Journal of Integer Sequences, Vol. 25 (2022), Article 22.9.8 |

Physics Department

Faculty of Science

Mahidol University

Rama 6 Road

Bangkok 10400

Thailand

**Abstract:**

We consider a two-parameter family of triangles whose -th entry
(counting the initial entry as the -th entry) is the number of
tilings of -boards (which are linear arrays of unit square cells for any
nonnegative integer ) with unit squares and -combs for
some fixed
and
that use tiles in total
of which are combs. A -comb is a tile composed of
unit square sub-tiles (referred to as teeth) placed so that each tooth
is separated from the next by a gap of width . We show that the
entries in the triangle are coefficients of the product of two
consecutive generalized Fibonacci polynomials each raised to some
nonnegative integer power. We also present a bijection between the
tiling of an
-board with -combs with the
remaining cells filled with squares and the -subsets of
such that no two elements of the subset differ by a
multiple of up to . We can therefore give a combinatorial
proof of how the number of such -subsets is related to the
coefficient of a polynomial. We also derive a recursion relation for
the number of closed walks from a particular node on a class of
directed pseudographs and apply it obtain an identity concerning the
, instance of the family of triangles. Further identities
of the triangles are also established mostly via combinatorial proof.

(Concerned with sequences A000045 A000930 A003269 A003520 A005578 A005708 A005709 A005710 A007318 A011782 A031923 A059259 A099163 A224808 A224809 A224811 A350110 A350111 A350112 A354665 A354666 A354667 A354668.)

Received September 3 2022; revised version received December 8 2022.
Published in *Journal of Integer Sequences*,
December 13 2022.

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