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

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

0,3

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Formula

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)

Example

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 + ...

Mathematica

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}];

Prog

(PARI) {a(n) = if( n<1, n==0, n%2, n, 2*n)};
(PARI) {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))};
(Haskell)
import Data.List (transpose)
a257088 n = a257088_list !! n
a257088_list = concat $ transpose [a008574_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

Crossrefs

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

CF. A257083 (partial sums), A246695.

Sequence in context: A021232 A263616 A280166 * A022998 A082895 A086938

Adjacent sequences: A257085 A257086 A257087 * A257089 A257090 A257091

Keyword

nonn,mult,easy

Author

Michael Somos, Apr 16 2015

Status

approved