A316631 Expansion of A(x) = x*(1+3*x^2+x^3+3*x^4+x^6)/(1-x^4)^2.

0, 1, 0, 3, 1, 5, 0, 7, 2, 9, 0, 11, 3, 13, 0, 15, 4, 17, 0, 19, 5, 21, 0, 23, 6, 25, 0, 27, 7, 29, 0, 31, 8, 33, 0, 35, 9, 37, 0, 39, 10, 41, 0, 43, 11, 45, 0, 47, 12, 49, 0, 51, 13, 53, 0, 55, 14, 57, 0, 59, 15, 61, 0, 63, 16, 65, 0, 67, 17, 69, 0, 71, 18, 73, 0, 75, 19, 77, 0, 79, 20

Offset

0,4

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

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

Formula

a(n) = n/4 if n mod 4 = 0, and a(n) = 0 if n mod 4 = 2, and a(n) = n if n mod 2 = 1.

Linear recurrence: a(n) = 2*a(n-4) - a(n-8) for n > 7.

a(n) for n > 0 is multiplicative with a(2^e) = 1 - e if e < 2 and a(2^e) = 2^(e-2) if e > 1 otherwise a(p^e) = p^e for prime p > 2 and e >= 0.

Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (1-1/2^s)^2 * zeta(s-1).

Dirichlet inverse b(n) for n > 0 is multiplicative with b(2^e) = 1 - e and for prime p > 2: b(p) = -p and b(p^e) = 0 if e > 1.

Dirichlet convolution with A104117(n) yields A000027(n).

Dirichlet convolution with A115364(n) yields A000203(n).

Sum_{k=1..n} a(k) ~ (9/32) * n^2. - Amiram Eldar, Nov 20 2022

Example

a(22) = 0 since 22 mod 4 = 2; a(23) = 23 for 23 mod 2 = 1; a(24) = 6 because 24 mod 4 = 0 and 24/4 = 6.

Maple

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

Mathematica

CoefficientList[Series[x (1 + 3 x^2 + x^3 + 3 x^4 + x^6)/(1 - x^4)^2, {x, 0, 80}], x] (* Michael De Vlieger, Jul 20 2018 *)
LinearRecurrence[{0, 0, 0, 2, 0, 0, 0, -1}, {0, 1, 0, 3, 1, 5, 0, 7}, 81] (* Robert G. Wilson v, Jul 21 2018 *)

Prog

(PARI) concat(0, Vec((x*(1+3*x^2+x^3+3*x^4+x^6)/(1-x^4)^2) + O(x^80))) \\ Felix Fröhlich, Jul 09 2018
(PARI) {my(N=79); concat([0], dirdiv(vector(N, n, n), vector(N, n, my(k=valuation(n, 2)); if(n==2^k, k+1, 0))))} \\ Andrew Howroyd, Jul 09 2018
(GAP) a:=[0, 1, 0, 3, 1, 5, 0, 7];; for n in [9..85] do a[n]:=2*a[n-4]-a[n-8]; od; a; # Muniru A Asiru, Jul 20 2018

Crossrefs

Cf. A000027, A000203, A104117, A115364.

Sequence in context: A378597 A094919 A328373 * A197152 A337668 A176907

Adjacent sequences: A316628 A316629 A316630 * A316632 A316633 A316634

Keyword

mult,nonn,easy

Author

Werner Schulte, Jul 09 2018

Status

approved