A166517 a(n) = (3 + 5*(-1)^n + 6*n)/4.

2, 1, 5, 4, 8, 7, 11, 10, 14, 13, 17, 16, 20, 19, 23, 22, 26, 25, 29, 28, 32, 31, 35, 34, 38, 37, 41, 40, 44, 43, 47, 46, 50, 49, 53, 52, 56, 55, 59, 58, 62, 61, 65, 64, 68, 67, 71, 70, 74, 73, 77, 76, 80, 79, 83, 82, 86, 85, 89, 88, 92, 91, 95, 94, 98, 97, 101, 100, 104, 103, 107

Offset: 0

Author

Vincenzo Librandi, Oct 16 2009

Keywords

• nonn
• easy

Comments

A sequence defined by a(1)=1, a(n)=k*n-a(n-1), k a constant parameter, has recurrence a(n)= 3*a(n-1) -3*a(n-2) +a(n-3). Its generating function is x*(1+2*(k-1)*x+(1-k)*x^2)/((1+x)*(1-x)^2). The closed form is a(n) = k*n/2+k/4+(-1)^n*(3*k/4-1). This applies with k=3 to this sequence here, and for example to sequences A165033, and A166519-A166525. - R. J. Mathar, Oct 17 2009

From Paul Curtz, Feb 20 2010: (Start)

Also: A001651, terms swapped by pairs.

a(n) mod 9 defines a period-6 sequence which is a permutation of A141425. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A001651, A010685, A010716, A016777, A016945, A073010.

Programs

Magma

[(3 +5*(-1)^n+6*n)/4: n in [0..80]]; // Vincenzo Librandi, Sep 13 2013

Mathematica

CoefficientList[Series[(2 x^2 - x + 2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Harvey P. Dale, Mar 25 2011 *)
Table[(3 + 5 (-1)^n + 6 n) / 4, {n, 0, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

Formula

a(n) = 3*n - a(n-1).

From Paul Curtz, Feb 20 2010: (Start)

a(n+1)-a(n) = (-1)^(n+1)*A010685(n).

Second differences: |a(n+2)-2*a(n+1)+a(n)| = A010716(n).

a(2*n) + a(2*n+1) = A016945(n) = 6*n+3.

a(2*n) = A016945(n).

a(2*n+1) = A016777(n). (End)

G.f. ( 2-x+2*x^2 ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011

E.g.f.: (1/4)*exp(-x)*(5 + 3*exp(2*x) + 6*x*exp(2*x)). - G. C. Greubel, May 15 2016

Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 24 2023

Extensions

a(0)=2 added by Paul Curtz, Feb 20 2010