A026796 -id:A026796 - cp's OEIS frontend

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 4, 5, 3, 1, 1, 0, 0, 2, 0, 16, 0, 4, 0, 1, 0, 0, 2, 15, 58, 59, 21, 5, 1, 1, 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1, 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1, 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1

Offset: 1

Jason Kimberley, Feb 07 2013

			01: 0;
02: 0, 0;
03: 0, 0, 1;
04: 0, 0, 0, 1;
05: 0, 0, 0, 0, 1;
06: 0, 0, 1, 1, 1, 1;
07: 0, 0, 1, 0, 2, 0, 1;
08: 0, 0, 1, 4, 5, 3, 1, 1;
09: 0, 0, 2, 0, 16, 0, 4, 0, 1;
10: 0, 0, 2, 15, 58, 59, 21, 5, 1, 1;
11: 0, 0, 3, 0, 264, 0, 266, 0, 6, 0, 1;
12: 0, 0, 4, 71, 1535, 7848, 7848, 1547, 94, 9, 1, 1;
13: 0, 0, 5, 0, 10755, 0, 367860, 0, 10786, 0, 10, 0, 1;
14: 0, 0, 6, 428, 87973, 3459379, 21609300, 21609300, 3459386, 88193, 540, 13, 1, 1;
15: 0, 0, 9, 0, 803973, 0, 1470293675, 0, 1470293676, 0, 805579, 0, 17, 0, 1;
16: 0, 0, 10, 3406, 8020967, 2585136353, 113314233804, 733351105934, 733351105934, 113314233813, 2585136741, 8037796, 4207, 21, 1, 1;

Jason Kimberley, Table of i, a(i)=E(n,k) for i = 1..136 (n = 1..16)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A198313(n).

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), this sequence (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), A185163 (k=6).

E(n,k) = A186733(n,k) + A210703(n,k), noting that A210703 is a tabf.

E(n,k) = A051031(n,k) - A185304(n,k), noting that A185304 is a tabf.

A339164 Number of compositions (ordered partitions) of n into distinct parts, the least being 3.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 50, 50, 80, 104, 134, 158, 212, 356, 410, 578, 752, 1040, 1238, 1646, 1964, 3236, 3674, 5066, 6368, 8720, 10862, 14078, 17180, 22076, 31802, 38378, 49784, 63824, 82670, 104150, 136220, 165980

Offset: 0

Ilya Gutkovskiy, Nov 25 2020

nonn

			a(12) = 8 because we have [9, 3], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 9], [3, 5, 4] and [3, 4, 5].

Index entries for sequences related to compositions

Cf. A026796, A026824, A339101, A339162, A339163, A339165, A339166.

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-3)*(i+4)/2 `if`(n<3, 0, b(n-3$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020

nmax = 49; CoefficientList[Series[Sum[k! x^(k (k + 5)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=1} k! * x^(k*(k + 5)/2) / Product_{j=1..k-1} (1 - x^j).

A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 59, 7848, 3459379, 2585136353, 2807104852102

Offset: 0

Jason Kimberley, Mar 12 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), this sequence (k=5), A185163 (k=6).

a(n) = A165626(n) - A185354(n).

a(n) = A184953(n) + A185053(n).

A185163 Number of not necessarily connected 6-regular simple graphs on n vertices with girth exactly 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7848, 367860, 21609300, 1470293675, 113314233804, 9799685588955

Offset: 0

Jason Kimberley, Mar 21 2012

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Not necessarily connected k-regular simple graphs girth exactly 3: A198313 (any k), A185643 (triangle); fixed k: A026796 (k=2), A185133 (k=3), A185143 (k=4), A185153 (k=5), this sequence (k=6).

a(n) = A165627(n) - A185364(n).

a(n) = A184953(n) + A185053(n).

A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 31, 36, 45, 54, 66, 78, 97, 114, 138, 164, 197, 232, 280, 328, 392, 461, 546, 639, 758, 884, 1041, 1215, 1425, 1657, 1941, 2250, 2624, 3041, 3534, 4084, 4740, 5465, 6321, 7280, 8399

Offset: 1

Clark Kimberling

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026796, A027189.

b:= proc(n, i, t) option remember; `if`(n=0, t,
     `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
    end:
a:= n-> `if`(n<3, 0, b(n-3, 3, 0)):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 18 2019

b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
a[n_] := If[n < 3, 0, b[n - 3, 3, 0]];
Array[a, 100] (* Jean-François Alcover, May 17 2020, after Alois P. Heinz *)

a(n) + A027189(n) = A026796(n). - R. J. Mathar, Oct 18 2019
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (8 * 3^(3/2) * n^2). - Vaclav Kotesovec, May 17 2020
G.f.: x^6 * Sum_{k>=0} x^(6*k)/Product_{j=1..2*k+1} (1-x^j). - Seiichi Manyama, May 15 2023

A182806 Number of partitions of 3n into parts >= 3.

Original entry on oeis.org

1, 2, 4, 9, 17, 33, 60, 110, 191, 331, 556, 927, 1510, 2438, 3872, 6095, 9465, 14578, 22210, 33581, 50305, 74831, 110441, 161955, 235858, 341474, 491365, 703263, 1001014, 1417812, 1998184, 2803342

Offset: 1

Omar E. Pol, Dec 05 2010

nonn

Essentially a trisection of A008483.

Cf. A026796, A008483, A135010, A138121, A182746, A182747, A182807, A182808.

More terms from D. S. McNeil, Dec 05 2010

A182807 Number of partitions of 3n+1 into parts >= 3.

Original entry on oeis.org

1, 2, 5, 10, 21, 39, 73, 130, 230, 391, 660, 1087, 1775, 2842, 4510, 7056, 10945, 16779, 25519, 38438, 57480, 85241, 125577, 183669, 267016, 385714, 554102, 791483, 1124831, 1590370, 2238095, 3134927

Offset: 1

Omar E. Pol, Dec 05 2010

nonn

Essentially a trisection of A008483.

Cf. A026796, A008483, A135010, A138121, A182746, A182747, A182806, A182808.

More terms from D. S. McNeil, Dec 05 2010

A182808 Number of partitions of 3n+2 into parts >= 3.

Original entry on oeis.org

1, 3, 6, 13, 25, 49, 88, 158, 273, 468, 779, 1284, 2075, 3323, 5237, 8182, 12625, 19323, 29269, 44004, 65585, 97084, 142627, 208233, 302008, 435525, 624363, 890414, 1263105, 1783200, 2505329

Offset: 1

Omar E. Pol, Dec 05 2010

nonn

Essentially a trisection of A008483.

Cf. A008483, A026796, A135010, A138121, A182746, A182747, A182806, A182807.

More terms from D. S. McNeil, Dec 05 2010

cp's OEIS Frontend

A185643 Triangular array E(n,k) counting, not necessarily connected, k-regular simple graphs on n vertices with girth exactly 3.

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A339164 Number of compositions (ordered partitions) of n into distinct parts, the least being 3.

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A185153 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth exactly 3.

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A185163 Number of not necessarily connected 6-regular simple graphs on n vertices with girth exactly 3.

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A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

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A182806 Number of partitions of 3n into parts >= 3.

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A182807 Number of partitions of 3n+1 into parts >= 3.

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A182808 Number of partitions of 3n+2 into parts >= 3.

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