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

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

Kyler L. Beaumont and Harold J. Smith, Exploding dice combinatorics, Parabola 61(3) (2025) 2. See p. 4.

Michael Somos, Rational Function Multiplicative Coefficients, 2014.

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

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

E.g.f.: 1 + x*(2*cosh(x) + 5*sinh(x)/2). - Amiram Eldar, Nov 26 2025

Sum_{n>0} 1/a(n)^2 = 91*Pi^2/2400. - Stefano Spezia, Jan 03 2026

Example

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 + ...

Mathematica

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

Prog

(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)))};
(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

Crossrefs

Cf. A010710, A040001.

Sequence in context: A295741 A007503 A337298 * A112967 A375739 A244731

Adjacent sequences: A184415 A184416 A184417 * A184419 A184420 A184421

Keyword

nonn,easy

Author

Michael Somos, Feb 14 2011

Status

approved