A262672 Expansion of (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).

3, 2, 2, 1, 4, 3, 3, 2, 5, 4, 4, 3, 6, 5, 5, 4, 7, 6, 6, 5, 8, 7, 7, 6, 9, 8, 8, 7, 10, 9, 9, 8, 11, 10, 10, 9, 12, 11, 11, 10, 13, 12, 12, 11, 14, 13, 13, 12, 15, 14, 14, 13, 16, 15, 15, 14, 17, 16, 16, 15, 18, 17, 17, 16, 19, 18, 18, 17, 20, 19, 19, 18, 21

Offset: 0

Author

Wesley Ivan Hurt, Sep 26 2015

Keywords

• nonn
• easy

Comments

From Altug Alkan, Sep 29 2015: (Start)

Sequence can be defined as a composition of arithmetic sequences and it can be generated by the equations: a(4k) = k+3, a(4k+3) = k+1, a(4k+1) = a(4k+2) = k+2, with k >= 0.

Obviously, minimum value of sequence is a(3), which is also unique value that has no repetition in sequence. (End)

Examples

			G.f. = 3 + 2*x + 2*x^2 + x^3 + 4*x^4 + 3*x^5 + 3*x^6 + 2*x^7 + 5*x^8 + 4*x^9 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Row 2 of the example in A238340 (read from bottom).

Cf. A002265, A028242, A118402.

Programs

Magma

[(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n) div 4))/8 : n in [0..100]];

Magma

&cat[[3+n,2+n,2+n,1+n]: n in [0..20]]; // Bruno Berselli, Sep 29 2015

Magma

I:=[3,2,2,1,4]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..100]]; // Vincenzo Librandi, Sep 29 2015

Maple

A262672:=n->(2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8: seq(A262672(n), n=0..100);

Mathematica

Table[(2n + 13 + 5 (-1)^n - 6 (-1)^((2n + 3 + (-1)^n)/4))/8, {n, 0, 100}]
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 2, 2, 1, 4}, 100] (* Vincenzo Librandi, Sep 29 2015 *)
CoefficientList[Series[(3-x-x^3)/((x-1)^2(1+x+x^2+x^3)),{x,0,100}],x] (* Harvey P. Dale, May 26 2023 *)

PARI

a(n) = (2*n+13+5*(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/8;
vector(80, n, a(n-1)) \\ Altug Alkan, Sep 29 2015

PARI

{a(n) = my(k = n\4); [ 3, 2, 2, 1][n%4 + 1] + k}; /* Michael Somos, Oct 02 2015 */

Formula

G.f.: (3-x-x^3) / ((x-1)^2*(1+x+x^2+x^3)).

a(n) = a(n-1) + a(n-4) - a(n-5), for n>4

a(n) = ( 2*n + 13+5*(-1)^n - 6*(-1)^((2*n + 3 + (-1)^n)/4) )/8.

a(n) = 3*A002265(n+4) - A002265(n+3) - A002265(n+1).

a(2n) = A028242(n+4), a(2n+1) = A028242(n+2).

4*a(n) = |A118402(n+8)| + 3*i^(n*(n-1)), where i=sqrt(-1). [Bruno Berselli, Sep 29 2015]

E.g.f.: (1/8)*(2*x*exp(x) + 5*exp(-x) + 13*exp(x) + 6*sin(x) + 6*cos(x)). - G. C. Greubel, Sep 29 2015

a(n) = -a(-13 - n) for all n in Z. - Michael Somos, Oct 02 2015

Extensions

Edited by Bruno Berselli, Sep 30 2015