A357266 -id:A357266 - cp's OEIS frontend

A357248 Number of n-node tournaments that have exactly four circular triads.

280, 6240, 75600, 954240, 12579840, 175392000, 2594592000, 40721049600, 677053977600, 11901451161600, 220690229760000, 4307253350400000, 88289523818496000, 1896762491559936000, 42625344258072576000, 1000193047805952000000, 24463730767033958400000, 622724156293184225280000

Offset: 5

Ian R Harris, Ryan P. A. McShane, Sep 22 2022

nonn

			For n=5, the a(5)=280 solution is 5!*((7/3)*(5-4)+4*(5-5)+(7/6)(5-6)(5-7)[5>5]+(1/18)*(5-7)(5-8)(5-9)[5>6]+(1/1944)[5>7]*(5-8)!/(5-12)!)=5!*(7/3)*(5-4)=280.

Ian R. Harris and Ryan P. A. McShane, Counting Tournaments with a Specified Number of Circular Triads, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

Cf. A357242, A357257, A357266.

CoefficientList[Series[(x^7-27*x^6+216*x^5-702*x^4+972*x^3-405*x^2-243*x+189)*x^5/((3^4)*(1-x)^5), {x,0,22}], x]Table[n!, {n,0,22}] (* Stefano Spezia, Sep 27 2022 *)

a(n) = n!*((7/3)*(n-4)+4*(n-5)+(7/6)(n-6)(n-7)[n>5]+(1/18)*(n-7)(n-8)(n-9)[n>6]+(1/1944)[n>7]*(n-8)!/(n-12)!) (see Kadane).

E.g.f.: (x^7-27*x^6+216*x^5-702*x^4+972*x^3-405*x^2-243*x+189)*x^5/((3^4)*(1-x)^5).

A357257 Number of n-node tournaments that have exactly three circular triads.

Original entry on oeis.org

240, 2880, 33600, 403200, 5093760, 68275200, 972787200, 14724864000, 236396160000, 4016659046400, 72067387392000, 1362306097152000, 27071765360640000, 564357385912320000, 12317692759916544000, 280955128203509760000

Offset: 5

Ian R Harris, Ryan P. A. McShane, Sep 20 2022

nonn

			a(6) = 6!*(2*(6-4) + (1/3)*(6-5)*(6-6) + (1/162)*(6-6)*(6-7)*(6-8)*[6>5]) = 2880.

Ian R. Harris and Ryan P. A. McShane, Counting Tournaments with a Specified Number of Circular Triads, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

Cf. A357242, A357248, A357266.

Table[n!*(2*(n-4) + (1/3)*(n-5)*(n-6) + (1/162)*(n-6)*(n-7)*(n-8)*Boole[n>5]), {n,5,20}] (* Stefano Spezia, Sep 27 2022 *)

a(n) = n!*(2*(n-4) + (1/3)*(n-5)*(n-6) + (1/162)*(n-6)*(n-7)*(n-8)*[n>5]) (see Kadane).

E.g.f.: (x^4 - 18*x^3 + 72*x^2 - 108*x + 54)*x^5/((3^3)*(1-x)^4).

A367265 Numbers k such that there exists i >= 1 such that k divides 3^3^i - 1.

Original entry on oeis.org

1, 2, 13, 26, 109, 218, 433, 757, 866, 1417, 1514, 2834, 3889, 5629, 7778, 8209, 9841, 11258, 16418, 17497, 19682, 34994, 47197, 50557, 52489, 58321, 70957, 82513, 94394, 101114, 104978, 106717, 116642, 141914, 165026, 213434, 227461, 327781, 423901, 454922, 613561, 655562, 682357, 758173, 847802, 894781, 922441

Offset: 1

Jianing Song, Nov 11 2023

nonn

Note that 3^3^i - 1 divides 3^3^(i+1) - 1, so this sequence is also numbers k such that k divides 3^3^i - 1 for all sufficiently large i.
Also numbers k such that there exists i >= 1 such that k divides 3^^i - 1, where 3^^i = 3^3^...^3 (i times) = A014220(i-1).
Also numbers k such that ord(3,k) is a power of 3, where ord(a,k) is the multiplicative order of a modulo k: 3^3^i == 1 (mod k) if and only if ord(3,k) divides 3^i, so such i exists if and only if ord(3,k) is a power of 3.
If a term k is not squarefree, then it is divisible by p^2, where p is a Wieferich prime to base 3 (A014127) such that ord(3,p) is a power of 3. No such p is known.

			Suppose that q is an odd prime power such that ord(3,q) = 3^e. e = 1 gives q = 13; e = 2 gives q = 757; e = 3 gives q = 109, 433, 8209; e = 4 gives q = 3889, 1190701, 12557612956332313.

Cf. A094358 (squarefree divisors of 2^2^i - 1), A357266 (divisors of 3^3^i + 1), A014127.

The subsequence of primes is given by A367648.

isA357265(k) = (k%3!=0) && isprimepower(3*znorder(Mod(3,k)))

cp's OEIS Frontend

A357248 Number of n-node tournaments that have exactly four circular triads.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A357257 Number of n-node tournaments that have exactly three circular triads.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A367265 Numbers k such that there exists i >= 1 such that k divides 3^3^i - 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI