A184418 Convolution square of A040001.

1, 2, 5, 6, 10, 10, 15, 14, 20, 18, 25, 22, 30, 26, 35, 30, 40, 34, 45, 38, 50, 42, 55, 46, 60, 50, 65, 54, 70, 58, 75, 62, 80, 66, 85, 70, 90, 74, 95, 78, 100, 82, 105, 86, 110, 90, 115, 94, 120, 98, 125, 102, 130, 106, 135, 110, 140, 114, 145, 118, 150, 122, 155, 126

Offset: 0

Author

Michael Somos, Feb 14 2011

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 5*x^2 + 6*x^3 + 10*x^4 + 10*x^5 + 15*x^6 + 14*x^7 + 20*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Michael Somos, Rational Function Multiplicative Coefficients
• Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Programs

Magma

I:=[2,5,6,10]; [1] cat [n le 4 select I[n] else 2*Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Aug 14 2018

Mathematica

LinearRecurrence[{0,2,0,-1},{1,2,5,6,10},80] (* Harvey P. Dale, Jul 03 2017 *)

PARI

{a(n) = (n==0) + n * ([5/2, 2] [n%2 + 1])};

PARI

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

Formula

G.f.: (1 + x + x^2)^2 / (1 - x^2)^2 = 1 + x * (x + 2) * (2*x + 1) / (1 - x^2)^2. a(-n) = -a(n) except a(0) = 2.

Euler transform of length 3 sequence [2, 2, -2].

a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 5 * 2^(e-2) if e>0, b(p^e) = p^e if p>2.

a(2*n + 1) = 4*n + 2, a(2*n) = 5*n except a(0) = 2.

a(n) = (9+(-1)^n)*n/4 = (n/2)*A010710(n+1) for n>0. - Bruno Berselli, Mar 24 2011