Doron Zeilberger - cp's OEIS frontend

A381508 Pisano period of Hexanacci numbers (A001592) mod n.

1, 7, 728, 14, 208, 728, 342, 28, 2184, 1456, 354312, 728, 9520, 2394, 1456, 56, 709928, 2184, 5227320, 1456, 124488, 354312, 279840, 728, 1040, 9520, 6552, 2394, 243880, 1456, 71040, 112, 4606056, 4969496, 35568, 2184, 20362908, 5227320, 123760, 1456, 201840

Offset: 1

Martin Guerra and Doron Zeilberger, Apr 24 2025

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..222
Martin Guerra and Doron Zeilberger, Maple program

Cf. A001175, A001592.

# load programs from linked file:
seq(Pis([[0$5, 1],[1$6]],n,400000), n=1..16);

from math import lcm
from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A381508(n):
    if n == 1:
        return 1
    f = factorint(n).items()
    if len(f) > 1:
        return lcm(*(A381508(a**b) for a,b in f))
    else:
        k, x = 1, (0,0,0,0,1,1)
        while x != (0,0,0,0,0,1):
            k += 1
            x = x[1:]+(sum(x) % n,)
        return k # Chai Wah Wu, Apr 25 2025

a(17)-a(41) from Alois P. Heinz, Apr 25 2025

A359126 A000168(n+1) - A000139(n).

Original entry on oeis.org

0, 8, 52, 372, 2894, 23966, 208086, 1874508, 17390158, 165248499, 1601857338, 15790898316, 157915304928, 1598927475749, 16365689821454, 169113248927772, 1762344520554606, 18504654979649615, 195620858324078190, 2080695883684277190, 22254407183551916850

Offset: 0

N. J. A. Sloane, Dec 23 2022, following a suggestion from Doron Zeilberger

nonn

Number of separable rooted planar maps with n+1 edges. - Noam Zeilberger, Dec 26 2022

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Cf. A000139, A000168.

a := n -> 2*(3^(n + 1)*(2*n + 2)!/(n + 3)! - (3*n)!/(2*n + 1)!)/(n + 1)!:
seq(a(n), n = 0..20); # Peter Luschny, Dec 26 2022

a(n) ~ ((48^(n + 1) - 3^(3*n + 1/2)))/(2^(2*n + 1)*sqrt(Pi)*n^(5/2)). - Peter Luschny, Dec 26 2022

D-finite with recurrence -2*(389*n-1012)*(2*n+1)*(n+3)*(n+1)*a(n) +3*(14101*n^4-20062*n^3-56389*n^2+45022*n-6072)*a(n-1) +18*(-20677*n^4+100317*n^3-137223*n^2+14267*n+52524)*a(n-2) +108*(547*n-956)*(3*n-7)*(2*n-3)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Jan 25 2023

A339634 Number of final Tic-Tac-Toe positions on a (2*n+1) X 3 board that result in a tie.

Original entry on oeis.org

3, 16, 30, 72, 178, 444, 1114, 2808, 7098, 17984, 45656, 116106, 295718, 754226, 1926060, 4924188, 12602416, 32284214, 82777240, 212415744, 545495716, 1401849594, 3604921774, 9275890122, 23881602058, 61518226734, 158548607640, 408814563524, 1054590179342

Offset: 0

Doron Zeilberger, Taerim Kim, Karnaa Mistry, Weiji Zheng, Dec 10 2020

nonn

The number of (2*n+1) X 3 0,1-matrices with 3*n+2 1's and 3*n+1 0's and no consecutive horizontal, vertical, nor diagonal triples of 111 or 000.

			For n = 0 it is a 3 X 1 matrix, and there are 3 arrangements of 2 1's and a single 0 such that there are no 3-streaks of 1's nor 0's in the matrix.
For n = 1 it is the classic 3 X 3 Tic-Tac-Toe board, having 1's as X's and 0's as O's.

Doron Zeilberger, Class Projects for Combinatorics Fall 2020 (Rutgers University); Local copy [Pdf file only, no active links]
Doron Zeilberger, Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University); Local copy
Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University); Local copy [Pdf file only, no active links]

Bisection of A339631 (odd part).

Cf. A339633 (even part).

# Maple program adapted from OddTTT3(N) in Project 5 of Doron Zeilberger's Combinatorics Class Fall 2020 (Rutgers University).
A339634List:=proc(n) local f,i,t,x,N:
f:=(4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1):
N:=n-1:
#Take the Taylor expansion up to x^(2*N+2)
f:=taylor(f,x=0,2*N+3):
#Extract the coefficients of x^(2*i+1)*t^(3*i+2)
[3,seq(coeff(coeff(f,x,2*i+1),t,3*i+2),i=1..N)]:
end:

a(n) = [x^(2*n+1)*t^ceiling(3*(2*n+1)/2)] (4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1) for n >= 1.

A339633 Number of final Tic-Tac-Toe positions on a (2*n) X 3 board that resulted in a tie.

Original entry on oeis.org

1, 18, 28, 58, 140, 334, 824, 2038, 5084, 12730, 32004, 80694, 204004, 516902, 1312336, 3337682, 8502132, 21688182, 55395140, 141651742, 362601356, 929084578, 2382677360, 6115461118, 15707982020, 40375223374, 103846409504, 267258086338, 688201711116, 1773088924494

Offset: 0

Doron Zeilberger, Taerim Kim, Karnaa Mistry, Weiji Zheng, Dec 10 2020

nonn

The number of (2*n) X 3 0,1 matrices with 3*n 1's and 3*n 0's and no consecutive horizontal, vertical, nor diagonal triples of 111 or 000.

			For n = 1 it is a 3 X 2 matrix, and so there are 18 ways to have three 1's and three 0's such that there are no 3-streaks of 1's nor 0's in the matrix.

Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Doron Zeilberger, Class Projects for Combinatorics Fall 2020 (Rutgers University).
Doron Zeilberger, Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University).
Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Bisection of A339631 (even part).

Cf. A339634 (odd part).

# Maple program adapted from EvenTTT3(N) in Project 5 of Doron Zeilberger's Combinatorics Class Fall 2020 (Rutgers University).
A339633List:=proc(n) local f,i,t,x,N:
f:=(4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1):
N:=n-1:
#Take the Taylor expansion up to x^(2*N+2)
f:=taylor(f,x=0,2*N+3):
#Extract the coefficients of x^(2*i)*t^(3*i)
[1,seq(coeff(coeff(f,x,2*i),t,3*i),i=1..N)]:
end:

a(n) = [x^(2*n)*t^(3*n)] (4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1) for n >= 1.

A339631 Number of 3 X n matrices with 3n/2 1's (if n is even) or (3n+1)/2 1's (if n is odd) that do not have a horizontal nor vertical nor diagonal 3-streak of 1's.

Original entry on oeis.org

1, 3, 18, 16, 28, 30, 58, 72, 140, 178, 334, 444, 824, 1114, 2038, 2808, 5084, 7098, 12730, 17984, 32004, 45656, 80694, 116106, 204004, 295718, 516902, 754226, 1312336, 1926060, 3337682, 4924188, 8502132, 12602416, 21688182, 32284214, 55395140, 82777240, 141651742, 212415744

Offset: 0

Doron Zeilberger, Taerim Kim, Karnaa Mistry, Weiji Zheng, Dec 10 2020

nonn

Provided by D. Zeilberger's Maple package (ComboProject5.txt) for Combinatorics Fall 2020 at Rutgers University (see links). Generated using alternating procedures EvenTTT3() and OddTTT3() from this Maple package.

			For n = 1 it is a 3 X 1 matrix, and there is no way to have a 3-streak of 1's or 0's since there must be 2 1's and 1 0, so there are three matrices [110],[011],[101].
For n = 3 it is the classic Tic-Tac-Toe board, with 1's being X's and 0's being O's.

Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Doron Zeilberger, Class Projects for Combinatorics Fall 2020 (Rutgers University).
Doron Zeilberger, Initial Maple Package for Project 5 Combinatorics Fall 2020 (Rutgers University).
Doron Zeilberger, Math 454, Section 02 (Combinatorics) Fall 2020 (Rutgers University).

Bisections give: A339633 (even part), A339634 (odd part).

using Nemo
function A339631List(prec)
    R, t = PolynomialRing(ZZ, "t")
    S, x = PowerSeriesRing(R, prec+1, "x")
    num = (4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2)
    den = (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1)
    ser = divexact(num, den)
    C = [coeff(coeff(ser, n), div(3*n, 2)) for n in 0:prec]
    C[1] = 1; C[2] = 3
    return C
end
A339631List(39) |> println # Peter Luschny, Dec 19 2020

a(n) = [x^n*t^ceiling(3*n/2)] (4*t^17*x^11 + 4*t^16*x^11 + 8*t^16*x^10 + 12*t^15*x^10 + 6*t^15*x^9 + 8*t^14*x^10 + 8*t^14*x^9 + 8*t^13*x^9 - 2*t^13*x^8 + 6*t^12*x^9 - 16*t^12*x^8 - 2*t^11*x^8 - 26*t^11*x^7 - 26*t^10*x^7 - 19*t^10*x^6 - 38*t^9*x^6 - 7*t^9*x^5 - 19*t^8*x^6 - 13*t^8*x^5 - 13*t^7*x^5 - t^7*x^4 - 7*t^6*x^5 + 10*t^6*x^4 - t^5*x^4 + 16*t^5*x^3 + 16*t^4*x^3 + 9*t^4*x^2 + 18*t^3*x^2 + 9*t^2*x^2) / (t^12*x^8 + t^11*x^7 + t^10*x^7 + t^9*x^6 - 2*t^6*x^4 - t^5*x^3 - t^4*x^3 - t^3*x^2 + 1) for n >= 2.

A317955 Apply the morphism 1 -> 12, 2 -> 13, 3 -> 14, 4 -> 1 n times to 1, and concatenate the resulting string.

Original entry on oeis.org

1, 12, 1213, 12131214, 121312141213121, 12131214121312112131214121312, 12131214121312112131214121312121312141213121121312141213

Offset: 1

N. J. A. Sloane and Doron Zeilberger, Aug 21 2018

nonn

a(n) is the concatenation a(n) = a(n-1).a(n-2).a(n-3).a(n-4) for n >= 5.

Seiichi Manyama, Table of n, a(n) for n = 1..11

Cf. A317953, A317954.

A317954 Fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 14, 4 -> 1, starting from a(1) = 1.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2

Offset: 1

N. J. A. Sloane and Doron Zeilberger, Aug 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003849, A092782, A100619, A317955.

A:= [1]:
for n from 1 while nops(A) < 1000 do
  A:= subs([1=(1,2),2=(1,3),3=(1,4),4=1],A)
od:
A; # Robert Israel, Aug 21 2018

Nest[Flatten[#/.{0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 0}}] &, {0}, 7] (* G. C. Greubel, Jan 02 2019 *)

A317191 Fill an n X n square array T(j,k), 1<=j<=n, 1=k<=n, by antidiagonals upwards in which each term is the least nonnegative integer satisfying the condition that no row, column, diagonal, or antidiagonal contains a repeated term; a(n) = T(n,n).

Original entry on oeis.org

0, 3, 5, 4, 1, 10, 7, 2, 6, 8, 15, 12, 19, 17, 22, 23, 12, 26, 11, 31, 32, 12, 35, 10, 37, 42, 40, 45, 33, 49, 18, 17, 20, 53, 16, 51, 59, 18, 59, 60, 58, 64, 69, 69, 38, 29, 74, 26, 68, 78, 80, 36, 30, 33, 41, 39, 32, 33, 92, 41, 38, 89, 32, 35

Offset: 1

N. J. A. Sloane and Doron Zeilberger, Jul 30 2018

nonn

This is the analog for an n X n board of the sequence A317190 (which is the main diagonal when we fill in the whole of the fourth quadrant in this way).

			For n=3 the array T is
0 2 1
1 3 4
2 0 5
so a(3) = T(3,3) = 5.
For n=6 the array T is
0 2 1 5 3 4
1 3 4 0 7 2
2 0 5 1 6 9
3 1 2 4 0 5
4 6 0 3 1 7
5 7 8 6 4 10
so a(6) = T(6,6) = 10. This is the first time this sequence differs from A317190.

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A317190, A274318, A269529, A274528.

A316333 a(n) = A000085(4*n+3)/2^(n+2).

Original entry on oeis.org

1, 29, 2231, 323423, 75151585, 25451905333, 11799518538967, 7161375112402823, 5503252915369107329, 5217568316626716585485, 5977663757214838174587319, 8136442760259565566724537711, 12972630954295515566319694027489, 23938932303527622015634131021262757

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316330, A316331, A316332.

A316332 a(n) = A000085(4*n+2)/2^(n+1).

Original entry on oeis.org

1, 19, 1187, 149405, 31166057, 9670072483, 4163946939067, 2370770585582221, 1722046856020416785, 1552401874990891104371, 1699257737580930574489619, 2218555640616875773883091901, 3404174268230266459851637679353, 6062646848508401565245592651382915, 12398960005973049406349011215379703723

Offset: 0

N. J. A. Sloane, Jul 09 2018, following a suggestion from Doron Zeilberger

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..210
S. Chowla, I. N. Herstein and W. K. Moore, On recursions connected with symmetric groups I, Canad. J. Math. 3 (1951) 328-334.
Dongsu Kim and Jang Soo Kim, A combinatorial approach to the power of 2 in the number of involutions, J. Comb. Theory Ser. A 117 (8) (2010): 1082-1094.

Cf. A000085, A316330, A316331, A316333.

cp's OEIS Frontend

User: Doron Zeilberger

A381508 Pisano period of Hexanacci numbers (A001592) mod n.

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Maple

Python

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A359126 A000168(n+1) - A000139(n).

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Maple

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A339634 Number of final Tic-Tac-Toe positions on a (2*n+1) X 3 board that result in a tie.

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Maple

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A339633 Number of final Tic-Tac-Toe positions on a (2*n) X 3 board that resulted in a tie.

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Maple

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A339631 Number of 3 X n matrices with 3*n/2 1's (if n is even) or (3*n+1)/2 1's (if n is odd) that do not have a horizontal nor vertical nor diagonal 3-streak of 1's.

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Julia

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A317955 Apply the morphism 1 -> 12, 2 -> 13, 3 -> 14, 4 -> 1 n times to 1, and concatenate the resulting string.

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A317954 Fixed point of the morphism 1 -> 12, 2 -> 13, 3 -> 14, 4 -> 1, starting from a(1) = 1.

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Maple

Mathematica

A317191 Fill an n X n square array T(j,k), 1<=j<=n, 1=k<=n, by antidiagonals upwards in which each term is the least nonnegative integer satisfying the condition that no row, column, diagonal, or antidiagonal contains a repeated term; a(n) = T(n,n).

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A339631 Number of 3 X n matrices with 3n/2 1's (if n is even) or (3n+1)/2 1's (if n is odd) that do not have a horizontal nor vertical nor diagonal 3-streak of 1's.