A281362 a(0) = 1, a(1) = 2; for n>1, a(n) = a(n-1) + a(n-2) + floor(n/2).

1, 2, 4, 7, 13, 22, 38, 63, 105, 172, 282, 459, 747, 1212, 1966, 3185, 5159, 8352, 13520, 21881, 35411, 57302, 92724, 150037, 242773, 392822, 635608, 1028443, 1664065, 2692522, 4356602, 7049139, 11405757, 18454912, 29860686, 48315615, 78176319, 126491952

Offset: 0

Author

N. J. A. Sloane, Jan 27 2017, based on a comment in A215004 from Alex Ratushnyak, Jul 31 2012

Keywords

• nonn
• easy

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 13*x^4 + 22*x^5 + 38*x^6 + 63*x^7 + ...
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-3,0,1).

Crossrefs

Cf. A000045, A215004, A215005.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!((1 - x^2+x^4)/((1-x)^2*(1+x)*(1-x-x^2)))); // G. C. Greubel, Jul 29 2018

Mathematica

CoefficientList[Series[(1-x^2+x^4)/((1-x)^2*(1+x)*(1-x-x^2)), {x,0,50}], x] (* G. C. Greubel, Jul 29 2018 *)

PARI

Vec((1-x^2+x^4)/((1-x)^2*(1+x)*(1-x-x^2)) + O(x^50))

PARI

{a(n) = my(m=n); if( n<0, m=-1-n); polcoeff( (1 - x^2 + x^4) / ((1 - x - x^2 + x^3) * (1 - (-1)^(n<0)*x - x^2)) + x * O(x^m), m)}; /* Michael Somos, Jan 27 2017 */

SageMath

[2*fibonacci(n+2) -(n+2+(n%2))//2 for n in range(51)] # G. C. Greubel, Apr 05 2024

Formula

From Colin Barker, Jan 27 2017: (Start)

a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) + a(n-5) for n>4.

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

0 = 3 + a(n)*(-4 - 2*a(n) - a(n+1) + 6*a(n+2) - a(n+3)) + a(n+1)*(-1 + a(n+1) + a(n+2) - 2*a(n+3)) + a(n+2)*(+11 - 5*a(n+2) + 3*a(n+3)) + a(n+3)*(-6) for all n in Z. - Michael Somos, Jan 27 2017

From G. C. Greubel, Apr 05 2024: (Start)

a(n) = 2*Fibonacci(n+2) - (1/4)*(2*n + 5 - (-1)^n).

E.g.f.: (1/2)*(4*exp(x/2)*(cosh(sqrt(5)*x/2) + (3/sqrt(5))*sinh( sqrt(5)*x/2)) - (x+2)*cosh(x) - (x+3)*sinh(x)). (End)