Robert FERREOL - cp's OEIS frontend

A378683 a(0) = 1, a(n+1) = 6a(n)^3 - 3a(n).

1, 3, 153, 21489003, 59538796254981950751153, 1266343134315970349117919634635229303292221774134557782012151266098003

Offset: 0

Robert FERREOL, Dec 03 2024

nonn

If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = A238799(n) / a(n) ; this is Halley's method to calculate sqrt(3).

Wikipedia, Halley's method.

Cf. A002530, A002605, A238799.

a:=1 : A:=NULL : for k to 5 do a:=6*a^3-3*a : A:=A,a od : A;

NestList[6#^3-3# &, 1, 5] (* Stefano Spezia, Dec 06 2024 *)

a(n) = ((1 + sqrt(3))^(3^n) - (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) / sqrt(3).
a(n) = A002530(3^n).
a(n) = A002605(3^n)/2^((3^n+1)/2).

A371805 Composite numbers k that divide A001644(k) - 1.

Original entry on oeis.org

182, 25201, 54289, 63618, 194390, 750890, 804055, 1889041, 2487941, 3542533, 3761251, 6829689, 12032021, 12649337, 18002881

Offset: 1

Robert FERREOL, Apr 06 2024

If k is prime, k divides A001644(k) - 1; and since A001644(k) satisfies a tribonacci recurrence, these numbers could be called tribonacci pseudoprimes.

			(A001644(182)-1)/182 = 8056145960961609628091266244940745410646318417.

Cf. A001644.
Cf. A005845 (composite k that divide Lucas(k) - 1).
Cf. A013998 (composite k that divide Perrin(k) - 1).

A001644:=proc(n) option remember: if n=0 then 3 elif n=1 then 1 elif n=2 then 3 else A001644(n-1)+A001644(n-2)+A001644(n-3) fi end:
test:=n->A001644(n) mod n = 1:select(test and not isprime, [seq(n, n=1..100000)]);

seq[kmax_] := Module[{x = 1, y = 3, z = 7, s = {}, t}, Do[t = x + y + z; If[Mod[t, k] == 1 && CompositeQ[k], AppendTo[s, k]]; x = y; y = z; z = t, {k, 4, kmax}]; s]; seq[200000] (* Amiram Eldar, Apr 06 2024 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    t0, t1, t2 = 3, 1, 3
    for k in count(1):
        t0, t1, t2 = t1, t2, t0+t1+t2
        if k > 1 and not isprime(k) and (t0-1)%k == 0:
            yield k
print(list(islice(agen(), 5))) # Michael S. Branicky, Apr 07 2024

a(13)-a(15) from Amiram Eldar, Apr 07 2024

A352615 Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.

Original entry on oeis.org

2, 0, 7, 7, 6, 8, 1, 4, 6, 0, 0, 2, 8, 1, 5, 8, 2, 0, 5, 7, 8, 3, 1, 2, 0, 5, 5, 6, 7, 8, 5, 5, 2, 9, 0, 1, 2, 8, 0, 3, 7, 7, 9, 0, 5, 7, 6, 2, 4, 7, 8, 2, 4, 1, 0, 0, 6, 3, 5, 0, 3, 8, 0, 4, 8, 2, 2, 6, 3, 5, 5, 3, 1, 4, 6, 3, 2, 0, 3, 8, 4, 6, 3, 3, 0, 1, 6, 0, 0, 0, 0, 9, 6, 9, 2, 1, 9, 0, 7, 5, 2, 3, 4, 0, 5

Offset: 1

Robert FERREOL, Mar 23 2022

			2.0776814600281582057831205567855290128037790576247...

Robert Ferréol, Bohemian dome, Mathcurve.

Cf. A091670 ((1/Pi^2)*Integral_{0<=x,y<=Pi} 1/sqrt(1-cos^2(x)*cos^2(y)) dx dy).

a:=1/sqrt(2):evalf((EllipticE(a)-EllipticK(a))^2+EllipticE(a)^2,50);

RealDigits[(EllipticE[1/2] - EllipticK[1/2])^2 + EllipticE[1/2]^2, 10, 105][[1]] (* Amiram Eldar, Mar 24 2022 *)

Equals Sum _{n>=0} (Pi^2/(4*(2*n-1))*(binomial(2*n,n)/4^n)^3).

Equals (E(a) - K(a))^2 + E(a)^2 where a = 1/sqrt(2) and E (resp. K) is the complete elliptic integral of the second (resp. first) kind.

A307768 Number of n-step random walks on a line starting from the origin and returning to it at least once.

Original entry on oeis.org

0, 0, 2, 4, 10, 20, 44, 88, 186, 372, 772, 1544, 3172, 6344, 12952, 25904, 52666, 105332, 213524, 427048, 863820, 1727640, 3488872, 6977744, 14073060, 28146120, 56708264, 113416528, 228318856, 456637712, 918624304, 1837248608, 3693886906, 7387773812, 14846262964, 29692525928

Offset: 0

Robert FERREOL, Apr 27 2019

a(n)/2^n tends to 1 as n goes to infinity; this means that on the line any random walk returns sooner or later to its starting point with a probability 1.

a(n) is also the number of heads-or-tails games of length n during which at some point there are as many heads as tails.

			The a(3)=4 three-step walks returning to 0 are [0, -1, 0, -1], [0, -1, 0, 1], [0, 1, 0, -1], [0, 1, 0, 1].
The a(4)=10 three-step walks returning to 0 are [0, -1, -2, -1, 0], [0, -1, 0, -1, -2], [0, -1, 0, -1, 0], [0, -1, 0, 1, 0], [0, -1, 0, 1, 2], [0, 1, 0, -1, -2], [0, 1, 0, -1, 0], [0, 1, 0, 1, 0], [0, 1, 0, 1, 2], [0, 1, 2, 1, 0].

Robert Israel, Table of n, a(n) for n = 0..3318
Monique Laurent, Sven Polak, and Luis Felipe Vargas, Semidefinite approximations for bicliques and biindependent pairs, arXiv:2302.08886 [math.CO], 2023.

Cf. A027306, A063886, A045621, A128014, A284016.

b:=n->piecewise(n mod 2 = 0,binomial(n,n/2),2*binomial(n-1,(n-1)/2)):
seq(2^n-b(n),n=0..20);
# second program:
A307768 := series(exp(2*x) - int((1/x + 2)*BesselI(1,2*x),x) - BesselI(1,2*x), x = 0, 36): seq(n!*coeff(A307768, x, n), n = 0 .. 35); # Mélika Tebni, Jun 19 2024

a[n_] := If[n == 0, 0, 2^n - 2*Binomial[n-1, Floor[(n-1)/2]]];
Array[a, 36, 0] (* Jean-François Alcover, May 05 2019 *)

a(n) = 2^n - A063886(n).
a(n+1) = 2*A045621(n) = 2*(2^n - binomial(n,floor(n/2))).
a(2n) = 2^(2n) - binomial(2n,n); a(2n+1) = 2*a(2n).
G.f.: (1-sqrt(1-4*x^2))/(1-2*x). - Alois P. Heinz, May 05 2019
n*(a(n)-2*a(n-1)) - 4*(n-3)*(a(n-2)-2*a(n-3)) = 0. - Robert Israel, May 06 2019
a(2n+2) - 2*a(2n+1) = A284016(n) = 2*Catalan(n). - Robert FERREOL, Aug 26 2019
From Mélika Tebni, Jun 19 2024: (Start)
E.g.f.: exp(2*x) - Integral_{x=-oo..oo} (1/x + 2)*BesselI(1, 2*x) dx - BesselI(1, 2*x).
a(n) = 2*(A027306(n) - A128014(n)). (End)

A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).

Original entry on oeis.org

2, 3, 7, 43, 13, 3263443, 547, 29881, 5295435634831, 181, 2287, 73

Offset: 0

Robert FERREOL, Jan 19 2019

a(n) is also the smallest prime divisor of A007018(n+1) that is not a divisor of A007018(n).
The prime numbers a(n) are all distinct, which proves the infinitude of the prime numbers (Saidak's proof).
a(12) <= 2589377038614498251653. - Daniel Suteu, Jan 20 2019
a(12)..a(50) = [?, 52387, 13999, 17881, 128551, 635263, ?, ?, 352867, 387347773, ?, 74587, ?, ?, 27061, 164299, 20929, 1171, ?, 1679143, ?, ?, 120823, 2408563, 38218903, 333457, 30241, 4219, 1085443, 7603, 1861, ?, 23773, 51769, 1285540933, 429547, ?, 8323570543, ?], where ? denote unknown values > 10^10. - Max Alekseyev, Oct 11 2023

Filip Saidak, A new proof of Euclid's theorem, Amer. Math. Monthly, 113:10 (2006) 937-938.
Wikipedia, Sylvester's sequence: Divisibility and factorizations.

Cf. A007018, A000058, A007996, A091335, A180871.

with(numtheory):
u:=1: P:=NULL: to 9 do P:=P,sort([op(divisors(u+1))])[2]: u:=u*(u+1) od:
P;

f(n)=if(n<1, n>=0, f(n-1)+f(n-1)^2); \\ A007018
a(n)=divisors(f(n)+1)[2]; \\ Michel Marcus, Jan 20 2019

a(n) = A007996(m), where m is the smallest index such that A180871(m) = n. - Max Alekseyev, Oct 11 2023

a(10)-a(11) from Daniel Suteu, Jan 20 2019

A322419 Number of n-step self-avoiding walks on L-lattice.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 564, 904, 1448, 2320, 3684, 5872, 9376, 14960, 23688, 37652, 59912, 95316, 150744, 239080, 379528, 602424, 951788, 1507136, 2388252, 3784344, 5973988, 9447880, 14950796, 23658540, 37321752, 58965260, 93206864, 147333080, 232286272

Offset: 0

Robert FERREOL, Dec 07 2018

The L-lattice is an oriented square lattice in which each step must be followed by a step perpendicular to the preceding one.

			a(1) = 2 because there are only two possible directions at each intersection; for the same reason a(2) = 2*2 and a(3) = 2*4 ; but a(4) = 12 (not 16) because four paths return to the starting point and are not self-avoiding. See the 12 paths under "links".

Sean A. Irvine, Table of n, a(n) for n = 0..50
Robert FERREOL, The a(4)=12 walks in L-lattice
Keh-Ying Lin and Yee-Mou Kao, Universal amplitude combinations for self-avoiding walks and polygons on directed lattices, J. Phys. A: Math. Gen. 32 (1999), page 6929.
A. Malakis, Self-avoiding walks on oriented square lattices, Journal of Physics A: Mathematical and General, Volume 8, Number 12 (1975), page 1890.
Wikipedia, Connective constant

Cf. A001411 (square lattice), A117633 (Manhattan lattice), A189722, A004277 (coordination sequence), A151798.

walks:=proc(n)
    option remember;
    local i,father,End,X,walkN,dir,u,x,y;
    if n=1 then [[[0,0]]] else
         father:=walks(n-1):
         walkN:=NULL:
         for i to nops(father) do
            u:=father[i]:End:=u[n-1]:if n mod 2 = 0 then
            dir:=[[1,0], [-1, 0]] else dir := [[0,1], [0, -1]] fi:
            for X in dir do
             if not(member(End+X,u)) then walkN:=walkN,[op(u),End+X] fi;
             od od:
         [walkN] fi end:
n:=5:L:=walks(n):N:=nops(L);
# This program explicitly gives the a(n) walks.

mo = {{1, 0}, {-1, 0}}; moo = {{0, 1}, {0, -1}}; a[0] = 1;
a[tg_, p_: {{0, 0}}] := Module[{e, mv},
If[Mod[tg, 2] == 0, mv = Complement[Last[p] + # & /@ mo, p],
mv = Complement[Last[p] + # & /@ moo, p]];
If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]];
a /@ Range[0, 20] (* after the program from Giovanni Resta at A001411 *)

def add(L, x):
    M = [y for y in L]
    M.append(x)
    return M
plus = lambda L, M: [x + y for x, y in zip(L, M)]
mo = [[1, 0], [-1, 0]]
moo = [[0, 1], [0, -1]]
def a(n, P=[[0, 0]]):
    if n == 0:
        return 1
    if n % 2 == 0:
        mv1 = [plus(P[-1], x) for x in mo]
    else:
        mv1 = [plus(P[-1], x) for x in moo]
    mv2 = [x for x in mv1 if x not in P]
    if n == 1:
        return len(mv2)
    else:
        return sum(a(n - 1, add(P, x)) for x in mv2)
[a(n) for n in range(21)]

a(n) = 4*A189722(n) for n >= 2.

It is proved that a(n)^(1/n) has a limit mu called the "connective constant" of the L-lattice; approximate value of mu: 1.5657. It is only conjectured that a(n + 1) ~ mu * a(n).

A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

Original entry on oeis.org

1, 1, 0, 1, 2, 6, 1, 0, 14, 0, 1, 4, 37, 154, 1072, 1, 0, 92, 0, 5320, 0, 1, 8, 236, 1696, 32675, 301384, 4638576, 1, 0, 596, 0, 175294, 0, 49483138, 0, 1, 16, 1517, 18684, 1024028, 17066492, 681728204, 13916993782, 467260456608

Offset: 2

Robert FERREOL, Jan 10 2019

Orientation of the path is not important; you can start going either clockwise or counterclockwise. Paths related by symmetries are considered distinct.
The m X n grid of points when drawn forms a (m-1) X (n-1) rectangle of cells, so m-1 and n-1 are sometimes used as indices instead of m and n (see, e. g., Kreweras' reference below).
These cycles are also called "closed non-intersecting rook's tours" on m X n chess board.

			T(5,4)=14 is illustrated in the links above.
Table starts:
=================================================================
m\n|  2    3      4       5         6           7            8
---|-------------------------------------------------------------
2  |  1    1      1       1         1           1            1
3  |  1    0      2       0         4           0            8
4  |  1    2      6      14        37          92          236
5  |  1    0     14       0       154           0         1696
6  |  1    4     37     154      1072        5320        32675
7  |  1    0     92       0      5320           0       301384
8  |  1    8    236    1696     32675      301384      4638576
The table is symmetric, so it is completely described by its lower-left half.

Huaide Cheng, Rows n=2..17 of triangle, flattened
Robert Ferréol, The T(4,5)=14 hamiltonian cycles on 4 X 5 square grid of points; the T(5,6) = 154 cycles on 5 X 6 grid are reduced to 44 different forms.
Germain Kreweras, Dénombrement des cycles hamiltoniens dans un rectangle quadrillé, European Journal of Combinatorics, Volume 13, Issue 6 (1992), page 476.
Robert Stoyan and Volker Strehl, Enumeration of Hamiltonian Circuits in Rectangular Grids, Séminaire Lotharingien de Combinatoire, B34f (1995), 21 pp.
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.

Row/column k=4..12 are: (with interspersed zeros for odd k): A006864, A006865, A145401, A145416, A145418, A160149, A180504, A180505, A213813.
Cf. A003763 (bisection of main diagonal), A222200 (subdiagonal), A231829, A270273, A332307.
T(n,2n) gives A333864.

# Program due to Laurent Jouhet-Reverdy
def cycle(m, n):
     if (m%2==1 and n%2==1): return 0
     grid = [[0]*n for _ in range(m)]
     grid[0][0] = 1; grid[1][0] = 1
     counter = [0]; stop = m*n-1
     def run(i, j, nb_points):
         for ni, nj in [(i-1, j), (i+1, j), (i, j+1), (i, j-1)] :
             if  0<=ni<=m-1 and 0<=nj<=n-1 and grid[ni][nj]==0 and (ni,nj)!=(0,1):
                 grid[ni][nj] = 1
                 run(ni, nj, nb_points+1)
                 grid[ni][nj] = 0
             elif (ni,nj)==(0,1) and nb_points==stop:
                 counter[0] += 1
     run(1, 0, 2)
     return counter[0]
L=[];n=7#maximum for a time < 1 mn
for i in range(2,n):
    for j in range(2,i+1):
       L.append(cycle(i,j))
print(L)

T(m,n) = T(n,m).
T(2m+1,2n+1) = 0.
T(2n,2n) = A003763(n).
T(n,n+1) = T(n+1,n) = A222200(n).
G. functions , G_m(x)=Sum {n>=0} T(m,n-2)*x^n after Stoyan's link p. 18 :
G_2(x) = 1/(1-x) = 1+x+x^2+...
G_3(x) = 1/(1-2*x^2) = 1+2*x^2+4*x^4+...
G_4(x) = 1/(1-2*x-2*x^2+2*x^3-x^4) = 1+2*x+6*x^2+...
G_5(x) = (1+3*x^2)/(1-11*x^2-2*x^6) = 1+14*x^2+154*x^4+...

More terms from Pontus von Brömssen, Feb 15 2021

A320122 Numbers that are not Keith numbers in any base.

Original entry on oeis.org

12, 30, 390, 1170, 1200, 1560, 2340, 2760, 3120, 3900, 4680, 6120, 6240, 7680, 7800, 8460, 10020, 10140, 10950, 11580, 15090, 15480, 17160, 17580, 18360, 19140, 20280, 20700, 20940, 21480, 23040, 23280, 24060, 24210, 24960, 26550, 28740, 29250, 29520, 29670, 30060, 31080, 32400

Offset: 1

Robert FERREOL, Oct 06 2018

A number N >= 2 is a Keith number in a base b <= N if the Fibonacci sequence u(i) whose initial terms are the t digits of N in the base b, and later terms are given by rule that u(i) = sum of t previous terms, contains N itself. Here a(n) is the n-th number N that is not a Keith number in any base b <= N.

			a(1) = 12 because 12 is not a Keith number in any base from 2 to 12, while all previous numbers are in some base.
For example, with b = 2, the sequence is : 1, 1, 0, 0, 2, 3, 5, 10, 20, ...; it doesn't contain 12. See A251703.

Arie Bos, Inventory of n-step Fibonacci sequences, 2015.

Cf. A007629 (Keith numbers in base 10).

fibo:=proc(n, b) local L,m,M,k:
L:=convert(n,base,b):m:=nops(L):M:=seq(L[m+1-k],k=1..m):
while M[m]

iskb(n, b) = if(nA007629
isok(n) = if (n<=2, 0, for(b=2, n-1, if (iskb(n, b), return(0))); return (1)); \\ Michel Marcus, Oct 08 2018

def digits(n, b):
    r = []
    m = n
    while m > 0:
        r = [m % b] + r
        m = m // b
    return r
def fibo(n, b):
    L = digits(n, b)
    m = len(L) - 1
    while L[m] < n:
        L.append(sum(k for k in L))
        L.pop(0)
    return L[m] == n
def test(n):
    for b in range(2, n + 1):
        if fibo(n, b):
            return True
    return False
print([n for n in range(2, 2001) if not test(n)])

More terms from Michel Marcus, Oct 08 2018

A284458 Number of pairs (f,g) of endofunctions on [n] such that the composite function gf has no fixed point.

Original entry on oeis.org

1, 0, 2, 156, 16920, 2764880, 650696400, 210105425628, 89425255439744, 48588905856409920, 32845298636854828800, 27047610425293718239100, 26664178085975252011318272, 31009985808408471580603417296, 42017027730087624384021945067520

Offset: 0

Robert FERREOL, Mar 27 2017

nonn

Consider n boys and n girls, each boy secretly loving a girl and vice versa. The probability that no couple could be formed is a(n)/n^(2n).
a(n) is the number of pairs of binary matrices n X n (M, N), M having only one 1 per row, N having only one 1 per column, such that M + N has no coefficient equal to 2.
Limit_{n -> infinity} a(n)/n^(2n) = 1/e.

			For two boys A,B and two girls A',B', the a(2) possibilities are:
A loves A' who loves B who loves B' who loves A;
A loves B' who loves B who loves A' who loves A.

Robert Israel, Table of n, a(n) for n = 0..214
Math StackExchange, An unexpected application of non-trivial combinatorics, 2014.
Math StackExchange, Couple Probability, 2014.
Eric Weisstein's MathWorld, Confluent Hypergeometric Function of the Second Kind

Cf. A065440, A350558, A343700.

a:=n->add((-1)^k*binomial(n,k)^2*k!*n^(2*(n-k)),k=0..n):
seq(a(n),n=0..15);

Table[Sum[(-1)^k Binomial[n,k]^2 * k! * n^(2*(n - k)), {k, 0, n}], {n, 1, 15}] (* Indranil Ghosh, Mar 27 2017 *)
Table[HypergeometricU[-n, 1, n^2], {n, 1, 15}] (* Jean-François Alcover, Mar 29 2017 *)

a(n) = sum(k=0, n, (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k))); \\ Michel Marcus, Apr 04 2017

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * k! * n^(2*(n-k)).

a(n) = A343700(n)/A350558(n). - Dan Eilers, Jan 17 2023

A283654 Triangle T(n,m) read by rows: number of n X m binary matrices with no rows or columns in which all entries are the same (n >= 1, 1 <= m <= n).

Original entry on oeis.org

0, 0, 2, 0, 6, 102, 0, 14, 906, 22874, 0, 30, 6510, 417810, 17633670, 0, 62, 42666, 6644714, 622433730, 46959933962, 0, 126, 267582, 99044946, 20218802310, 3204360965106, 451575174961302, 0, 254, 1641786, 1430529674, 630917888610, 208308918928634, 60134626974122946, 16271255119687320314

Offset: 1

Robert FERREOL, Mar 14 2017

			The T(2,3)=6 matrices are
1 0 1
0 1 0
and the matrices obtained by permutations of rows and columns.
First values in triangle
0;
0, 2;
0, 6, 102;
0, 14, 906, 22874;
0, 30, 6510, 417810, 17633670;
0, 62, 42666, 6644714, 622433730, 46959933962;
0, 126, 267582, 99044946, 20218802310, 3204360965106, 451575174961302;

Diagonal gives A283624.

Cf. A183109.

T0:=(n,m)->add((-1)^(m+k)*binomial(n,k)*(2^k-1)^m, k=0..n):
T:=(n,m)->2*T0(n,m)+2^(n*m)+(2^n-2)^m+(2^m-2)^n-2*(2^m-1)^n-2*(2^n-1)^m:
seq(seq(T(n,m), m=1..n),n=1..10);

T[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}]; Flatten[Table[2*T[n, m] + 2^(n*m) + (2^n - 2)^m + (2^m - 2)^n - 2*(2^m - 1)^n - 2*(2^n - 1)^m, {n, 10}, {m, n}]] (* Indranil Ghosh, Mar 14 2017 *)

T(n, m) = sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
tabl(nn) = {for(n=1, nn, for(m=1, n, print1(2*T(n,m) + 2^(n*m) + (2^n - 2)^m + (2^m - 2)^n - 2*(2^m - 1)^n - 2*(2^n - 1)^m,", ");); print(););};
tabl(10); \\ Indranil Ghosh, Mar 14 2017

import math
f=math.factorial
def C(n,r): return f(n)//f(r)//f(n - r)
def T(n,m): return sum([(-1)**j*C(m,j)*(2**(m - j) - 1)**n for j in range (0, m+1)])
i=1
for n in range(1,11):
    for m in range(1, n+1):
        print(str(i)+" "+str(2*T(n, m) + 2**(n*m) + (2**n - 2)**m + (2**m - 2)**n - 2*(2**m - 1)**n - 2*(2**n - 1)**m))
        i+=1 # Indranil Ghosh, Mar 14 2017

T(n,m) = T(m,n) = 2*A183109(n,m) + 2^(n*m) + (2^n-2)^m + (2^m-2)^n - 2*(2^m-1)^n - 2*(2^n-1)^m.

T(n,1)=0, T(n,2)=2^n-2, T(n,3)=6^n-6*(3^n-2^n).

cp's OEIS Frontend

User: Robert FERREOL

A378683 a(0) = 1, a(n+1) = 6*a(n)^3 - 3*a(n).

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A371805 Composite numbers k that divide A001644(k) - 1.

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A352615 Decimal expansion of Integral_{0<=x,y<=Pi/2} sqrt(1-cos^2(x)*cos^2(y)) dx dy.

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A307768 Number of n-step random walks on a line starting from the origin and returning to it at least once.

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A323605 Smallest prime divisor of A000058(n) = A007018(n) + 1 (Sylvester's sequence).

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A322419 Number of n-step self-avoiding walks on L-lattice.

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A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

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A378683 a(0) = 1, a(n+1) = 6a(n)^3 - 3a(n).