washington Bomfim, Table of n, a(n) for n = 7..100
Washington Bomfim, The 32 unicyclic graphs of order 9 with a pentagon..

Cf. A058879, A005232, A000081, A002620.

m=n-4 x=m-4 a(n) = round((x^3+9*x^2+(32-9*(x%2))*x)/48+0.6)+2*floor((m-1)^2/4)+7*floor(m/2)+9

With m = n-4 and x = m-4, a(n) = <(x^3 +9x^2 +(32-9(x mod 2))x)/48 +0.6> + 2floor((m-1)^2/4) + 7floor(m/2) + 9. Empirically for n odd a(n) = (n^3 +9n^2 -n +87)/48 Empirically for n even a(n) = (n^3 +9n^2 +8n +192-n%4*6)/48.

Empirical g.f.: -x^7*(16*x^7-23*x^6-9*x^5+18*x^4-17*x^3+24*x^2+8*x-18) / ((x-1)^4*(x+1)^2*(x^2+1)). [Colin Barker, Feb 18 2013]

A367783 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to shifts.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 4, 0, 6, 0, 6, 0, 11, 0, 12, 8, 7, 0, 6, 0, 8, 0, 18, 0, 32, 0, 20, 0, 29, 0, 42, 8, 67, 16, 30, 0, 13, 0, 22, 0, 32, 0, 42, 0, 64, 0, 50, 0, 64

Offset: 1

Giedrius Alkauskas, Nov 30 2023

A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.
a(n) > 0 for even n >= 12.
a(n) > 0 for odd n with natural density 1 (among odd numbers).
For odd n, a(n) is divisible by 8.

			For n = 4 a(4) = 1 way to place 4 points is as follows:
  .xx.
  .xx.
For n = 8 a(8) = 2 ways to place 8 points are as follows:
  ..x.
  .xxx
  xxx.
  .x..
(and its reflection with respect to a vertical axis).
For n = 18 a(18) = 4 ways to place 18 points are as follows:
  ...x..
  ..xxx.
  .xxxxx
  xxxxx.
  .xxx..
  ..x...
(and its reflection with respect to a vertical axis), and
  .....x....
  ......x...
  .......x..
  ....x...x.
  ...xxx...x
  x...xxx...
  .x...x....
  ..x.......
  ...x......
  ....x.....
(and its reflection with respect to a vertical axis).

Giedrius Alkauskas, Friendly paths for finite subsets of plane integer lattice. I, arXiv:2302.01137 [math.CO], 2024.
Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
Giedrius Alkauskas, Friendly paths. Problem 11484, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168.

Cf. A000009, A000292, A005232, A369382.

a(36) corrected by Giedrius Alkauskas, Feb 02 2024

a(49)-a(60) from Giedrius Alkauskas, Feb 06 2024

A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 1, 3, 0, 2, 0, 3, 0, 6, 0, 10, 0, 6, 0, 9, 0, 12, 1, 18, 2, 9, 0, 5, 0, 7, 0, 8, 0, 12, 0, 18, 0, 14, 0, 17

Offset: 1

Giedrius Alkauskas, Jan 22 2024

A monotone path is a lattice path consisting of east and north unit steps or a path consisting of east and south unit steps. When counting, points lying on the path itself are discarded.
Related to A367783, only sets obtained by rotation and reflection are considered to be the same.
For odd n, a(n) = A367783(n)/8.
For even n, 8 * a(n) >= A367783(n).
a(n) > 0 for even n >= 12.
a(n) > 0 for odd n with natural density 1 (among odd numbers).

			For n = 4, a(4) = 1 way to place 4 points is as follows:
.xx.
.xx.
For n = 14, a(14) = 1 way to place 14 points is as follows:
  ...x..
  ..x.x.
  .xxx.x
  x.xxx.
  .x.x..
  ..x...
For n = 27, a(27) = 1 way to place 27 points is as follows:
  ....x....
  ...x.....
  ..x......
  .x..xx...
  x..xxxx..
  ..xxxxxxx
  ...xxxxx.
  ....xxx..
  .....x...

Giedrius Alkauskas, Friendly paths for finite subsets of plane integer lattice. I, arXiv:2302.01137 [math.CO], 2024.
Giedrius Alkauskas, Problem 11484, Problems and solutions, Amer. Math. Monthly, 117 (2) February (2010), p. 182.
Giedrius Alkauskas, Friendly paths. Problem 11484, Problems and solutions, Amer. Math. Monthly, 119 (2) February (2012), 167-168.

Cf. A000009, A000292, A005232, A367783.

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 5, 8, 10, 5, 1, 1, 1, 6, 10, 16, 10, 4, 1, 1, 1, 6, 12, 20, 16, 7, 1, 1, 1, 7, 14, 29, 26, 16, 4, 1, 1, 1, 7, 16, 35, 38, 26, 8, 1, 1, 1, 8, 19, 47, 57, 50

Offset: 2

Washington Bomfim, Aug 28 2008

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0 to floor(n/2) beads 0, i.e., 0 <= k <= floor(n/2). If n is even, the bracelet 0101...01 with n/2 beads of each kind does not have 00 and we cannot change any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet 0101...011 with (n-1)/2 beads 0.
The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0 with 00 prohibited, is equal to the number of binary bracelets with n-k beads, k of them 0. See below.
Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1 (circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.
If we do this insertion k times, each time after a distinct 0 of B, we obtain a bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.
On the contrary, Let B be a binary bracelet with n beads, k of them 0, with 00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet of n-1 beads, k of them 0. (If not and we undo the removal, the configuration obtained cannot be a bracelet and this is absurd.) If we repeat this removal k times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

			Array begins
1 1
1 1
1 1 1
1 1 1
1 1 2 1
1 1 2 1
1 1 3 2 1
1 1 3 3 1
1 1 4 4 3 1
...
A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
A032279(5) = 1+1+4+4+3+1 = 14.

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.

cp's OEIS Frontend

A008733 Molien series for 3-dimensional group [2+, n] = 2*(n/2).

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A056204 Number of n X 5 binary matrices under row and column permutations and column complementations.

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A056205 Number of n X 6 binary matrices under row and column permutations and column complementations.

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A275416 Triangle read by rows: T(n,k) is the number of multisets of k odd numbers with a cap of the total sum set to n.

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A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

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A141782 Number of connected graphs with one cycle of length m = n-4 and n nodes.

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PARI

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A367783 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to shifts.

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A369382 Number of subsets of the integer lattice Z^2 of cardinality n such that there is no monotone lattice path which splits the set in half, up to lattice symmetry.

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A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

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