A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

-1, 0, -3, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837, 870, 899, 930, 957, 992, 1023, 1056, 1085, 1122, 1155, 1190

Offset

0,3

Comments

Let a(n)/A000290(n) = [-1/0, 0/1, -3/4, 2/9, 3/16, 6/25, 5/36, 12/49, 15/64, 20/81, 21/100, 30/121, ...] = a(n)/b(n) (say).

Then b(n)-4*a(n)=4, 1, 16, 1 (period of length 4).

Permutation from a(n) to A061037(n): 1, 3, 2, 7, 5, 11, 4, 15, 9, 19, 6, ... = shifted A145979 + 1.

A061037(n) - a(n) = 0, 3, -3, -3, 0, -15, 3, -33, 0 -57, 15, -87, 0, -123, ...

First 3 rows:

-1 0 -3 2 3 6 5 12 15 20 21 30 35

1 -3 5 1 3 -1 7 3 5 1 9 5 7

-4 8 -4 2 -4 8 -4 2 -4 8 -4 2 -4.

Note that the terms of a(n) increase from 12. Compare to increasing terms permutation of A061037(n): -3,-1,0,2,3,5,6,12,15, .... and A129647.

c(n) = 0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, ... (cf. A134967)

d(n) = -1, 1, 1, 3, 1, 3, 3, 5, 3, 5, 5, 7, 5, 7, 7, 9, ..., hence:

a(n) = c(n+1) * d(n+1).

Links

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

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

Formula

a(k+4) - a(k) = 2*k + 4.

a(k+2) - a(k-2) = 2*k.

a(k+6) - a(k-6) = 6*k.

a(k+10) - a(k-10) = 10*k.

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

a(2*k) = -1, -3, followed by 3, 5, 15, 21, 35, 45, ... (A142717);

a(2*k+1) = k*(k+1) (see A002378).

A198442(n) = -1,0,0,2,3,6,8,12, minus 3 at A198442(4*n+2).

G.f. -( 1-2*x+4*x^2-8*x^3+3*x^4 )/( (1-x)^2*(1-x^4) ). - R. J. Mathar, Jul 17 2012; edited by N. J. A. Sloane, Jul 22 2012

From R. J. Mathar, Jun 28 2013: (Start)

a(4*k) = A000466(k);

a(4*k+1) = A002943(k);

a(4*k+2) = A078371(k-1) for k>0;

a(4*k+3) = A002939(k+1). (End)

a(n) = (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8. - Luce ETIENNE, Oct 27 2016

Maple

A214297 := proc(n)
    option remember;
    if n <=5 then
        op(n+1, [-1, 0, -3, 2, 3, 6]) ;
    else
        2*procname(n-1)-procname(n-2)+procname(n-4)-2*procname(n-5)+procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

Mathematica

Table[(2 n^2 - 11 - 9 (-1)^n + 6 ((-1)^((2 n + 1 - (-1)^n)/4) + (-1)^((2 n - 1 + (-1)^n)/4)))/8, {n, 0, 69}] (* or *)
CoefficientList[Series[-(1 - 2 x + 4 x^2 - 8 x^3 + 3 x^4)/((1 - x)^2*(1 - x^4)), {x, 0, 69}], x] (* Michael De Vlieger, Mar 24 2017 *)

Prog

(PARI) vector(100, n, n--; (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8) \\ G. C. Greubel, Sep 19 2018
(Magma) [(2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018

Crossrefs

Sequence in context: A281667 A368102 A336749 * A022472 A014679 A208454

Adjacent sequences: A214294 A214295 A214296 * A214298 A214299 A214300

Keyword

sign,easy

Author

Paul Curtz, Jul 11 2012

Extensions

Edited by N. J. A. Sloane, Jul 22 2012

Status

approved