A050531 Number of multigraphs with loops on 3 nodes with n edges.

1, 2, 6, 14, 28, 52, 93, 152, 242, 370, 546, 784, 1103, 1512, 2040, 2706, 3534, 4554, 5803, 7304, 9108, 11252, 13780, 16744, 20205, 24206, 28826, 34126, 40176, 47056, 54857, 63648, 73542, 84630, 97014, 110808, 126139, 143108, 161868, 182546, 205282

Offset: 0

Author

Vladeta Jovovic, Dec 29 1999

Keywords

• easy
• nonn

Comments

a(n) is also the number of multigraphs (no loops allowed) on 3 nodes with n edges of two colors. - Geoffrey Critzer, Aug 10 2015

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Crossrefs

Column k=3 of A290428.

Cf. A076118, A002620, A290428.

Programs

Maple

a076118:= gfun:-rectoproc({a(n+4) = 2*a(n+3)-3*a(n+2)+2*a(n+1)-a(n), a(0)=0,a(1)=1,a(2)=1,a(3)=-1}, a(n), remember):
f:= n -> ceil((-1)^n*a076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5):
map(f, [$0..100]); # Robert Israel, Aug 07 2015

Mathematica

< 1/(1 - x^i), {i, 1, n^2 - n}], {x, 0, nn}], x] (* Geoffrey Critzer, Aug 07 2015 *)
CoefficientList[Series[(x^6 + x^4 + 2 x^3 + x^2 + 1)/((x^3 - 1)^2 (x^2 - 1)^2 (x - 1)^2), {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2015 *)

PARI

Vec((x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2) + O(x^40)) \\ Colin Barker, Jul 07 2019

Formula

G.f.: (x^6+x^4+2*x^3+x^2+1)/((x^3-1)^2*(x^2-1)^2*(x-1)^2).

a(n) = ceiling((-1)^n*A076118(n+1)/9+(-1)^n*n/32+(4009/4320)*n+(1/2)*n^2+(5/36)*n^3+(1/48)*n^4+(1/720)*n^5). - Robert Israel, Aug 07 2015

a(n) = (A+B+C)/6 where A = binomial(n+5,5); B = (n+2)*(n+3)*(n+4)/8 if n even, B = (n+1)*(n+3)*(n+5)/8 if n odd; C = 2*((n/3) + 1) if n divisible by 3, C = 0 if n not divisible by 3. - David Pasino, Jul 06 2019

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12) for n>11. - Colin Barker, Jul 07 2019