A257143 a(2*n) = 3*n if n>0, a(2*n + 1) = 2*n + 1, a(0) = 1.

1, 1, 3, 3, 6, 5, 9, 7, 12, 9, 15, 11, 18, 13, 21, 15, 24, 17, 27, 19, 30, 21, 33, 23, 36, 25, 39, 27, 42, 29, 45, 31, 48, 33, 51, 35, 54, 37, 57, 39, 60, 41, 63, 43, 66, 45, 69, 47, 72, 49, 75, 51, 78, 53, 81, 55, 84, 57, 87, 59, 90, 61, 93, 63, 96, 65, 99

Offset: 0

Author

Michael Somos, Apr 16 2015

Keywords

• nonn
• mult
• easy

Examples

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 6*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 12*x^8 + ...
		

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).

Crossrefs

Cf. A080512, A111711 (partial sums), A188626.

Cf. A005408, A008486.

Programs

Haskell

import Data.List (transpose)
a257143 n = a257143_list !! n
a257143_list = concat $ transpose [a008486_list, a005408_list]
-- Reinhard Zumkeller, Apr 17 2015

Mathematica

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

PARI

{a(n) = if( n<0, 0, polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2) + x * O(x^n), n))};

Formula

a(n) is multiplicative with a(2^e) = 3 * 2^(e-1) if e>0, a(p^e) = p^e otherwise and a(0) = 1.

Euler transform of length 5 sequence [ 1, 2, 0, 0, -1].

G.f.: (1 - x^5) / ((1 - x) * (1 - x^2)^2).

G.f.: (1 + x + x^2 + x^3 + x^4) / (1 - 2*x^2 + x^4).

a(n) = A080512(n) if n>0.

First difference of A111711.

A188626(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.

From Amiram Eldar, Jan 03 2023: (Start)

Dirichlet g.f.: zeta(s-1)*(1+1/2^s).

Sum_{k=1..n} a(k) ~ (5/8) * n^2. (End)