A022998 -id:A022998 - cp's OEIS frontend

A118209 Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.

1, -3, -2, 5, -4, 6, -6, -11, 7, 12, -10, -10, -12, 18, 8, 21, -16, -21, -18, -20, 12, 30, -22, 22, 21, 36, -20, -30, -28, -24, -30, -43, 20, 48, 24, 35, -36, 54, 24, 44, -40, -36, -42, -50, -28, 66, -46, -42, 43, -63, 32, -60, -52, 60, 40, 66, 36, 84, -58, 40, -60, 90, -42, 85, 48, -60, -66, -80, 44, -72, -70, -77, -72, 108, -42

Offset: 1

Stuart Clary, Apr 15 2006

Related to the logarithmic derivative of A118207(x) and A118208(x).

Related to a signed variant of A022998 via Mobius inversion. - R. J. Mathar, Jul 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008836, A022998, A028260, A061019, A061020, A062157, A117212, A118207, A118208.

nmax = 80; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; Drop[ CoefficientList[ Series[ Sum[ lambda[k] k x^k/(1 + x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ]
f[p_, e_] := (p*(-p)^e+1)/(p+1); f[2, e_] := ((-1)^e*2^(e+2) - 1)/3; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Aug 12 2023 *)

a(n) = sumdiv(n, d, (-1)^(n/d - 1)*(-1)^vecsum(factor(d)[,2])*d) \\ Michel Marcus, Dec 10 2016

a(n) = Sum_{d|n} (-1)^(n/d - 1)*lambda(d)*d, Dirichlet convolution of A061019 and A062157.
G.f.: A(x) is x times the logarithmic derivative of A118207(x).
G.f.: A(x) = A061020(x) - 2 A061020(x^2).
Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-2^(1-s))/zeta(s-1). - R. J. Mathar, Jul 03 2011
a(n) > 0 for n in A028260. - Michel Marcus, Dec 10 2016
Multiplicative with a(2^e) = ((-1)^e*2^(e+2) - 1)/3, and a(p^e) = (p*(-p)^e+1)/(p+1) for an odd prime p. - Amiram Eldar, Aug 12 2023

A118431 Numerator of sum of reciprocals of first n 5-simplex numbers A000389.

Original entry on oeis.org

1, 7, 17, 69, 625, 209, 329, 247, 357, 1250, 341, 1819, 2379, 3059, 19375, 1211, 1496, 3657, 4427, 53125, 12649, 14949, 17549, 10237, 59375, 6851, 7866, 35959, 40919, 231875, 52359, 14726, 16511, 36907, 205625, 91389, 101269

Offset: 1

Jonathan Vos Post, Apr 28 2006

Denominators are A118432. Fractions are: 1/1, 7/6, 17/14, 69/56, 625/504, 209/168, 329/264, 247/198, 357/286, 1250/1001, 341/273, 1819/1456, 2379/1904, 3059/2448, 19375/15504, 1211/969, 1496/1197, 3657/2926, 4427/3542, 53125/42504, 12649/10120, 14949/11960, 17549/14040, 10237/8190, 59375/47502, 6851/5481, 7866/6293, 35959/28768, 40919/32736, 231875/185504, 52359/41888, 14726/11781, 16511/13209, 36907/29526, 205625/164502, 91389/73112, 101269/81016. The numerator of sum of reciprocals of first n triangular numbers is A026741. The numerator of sum of reciprocals of first n tetrahedral numbers is A118391. The numerator of sum of reciprocals of first n pentatope numbers is A118411.

			a(1) = 1 = numerator of 1/1.
a(2) = 7 = numerator of 7/6 = 1/1 + 1/6.
a(3) = 17 = numerator of 17/14 = 1/1 + 1/6 + 1/21.
a(4) = 69 = numerator of 69/56 = 1/1 + 1/6 + 1/21 + 1/56.
a(5) = 55 = numerator of 55/42 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126.
a(10) = 1250 = numerator of 1250/1001 = 1/1+ 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002.
a(20) = 53125 = numerator of 53125/42504 = 1/1 + 1/6 + 1/21 + 1/56 + 1/126 + 1/252 + 1/462 + 1/792 + 1/1287 + 1/2002 + 1/3003 + 1/4368 + 1/6188 + 1/8568 + 1/11628 + 1/15504 + 1/20349 + 1/26334 + 1/33649 + 1/42504.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000332, A000389, A022998, A026741, A118391, A118391, A118411, A118412, A118432.

Numerator[Accumulate[1/Binomial[Range[5, 50], 5]]] (* G. C. Greubel, Nov 21 2017 *)

A118411(n)/A118412(n) = Sum_{i=1..n} (1/A000389(n)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/C(n,5)).
A118411(n)/A118412(n) = Sum_{i=1..n} (1/(n*(n+1)*(n+2)*(n+3)*(n+4)/120)).

A166494 Irregular triangle T(n,k) = greatest common divisor of n-th row terms of A143753.

Original entry on oeis.org

3, 8, 5, 5, 12, 12, 7, 7, 7, 16, 16, 16, 9, 9, 9, 9, 20, 20, 20, 20, 11, 11, 11, 11, 11, 24, 24, 24, 24, 24, 13, 13, 13, 13, 13, 13, 28, 28, 28, 28, 28, 28, 15, 15, 15, 15, 15, 15, 15, 32, 32, 32, 32, 32, 32, 32, 17, 17, 17, 17, 17, 17, 17, 17, 36, 36, 36, 36, 36, 36, 36, 36, 19, 19, 19, 19, 19, 19, 19, 19, 19, 40, 40, 40, 40, 40, 40, 40, 40, 40, 21, 21, 21, 21, 21, 21, 21, 21

Offset: 1

Paul Curtz, Oct 15 2009

			3;
8;
5,5;
12,12;
7,7,7;
16,16,16;

Cf. A166866.

T(n,k) = A022998(n+2).

A174474 a(n) = (2n^2 - 2n - 3)/8 + 3(-1)^n(1-2*n)/8.

Original entry on oeis.org

0, 0, -1, 3, 0, 8, 3, 15, 8, 24, 15, 35, 24, 48, 35, 63, 48, 80, 63, 99, 80, 120, 99, 143, 120, 168, 143, 195, 168, 224, 195, 255, 224, 288, 255, 323, 288, 360, 323, 399, 360, 440, 399, 483, 440, 528, 483, 575, 528, 624, 575

Offset: 0

Paul Curtz, Nov 28 2010

First differences are (-1)^n*A022998(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

[(2*n^2-2*n-3)/8+3*(-1)^n*(1-2*n)/8: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011

LinearRecurrence[{1,2,-2,-1,1},{0,0,-1,3,0},60] (* Harvey P. Dale, Jun 01 2015 *)

Bisections: a(2n) = A067998(n). a(2n+1) = A005563(n).
a(2n) + a(2n+1) = A001105(n).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
G.f.: x^2*(1 - 4*x + x^2) / ( (1+x)^2*(x-1)^3 ).

A235355 0 followed by the sum of (1),(2), (3,4),(5,6), (7,8,9),(10,11,12) from the natural numbers.

Original entry on oeis.org

0, 1, 2, 7, 11, 24, 33, 58, 74, 115, 140, 201, 237, 322, 371, 484, 548, 693, 774, 955, 1055, 1276, 1397, 1662, 1806, 2119, 2288, 2653, 2849, 3270, 3495, 3976, 4232, 4777, 5066, 5679, 6003, 6688, 7049, 7810, 8210, 9051, 9492, 10417, 10901, 11914, 12443, 13548

Offset: 0

Paul Curtz, Jan 07 2014

Difference table for 0 followed by a(n):
  0,  0,  1,   2,   7,  11,  24,  33,...
  0,  1,  1,   5,   4,  13,   9,  25,... =A147685(n)
  1,  0,  4,  -1,   9,  -4,  16,  -9,... =interleave A000290(n+1),-A000290(n)
  -1, 4, -5,  10, -13,  20, -25,  34,...
  5, -9, 15, -23,  33, -45,  59, -75,... =(-1)^n*A027688(n+2).
a(-n) = -a(n-1).
From the second row, signature (0,3,0,-3,0,1).
Consider a(n+2k+1)+a(2k-n):
  1,     2,   6,   9,  17,  22,  34,...
  9,    12,  24,  33,  57,  72, 108,...
  35,   40,  60,  75, 115, 140, 200,...
  91,   98, 126, 147, 203, 238, 322,...
  189, 198, 234, 261, 333, 378, 486,... .
The first column is A005898(n).
The rows are successively divisible by 2*k+1. Hence
  1,   2,  6,  9, 17, 22, 34,...
  3,   4,  8, 11, 19, 24, 36,...
  7,   8, 12, 15, 23, 28, 40,...
  13, 14, 18, 21, 29, 34, 46,...
  21, 22, 26, 29, 37, 42, 54,...
The first column is A002061(n+1).
The main diagonal is A212965(n).
The first difference of every row is A022998(n+1).
Compare to the (2k+1)-sections of A061037 in A165943.

			a(1)=1, a(2)=2, a(3)=3+4=7, a(4)=5+6=11, a(5)=7+8+9=24, a(6)=10+11+12=33.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A075356, A234305.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,2,7,11,24,33},50] (* Harvey P. Dale, Nov 22 2014 *)

Vec(x*(x^2+1)*(x^2+x+1)/((x-1)^4*(x+1)^3) + O(x^100)) \\ Colin Barker, Jan 20 2014

a(n) = 4*a(n-2) -6*a(n-4) +4*a(n-6) -a(n-8), n>7.
a(2n) = 0 followed by A085786(n). a(2n+1) =  A081436(n).
a(2n) + a(2n+1) = A005898(n).
a(2n-1) + a(2n) = A061317(n).
a(n) = (-1)*((-1+(-1)^n-2*n)*(2+n+n^2))/16. a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). G.f.: x*(x^2+1)*(x^2+x+1) / ((x-1)^4*(x+1)^3). - Colin Barker, Jan 20 2014

More terms from Colin Barker, Jan 20 2014

A257088 a(2n) = 4n if n>0, a(2n + 1) = 2n + 1, a(0) = 1.

Original entry on oeis.org

1, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128

Offset: 0

Michael Somos, Apr 16 2015

			G.f. = 1 + x + 4*x^2 + 3*x^3 + 8*x^4 + 5*x^5 + 12*x^6 + 7*x^7 + 16*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A001082, A005408, A007310, A008574, A040001, A215947.

CF. A257083 (partial sums), A246695.

import Data.List (transpose)
a257088 n = a257088_list !! n
a257088_list = concat $ transpose [a008574_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], n, True, 2 n];
a[ n_] := SeriesCoefficient[ (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4), {x, 0, n}];

PARI
```
{a(n) = if( n<1, n==0, n%2, n, 2*n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4) + x * O(x^n), n))};

Euler transform of length 4 sequence [ 1, 3, -1, -1].
a(n) is multiplicative with a(2^e) = 2^(e+1) if e>0, otherwise a(p^e) = p^e.
G.f.: (1 + x + 2*x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).
G.f.: (1 - x^3) * (1 - x^4) / ((1 - x) * (1 - x^2)^3).
MOBIUS transform of A215947 is [1, 4, 3, 8, 5, ...].
a(n) = n * A040001(n) if n>0.
a(n) + a(n-1) = A007310(n) if n>0.
a(n) = A001082(n+1) - A001082(n) if n>0.
Binomial transform with a(0)=0 is A128543 if n>0.
a(2*n) = A008574(n). a(2*n + 1) = A005408(n).
a(n) = A022998(n) if n>0. - R. J. Mathar, Apr 19 2015
From Amiram Eldar, Jan 28 2025: (Start)
Dirichlet g.f.: (1+2^(1-s)) * zeta(s-1).
Sum_{k=1..n} a(k) ~ (3/4) * n^2. (End)
a(n) = gcd(n^n, 2*n). - Mia Boudreau, Jun 27 2025

A259445 Multiplicative with a(n) = n if n is odd and a(2^s)=2.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 9, 10, 11, 6, 13, 14, 15, 2, 17, 18, 19, 10, 21, 22, 23, 6, 25, 26, 27, 14, 29, 30, 31, 2, 33, 34, 35, 18, 37, 38, 39, 10, 41, 42, 43, 22, 45, 46, 47, 6, 49, 50, 51, 26, 53, 54, 55, 14, 57, 58, 59, 30, 61, 62, 63, 2, 65, 66, 67, 34

Offset: 1

José María Grau Ribas, Jun 27 2015

If n = 2^s*m with m odd and s > 0 then a(n) = 2*m.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Peter Bala, A signed Dirichlet product of arithmetical functions
Index to divisibility sequences

Cf. A000265, A022998, A018819, A046897, A160467, A321558.

A259445 := proc(n::integer)
    local a, pe, p,e ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 2 then
            a := 2*a ;
        else
            a := a*p^e  ;
        end if;
    end do:
    a;
end proc:
seq(A259445(n),n=1..80) ; # R. J. Mathar, Feb 21 2025

G[n_] := If[Mod[n, 2] == 0, n/2^(FactorInteger[n][[1, 2]] - 1), n]; Table[G[n], {n, 1, 70}]

a(n)=n>>max(valuation(n,2)-1,0) \\ Charles R Greathouse IV, Jun 28 2015

From Peter Bala, Feb 21 2019: (Start)
a(n) = n*gcd(n,2)/gcd(n,2^n).
a(2*n) = 2*A000265(2*n); a(2*n+1) = A000265(2*n+1).
O.g.f.: x*(1 + 4*x + x^2)/(1 - x^2)^2 - 2*( F(x^2) + F(x^4) + F(x^8) + ... ), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} (1/a(n))*x^n = (3/4)*L(x) - (1/4)*L(-x) + (1/4)*( L(x^2) + L(x^4) + L(x^8) + ... ), where L(x) = log(1/(1 - x)).
(End)
From Peter Bala, Mar 09 2019: (Start)
a(n) = (-1)^(n+1)*Sum_ {d divides n} (-1)^(d+n/d)*phi(d), where phi(n)  = A000010(n) is the Euler totient function. Cf. the identity n = Sum_ {d divides n} phi(d). Cf. A046897 and A321558.
O.g.f.: Sum_{n >= 1} phi(n)*x^n/(1 + (-x)^n). (End)
From Amiram Eldar, Nov 28 2022: (Start)
Dirichlet g.f.: zeta(s-1)*(1 + 1/2^(s-1) - 2/(2^s-1)).
Sum_{k=1..n} a(k) ~ (5/12) * n^2. (End)
a(n) = n /A160467(n). - R. J. Mathar, Feb 21 2025

A283971 a(n) = n except a(4n + 2) = 2n + 1.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67

Offset: 0

Paul Curtz, Mar 18 2017

From Federico Provvedi, Nov 13 2018: (Start)
For n > 1, a(n) is also the cycle length generated by the cycle lengths of the digital roots, in base n, of the powers of k, with k > 0.
Example for n=10 (decimal base): for every h >= 0, the digital roots of 2^h generate a periodic cycle {1,2,4,8,7,5} with period 6; 3^h generates {1,3,9,9,9,9,...} so the periodic cycle {9} has period 1; 4^h generates the periodic cycle {1,4,7} with period 3; etc. So, for n=10 (decimal base representation) the sequence generated by the periods of the digital roots of powers of k (with k > 0) is also periodic {1,6,1,3,6,1,3,2,1} with period 9, hence a(10) = 9. (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A005408, A010121, A022998, A051176, A060819, A186646.

a:=[0,1,1,3,4,5,3,7];; for n in [9..85] do a[n]:=2*a[n-4]-a[n-8]; od; a; # Muniru A Asiru, Jul 20 2018

seq(coeff(series(x*(1+x+3*x^2+4*x^3+3*x^4+x^5+x^6)/((1-x)^2*(1+x)^2*(1+x^2)^2), x,n+1),x,n),n=0..80); # Muniru A Asiru, Jul 20 2018

Table[If[Mod[n, 4] == 2, (n - 2)/2 + 1, n], {n, 67}] (* or *)
CoefficientList[Series[x (1 + x + 3 x^2 + 4 x^3 + 3 x^4 + x^5 + x^6)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 67}], x] (* Michael De Vlieger, Mar 19 2017 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {0, 1, 1, 3, 4, 5, 3, 7}, 70] (* Robert G. Wilson v, Jul 23 2018 *)
Table[Length[FindTransientRepeat[(Length[FindTransientRepeat[Mod[#1^Range[b]-1,b-1]+1,2][[2]]]&)/@Range[2, 2*b], 2][[2]]], {b, 2, 100}] (* Federico Provvedi, Nov 13 2018 *)

a(n)=if(n%4==2, n\4*2 + 1, n) \\ Charles R Greathouse IV, Mar 18 2017

concat(0, Vec(x*(1 + x + 3*x^2 + 4*x^3 + 3*x^4 + x^5 + x^6) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^40))) \\ Colin Barker, Mar 19 2017

def A283971(n): return n if (n-2)&3 else n>>1 # Chai Wah Wu, Jan 10 2023

a(2*n) = A022998(n), a(1+2*n) = 1 + 2*n.
a(n) = 2*a(n-4) - a(n-8).
From Colin Barker, Mar 19 2017: (Start)
G.f.: x*(1 + x + 3*x^2 + 4*x^3 + 3*x^4 + x^5 + x^6) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = -((-1)^n - (-i)^n - i^n - 7)*n/8, where i = sqrt(-1).
(End)
a(n) = A060819(n) * periodic sequence of length 4: repeat [4, 1, 1, 1].
a(n) = a(n-4) + periodic sequence of length 4: repeat [4, 4, 2, 4].
From Werner Schulte, Jul 08 2018: (Start)
For n > 0, a(n) is multiplicative with a(p^e) = p^e for prime p >= 2 and e >= 0 except a(2^1) = 1.
Dirichlet g.f.: (1 - 1/2^s - 1/2^(2*s-1)) * zeta(s-1).
(End)
a(n) = n*(7 + cos(n*Pi/2) - cos(n*Pi) + cos(3*n*Pi/2))/8. - Wesley Ivan Hurt, Oct 04 2018
E.g.f.: (1/4)*x*(4*cosh(x) - sin(x) + 3*sinh(x)). - Franck Maminirina Ramaharo, Nov 13 2018
Sum_{k=1..n} a(k) ~ (7/16) * n^2. - Amiram Eldar, Nov 28 2022

A317326 Multiples of 26 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 26, 3, 52, 5, 78, 7, 104, 9, 130, 11, 156, 13, 182, 15, 208, 17, 234, 19, 260, 21, 286, 23, 312, 25, 338, 27, 364, 29, 390, 31, 416, 33, 442, 35, 468, 37, 494, 39, 520, 41, 546, 43, 572, 45, 598, 47, 624, 49, 650, 51, 676, 53, 702, 55, 728, 57, 754, 59, 780, 61, 806, 63, 832, 65, 858, 67, 884, 69

Offset: 0

Omar E. Pol, Jul 25 2018

a(n) is the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 30-gonal numbers (A316729).
Partial sums give the generalized 30-gonal numbers.
More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
From Bruno Berselli, Jul 27 2018: (Start)
Also, this type of sequence is characterized by:
O.g.f.: x*(1 + m*x + x^2)/(1 - x^2)^2;
E.g.f.: x*(2 - m + (2 + m)*exp(2*x))*exp(-x)/4;
a(n) = -a(-n) = (2 + m - (2 - m)*(-1)^n)*n/4;
a(n) = (m/2)^((1 + (-1)^n)/2)*n;
a(n) = 2*a(n-2) - a(n-4), with signature (0,2,0,-1). (End)

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A252994 and A005408 interleaved.
Column 26 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16), A317313 (k=17), A317314 (k=18), A317315 (k=19), A317316 (k=20), A317317 (k=21), A317318 (k=22), A317319 (k=23), A317320 (k=24), A317321 (k=25), A317322 (k=26), A317323 (k=27), A317324 (k=28), A317325 (k=29), this sequence (k=30).
Cf. A316729.

[13^div(1+(-1)^n,2)*n for n in 0:70] |> println # Bruno Berselli, Jul 28 2018

Table[(7 + 6 (-1)^n) n, {n, 0, 70}] (* Bruno Berselli, Jul 27 2018 *)

a(2*n) = 26*n, a(2*n+1) = 2*n + 1.
From Bruno Berselli, Jul 27 2018: (Start)
O.g.f.: x*(1 + 26*x + x^2)/(1 - x^2)^2.
E.g.f.: x*(-6 + 7*exp(2*x))*exp(-x).
a(n) = -a(-n) = (7 + 6*(-1)^n)*n.
a(n) = 13^((1 + (-1)^n)/2)*n.
a(n) = 2*a(n-2) - a(n-4). (End)
Multiplicative with a(2^e) = 13*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 3*2^(3-s)). - Amiram Eldar, Oct 26 2023

A323712 Number of the following described shuffles required to return a deck of n cards to its original state. Create two piles by alternating a card from the top of the deck left-right-left-right until the deck is exhausted. Then, placing the left pile on top of the right pile constitutes one shuffle.

Original entry on oeis.org

1, 1, 3, 4, 4, 6, 6, 3, 9, 5, 5, 12, 12, 4, 12, 8, 8, 9, 9, 6, 6, 22, 22, 20, 20, 9, 27, 28, 28, 10, 10, 5, 15, 12, 12, 36, 36, 12, 12, 20, 20, 7, 7, 12, 36, 46, 46, 42, 42, 8, 24, 52, 52, 20, 20, 9, 9, 29, 29, 60, 60, 6, 18, 12, 12, 33, 33, 22, 66, 70, 70, 18

Offset: 1

David Lovler, Jan 24 2019

nonn

The card shuffling procedure is the same for even n and odd n.
Here are a few conjectures.
a(n) <= n for all n.
a(p)=a(p-1) and a(p)|p-1 when p is a prime >= 5.
a(n)=a(n-1) and a(n)|n-1 for nonprimes 341=31*11 and 22369621=8191*2731 and probably other pseudoprimes of the form p*((p+2)/3) where p is a Mersenne prime and (p+2)/3 is prime.
n cards are returned to their original state after n shuffles when n=1, 3, 4, 6, 9, 12, 22, 27, 28, 36, 46, 52, 60, 70, 78, 81, ... (A373461) . These values of n are either of the form p-1 where p is an odd prime number or 3^i, i >= 0.
a(c) is relatively small (compared with nearby values) when c is a Catalan number.
a(2n+1)=3*a(2n) or a(2n+1)=a(2n) for all n.

			For n=4, {a1,a2,a3,a4}-->{a3,a1,a4,a2}-->{a4,a3,a2,a1}-->{a2,a4,a1,a3}-->{a1,a2,a3,a4}, so a(4)=4.
For n=5, {a1,a2,a3,a4,a5}-->{a5,a3,a1,a4,a2}-->{a2,a1,a5,a4,a3}-->{a3,a5,a2,a4,a1}-->{a1,a2,a3,a4,a5}, so a(5)=4.

Alois P. Heinz, Table of n, a(n) for n = 1..1024

Cf. A024222, A022998, A163776, A373416 (fixed points).

pileShuf := proc(L::list)
    local i,n,shf ;
    shf := [] ;
    n := nops(L) ;
    if type(n,'odd') then
        for i from n to 1 by -2 do
            shf := [op(shf),op(i,L)] ;
        end do:
        for i from n-1 to 2 by -2 do
            shf := [op(shf),op(i,L)] ;
        end do:
    else
        for i from n-1 to 1 by -2 do
            shf := [op(shf),op(i,L)] ;
        end do:
        for i from n to 2 by -2 do
            shf := [op(shf),op(i,L)] ;
        end do:
    end if;
    shf ;
end proc:
A323712 := proc(n)
    local L,itr,isord,i;
    L := [seq(i,i=1..n)] ;
    for itr from 1 to n! do
        L := pileShuf(L) ;
        isord := true ;
        for i from 1 to nops(L) do
            if op(i,L) <> i then
                isord := false ;
                break ;
            end if;
        end do:
        if isord then
            return itr ;
        end if;
    end do:
    -1 ;
end proc:
seq(A323712(n),n=1..50) ; # R. J. Mathar, Aug 02 2024

perm(n, vn) = {my(va = List(), vb = List()); for (k=1, n, if (k % 2, listput(va, vn[k]), listput(vb, vn[k]));); Vec(concat(Vecrev(va), Vecrev(vb)));}
a(n) = {my(vn = vector(n,k,k), vs = perm(n, vn), nb = 1); while (vs != vn, vs = perm(n, vs); nb++); nb;} \\ Michel Marcus, Feb 06 2019

a(2^m) = m if m is odd, a(2^m) = 2m if m is even. - Alois P. Heinz, Feb 15 2019

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