A131735 -id:A131735 - cp's OEIS frontend

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 0

Paul Curtz, Jan 26 2008

Dirichlet series for the non-principal character modulo 6: L(s,chi) = Sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). - R. J. Mathar, Jul 31 2010

			G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
L. B. W. Jolley, Summation of Series, Dover (1961).

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.

&cat[[0, 1, 0, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
# Wesley Ivan Hurt, Jun 20 2016

a[ n_] := JacobiSymbol[-12, n]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 24 2014 *)
PadRight[{},120,{0,1,0,0,0,-1}] (* Harvey P. Dale, Aug 01 2021 *)

{a(n) = [0, 1, 0, 0, 0, -1][n%6+1]}; /* Michael Somos, Feb 10 2008 */

{a(n) = kronecker(-12, n)}; /* Michael Somos, Feb 10 2008 */

{a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))}; /* Michael Somos, Aug 11 2009 */

Euler transform of length 6 sequence [0, 0, 0, -1, 0, 1]. - Michael Somos, Feb 10 2008
G.f.: x * (1 - x^4) / (1 - x^6) = x*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos, Aug 11 2009
a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Aug 11 2009
a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.
sqrt(3)*a(n) = sin(Pi*n/3) + sin(2*Pi*n/3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - Ilya Gutkovskiy, Jun 21 2016

A372102 Number of permutations of [n] whose non-fixed points are not neighbors.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 19, 45, 107, 278, 728, 2033, 5749, 17105, 51669, 162674, 520524, 1724329, 5807143, 20146861, 71048431, 257139686, 945626800, 3558489633, 13599579817, 53060155137, 210124405097, 847904374466, 3470756061140, 14453943647561, 61023690771451

Offset: 0

Alois P. Heinz, Apr 18 2024

nonn

			a(3) = 2: 123, 321.
a(4) = 4: 1234, 1432, 3214, 4231.
a(5) = 9: 12345, 12543, 14325, 15342, 32145, 32541, 42315, 52143, 52341.
a(6) = 19: 123456, 123654, 125436, 126453, 143256, 143652, 153426, 163254, 163452, 321456, 325416, 326451, 423156, 423651, 521436, 523416, 621453, 623154, 623451.

Alois P. Heinz, Table of n, a(n) for n = 0..892
Wikipedia, Permutation.

Cf. A000142, A000166, A001629, A011973, A098825, A131735, A162969, A165961, A371995.

a:= proc(n) option remember; `if`(n<4, [2$3, 4][n+1],
      3*a(n-1)+(n-2)*a(n-2)+(n-1)*(a(n-4)-a(n-3)))/2
    end:
seq(a(n), n=0..30);

a[n_] := Sum[Binomial[n + 1 - k, k] * Subfactorial[k], {k, 0, (n + 1)/2}];
Table[a[n], {n, 0, 30}] (* Peter Luschny, Apr 24 2024 *)

a(n) = Sum_{j=0..floor((n+1)/2)} A000166(j)*A011973(n+1,j).
a(n) mod 2 = A131735(n+3).
Row sums of A371995(n+1), which are the antidiagonals of A098825. - Peter Luschny, Apr 24 2024
a(n) ~ sqrt(Pi) * exp(sqrt(n/2) - n/2 - 7/8) * n^(n/2 + 1) / 2^((n+3)/2). - Vaclav Kotesovec, Apr 25 2024

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A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

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A372102 Number of permutations of [n] whose non-fixed points are not neighbors.

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