OFFSET
0,2
REFERENCES
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
LINKS
Steven Kotlarz, Table of n, a(n) for n = 0..429
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Steven Kotlarz, Order-310 linear recurrence coefficients
FORMULA
a(n) = Sum_{j=1..310} c(j)*a(n-j), where the coefficients c(j) are given in the linked recurrence file (empirical). - Steven Kotlarz, May 30 2026
PROG
(Python)
# Number of perfect matchings in P_2 X C_7 X P_n, by cylindrical transfer matrix.
def A028459(n):
P, C = 2, 7
NV = P*C
E = [(p*C+c, p*C+(c+1)%C) for p in range(P) for c in range(C)] + \
[(c, C+c) for c in range(C)]
def layer(inc):
out = {}
def rec(v, used, og):
if v == NV:
out[og] = out.get(og, 0) + 1; return
if (used>>v)&1: rec(v+1, used, og); return
if (inc>>v)&1: rec(v+1, used|(1<<v), og); return
rec(v+1, used|(1<<v), og|(1<<v))
for a, b in E:
w = b if a==v else a if b==v else -1
if w>v and not (used>>w)&1 and not (inc>>w)&1:
rec(v+1, used|(1<<v)|(1<<w), og)
rec(0, 0, 0)
return out
if n == 0: return 1
vec = {0: 1}
for _ in range(n):
nv = {}
for s, cnt in vec.items():
for o, w in layer(s).items():
nv[o] = nv.get(o, 0) + cnt*w
vec = nv
return vec.get(0, 0)
print([A028459(n) for n in range(14)])
# Steven Kotlarz, May 30 2026
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved