Paul K. Stockmeyer - cp's OEIS frontend

A353581 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), with a(0) = 0 = a(1), a(2) = 2, and a(3) = 1.

0, 0, 2, 1, 2, 8, 20, 45, 108, 264, 638, 1537, 3710, 8960, 21632, 52221, 126072, 304368, 734810, 1773985, 4282778, 10339544, 24961868, 60263277, 145488420, 351240120, 847968662, 2047177441, 4942323542, 11931824528, 28805972600, 69543769725, 167893512048, 405330793824

Offset: 0

Paul K. Stockmeyer, May 04 2022

Paul K. Stockmeyer, Table of n, a(n) for n = 0..1000
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).

a(n) = (1/8) ((5-3*s)*(1+s)^n + (5+3*s)*(1-s)^n + 2*sin(n*Pi/2) - 10*cos(n*Pi/2)) where s = sqrt(2).

G.f.: x^2*(2 - 3*x)/((1 + x^2)*(1 - 2*x - x^2)). - Stefano Spezia, May 04 2022

A353582 a(n) = 2a(n-1) + 2a(n-3) + a(n-4) - 1, with a(0) = 0 = a(1), a(2) = 2, and a(3) = 3.

Original entry on oeis.org

0, 0, 2, 3, 5, 13, 33, 78, 186, 450, 1088, 2625, 6335, 15295, 36927, 89148, 215220, 519588, 1254398, 3028383, 7311161, 17650705, 42612573, 102875850, 248364270, 599604390, 1447573052, 3494750493, 8437074035, 20368898563, 49174871163, 118718640888, 286612152936, 691942946760

Offset: 0

Paul K. Stockmeyer, May 04 2022

Paul K. Stockmeyer, Table of n, a(n) for n = 0..1000
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (3,-2,2,-1,-1).

a(n) = (1/16)((4-s)*(1+s)^n + (4+s)*(1-s)^n - 8*sin(n*Pi/2) - 12*cos(n*Pi/2) + 4) where s = sqrt(2).

G.f.: x^2*(2 - 3*x)/((1 - x)*(1 + x^2)*(1 - 2*x - x^2)). - Stefano Spezia, May 04 2022

A353580 a(n) = 2*a(n-1) + a(n-2) - 1, with a(0) = 0 and a(1) = 2.

Original entry on oeis.org

0, 2, 3, 7, 16, 38, 91, 219, 528, 1274, 3075, 7423, 17920, 43262, 104443, 252147, 608736, 1469618, 3547971, 8565559, 20679088, 49923734, 120526555, 290976843, 702480240, 1695937322, 4094354883, 9884647087, 23863649056, 57611945198, 139087539451, 335787024099, 810661587648

Offset: 0

Paul K. Stockmeyer, May 03 2022

Paul K. Stockmeyer, Table of n, a(n) for n = 0..1000
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

nxt[{a_,b_}]:={b,2b+a-1}; NestList[nxt,{0,2},40][[;;,1]] (* or *) LinearRecurrence[{3,-1,-1},{0,2,3},40] (* Harvey P. Dale, Mar 14 2025 *)

a(n) = (1/4)((2*s - 1)*(1 + s)^n - (2*s + 1)*(1 - s)^n + 2) where s = sqrt(2).
G.f.: -x*(3*x-2)/((x-1)*(x^2+2*x-1)). - Alois P. Heinz, May 03 2022
E.g.f.: exp(x)*(1 - cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))/2. - Stefano Spezia, May 03 2022

A353579 Numbers in the smallest subset of N containing 1 and closed under the mappings k->2^n + k, k->32^{n+1} + k, and k->32^{n+2} + k where n = ceiling(log_2(k)).

Original entry on oeis.org

1, 2, 4, 7, 8, 13, 14, 15, 16, 26, 28, 29, 30, 31, 32, 52, 55, 56, 58, 60, 61, 62, 63, 64, 103, 104, 109, 110, 111, 112, 116, 119, 120, 122, 124, 125, 126, 127, 128, 205, 206, 207, 208, 218, 220, 221, 222, 223, 224, 231, 232, 237, 238, 239, 240, 244, 247, 248, 250, 252, 253, 254, 255, 256

Offset: 1

Paul K. Stockmeyer, May 03 2022

nonn

Paul K. Stockmeyer, Table of n, a(n) for n = 1..1155
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.

A353578 Numbers in the smallest subset of N containing 3 and closed under the mappings k->2k + 1, k->8k + 3, and k->16k + 3.

Original entry on oeis.org

3, 7, 15, 27, 31, 51, 55, 59, 63, 103, 111, 115, 119, 123, 127, 207, 219, 223, 231, 239, 243, 247, 251, 255, 411, 415, 435, 439, 443, 447, 463, 475, 479, 487, 495, 499, 503, 507, 511, 819, 823, 827, 831, 871, 879, 883, 887, 891, 895, 923, 927, 947, 951, 955, 959, 975, 987, 991, 999, 1007, 1011, 1015, 1019, 1023

Offset: 1

Paul K. Stockmeyer, May 03 2022

nonn

Paul K. Stockmeyer, Table of n, a(n) for n = 1..1155
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.

A351869 a(n) is the number of self-complementary score sequences that are possible for strong tournaments on n vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 9, 11, 30, 39, 103, 141, 363, 514, 1301, 1894, 4727, 7036, 17358, 26311, 64282, 98936, 239712, 373806, 899115, 1418130, 3389078, 5399133, 12828749, 20619565, 48739465, 78963217, 185769110, 303128971, 710067027, 1166206802, 2720959217, 4495461790

Offset: 0

Paul K. Stockmeyer, Feb 22 2022

nonn

See A000571 for the definition of a tournament and its score sequence. A tournament is strong if every two vertices are mutually reachable by directed paths. Alternatively, a tournament is strong if it contains a directed Hamiltonian cycle.

A sequence (s_1 <= s_2 <= ... <= s_n) of integers is the score sequence of a strong tournament iff Sum_{i=1..r} s_i > binomial(r,2) for 1 <= r < n and Sum_{i=1..n} s_i = binomial(n,2). It is self-complementary iff s_{n+1-i} = n-1-s_i for 1 <= i <= n/2.

			The 9 self-complementary strong score sequences of length seven are
  (1,1,2,3,4,5,5),
  (1,1,3,3,3,5,5),
  (1,2,2,3,4,4,5),
  (1,2,3,3,3,4,5),
  (1,3,3,3,3,3,5),
  (2,2,2,3,4,4,4),
  (2,2,3,3,3,4,4),
  (2,3,3,3,3,3,4),
  (3,3,3,3,3,3,3).

Paul K. Stockmeyer, Table of n, a(n) for n = 0..501
Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, arXiv:2202.05238 [math.CO], 2022.
Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, J. Integer Seq. 26 (2023), Article 23.5.2.

Cf. A000571, A351822, A345470.

For n >= 1, a(2*n) = Sum_{T=binomial(n,2)+1..n*(n-1)} Sum_{E=floor(n/2)..n-1} g_n(T,E) and a(2*n+1) = Sum_{T=binomial(n,2)+1..n^2} Sum_{E=floor(n/2)..n} g_n(T,E), where g_1(T, E)=[T=E], and for n>=2, g_n(T, E)=0 if T-E <= binomial(n-1, 2) and g_n(T, E) = Sum_{k=0..E} g_{n-1}(T-E, k) otherwise.

A351822 a(n) is the number of different score sequences that are possible for strong tournaments on n vertices.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 7, 21, 61, 184, 573, 1835, 5969, 19715, 66054, 223971, 767174, 2651771, 9240766, 32436016, 114596715, 407263306, 1455145050, 5224710466, 18843677124, 68243466611, 248090964092, 905092211818, 3312816854525, 12162610429661, 44780896121875, 165316324574671, 611819769698967

Offset: 0

Paul K. Stockmeyer, Feb 20 2022

nonn

See A000571 for the definition of a tournament and its score sequence. A tournament is strong if every two vertices are mutually reachable by directed paths. Alternatively, a tournament is strong if it contains a directed Hamiltonian cycle.

A sequence (s_1 <= s_2 <= ... <= s_n) of integers is the score sequence of a strong tournament iff Sum_{i=1..r} s_i > binomial(r,2) for 1 <= r < n and Sum_{i=1..n} s_i = binomial(n,2).

			The seven strong score sequences of length six are
  (1,1,2,3,4,4),
  (1,1,3,3,3,4),
  (1,2,2,2,4,4),
  (1,2,2,3,3,4),
  (1,2,3,3,3,3),
  (2,2,2,2,3,4),
  (2,2,2,3,3,3).

Paul K. Stockmeyer, Table of n, a(n) for n = 0..500
Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, arXiv:2202.05238 [math.CO], 2022.
Paul K. Stockmeyer, Counting Various Classes of Tournament Score Sequences, J. Integer Seq. 26 (2023), Article 23.5.2.

Cf. A000571.

For n >= 2, a(n) = Sum_{E=floor(n/2)..n-1} g_n(binomial(n, 2), E), where g_1(T, E) = [T=E]; g_n(T, E)=0 if T-E <= binomial(n-1, 2) and g_n(T, E) = Sum_{k=0..E} g_{n-1}(T-E, k) otherwise.

a(n) ~ c * 4^n / n^(5/2), where c = 0.202756471582408229... - Vaclav Kotesovec, Feb 21 2022

cp's OEIS Frontend

User: Paul K. Stockmeyer

A353581 a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), with a(0) = 0 = a(1), a(2) = 2, and a(3) = 1.

Views

Author

Keywords

Links

Formula

A353582 a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4) - 1, with a(0) = 0 = a(1), a(2) = 2, and a(3) = 3.

Views

Author

Keywords

Links

Formula

A353580 a(n) = 2*a(n-1) + a(n-2) - 1, with a(0) = 0 and a(1) = 2.

Views

Author

Keywords

Links

Programs

Mathematica

Formula

A353579 Numbers in the smallest subset of N containing 1 and closed under the mappings k->2^n + k, k->3*2^{n+1} + k, and k->3*2^{n+2} + k where n = ceiling(log_2(k)).

Views

Author

Keywords

Links

A353578 Numbers in the smallest subset of N containing 3 and closed under the mappings k->2k + 1, k->8k + 3, and k->16k + 3.

Views

Author

Keywords

Links

A351869 a(n) is the number of self-complementary score sequences that are possible for strong tournaments on n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A351822 a(n) is the number of different score sequences that are possible for strong tournaments on n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A353581 a(n) = 2a(n-1) + 2a(n-3) + a(n-4), with a(0) = 0 = a(1), a(2) = 2, and a(3) = 1.

A353582 a(n) = 2a(n-1) + 2a(n-3) + a(n-4) - 1, with a(0) = 0 = a(1), a(2) = 2, and a(3) = 3.

A353579 Numbers in the smallest subset of N containing 1 and closed under the mappings k->2^n + k, k->32^{n+1} + k, and k->32^{n+2} + k where n = ceiling(log_2(k)).