A217123 Number of possible ordered pairs (x, y) where x is the number of beads adjacent to at least one black bead and y the number of beads adjacent to at least one white bead in a binary necklace of length n.

2, 3, 4, 6, 8, 10, 14, 18, 22, 26, 32, 38, 44, 50, 58, 66, 74, 82, 92, 102, 112, 122, 134, 146, 158, 170, 184, 198, 212, 226, 242, 258, 274, 290, 308, 326, 344, 362, 382, 402, 422, 442, 464, 486, 508, 530, 554, 578, 602, 626, 652, 678

Offset

1,1

Links

Table of n, a(n) for n=1..52.

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

Formula

For p>0, 0<=q<4, a(4*p+q) = 4*p^2 + 2*p*q (+4 if q=3, +2 otherwise).

a(n) = (13-(-1)^n+2*(-i)^n+2*i^n+2*n^2)/8, where i=sqrt(-1). G.f.: x*(x^7-2*x^6-x^5+2*x^4-x^3+x-2)/((x-1)^3*(x+1)*(x^2+1)). [Colin Barker, Oct 04 2012]

Example

0: (0,0)
1: (1,0), (0,1)
2: (2,0), (1,1), (0,2)
3: (3,0), (3,2), (2,3), (0,3)
4: (4,4), (2,2), (0,4), (4,2), (2,4), (4,0)
5: (5,4), (5,0), (4,5), (5,2), (0,5), (4,3), (2,5), (3,4)
In the binary necklace 01010, four beads are next to a zero and three are next to a one, yielding the pair (4,3).

Maple

A217123 := proc(n)
    local p, q ;
    if n < 4 then
        return op(n, [2, 3, 4]) ;
    end if;
    q := n mod 4 ;
    p := floor(n/4) ;
    if q =3 then
        4*p^2+2*p*q +4;
    else
        4*p^2+2*p*q +2;
    end if;
end proc: # R. J. Mathar, Sep 27 2012

Crossrefs

Sequence in context: A220851 A028290 A003107 * A014977 A008583 A053253

Adjacent sequences: A217120 A217121 A217122 * A217124 A217125 A217126

Keyword

nonn,easy

Author

Andrew Woods, Sep 26 2012

Status

approved