A339852 Number of Hamiltonian circuits within parallelograms of size 5 X n on the triangular lattice.

1, 44, 549, 7104, 104100, 1475286, 20842802, 295671198, 4190083085, 59374628434, 841470846944, 11925007688342, 168996943899738, 2394974040514288, 33940795571394262, 480998063196253650, 6816550836218124869, 96601974078400509612, 1369012239935377295854, 19401203058253673198258

Offset

2,2

Links

Seiichi Manyama, Table of n, a(n) for n = 2..500

M. Peto, Studies of protein designability using reduced models, Thesis, 2007.

Index entries for linear recurrences with constant coefficients, signature (8,62,384,160,-1628,-11310,9700,-16019,102564, -98380,263340, -429661,174728,-361330,147404,284641,24764,182412,-156248, -138559,14756,14496,-3660,-2640,328,80,-8)

Formula

a(2)..a(31) = [1, 44, 549, 7104, 104100, 1475286, 20842802, 295671198, 4190083085, 59374628434, 841470846944, 11925007688342, 168996943899738, 2394974040514288, 33940795571394262, 480998063196253650, 6816550836218124869, 96601974078400509612, 1369012239935377295854, 19401203058253673198258, 274947636268050621400764, 3896469848341602644039976, 55219522831075639350876744, 782553393257523404353337072, 11090096073215866151573834374, 157165289898796544200350430624, 2227296155585971455156172389428, 31564527815820044279227403214372, 447322379530320420841684880901414, 6339309505792160540792742125116082] and

a(n) = 8*a(n-1) + 62*a(n-2) + 384*a(n-3) - 160*a(n-4) - 1628*a(n-5) - 11310*a(n-6) + 9700*a(n-7) - 16019*a(n-8) + 102564*a(n-9) - 98380*a(n-10) + 263340*a(n-11) - 429661*a(n-12) + 174728*a(n-13) - 361330*a(n-14) + 147404*a(n-15) + 284641*a(n-16) + 24764*a(n-17) + 182412*a(n-18) - 156248*a(n-19) - 138559*a(n-20) + 14756*a(n-21) + 14496*a(n-22) - 3660*a(n-23) - 2640*a(n-24) + 328*a(n-25) + 80*a(n-26) - 8*a(n-27) for n > 31.

Prog

(Python)
# Using graphillion
from graphillion import GraphSet
def make_T_nk(n, k):
    grids = []
    for i in range(1, k + 1):
        for j in range(1, n):
            grids.append((i + (j - 1) * k, i + j * k))
            if i < k:
                grids.append((i + (j - 1) * k, i + j * k + 1))
    for i in range(1, k * n, k):
        for j in range(1, k):
            grids.append((i + j - 1, i + j))
    return grids
def A339849(n, k):
    universe = make_T_nk(n, k)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles(is_hamilton=True)
    return cycles.len()
def A339852(n):
    return A339849(5, n)
print([A339852(n) for n in range(2, 21)])

Crossrefs

Row 5 of A339849.

Cf. A339202.

Sequence in context: A282645 A283541 A354608 * A202075 A297680 A264680

Adjacent sequences: A339849 A339850 A339851 * A339853 A339854 A339855

Keyword

nonn

Author

Seiichi Manyama, Dec 19 2020

Status

approved