A208556 -id:A208556 - cp's OEIS frontend

A230068 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element diagonally or antidiagonally next to at least one element with value 2-x(i,j).

0, 0, 0, 0, 9, 0, 0, 9, 9, 0, 0, 81, 45, 81, 0, 0, 225, 441, 441, 225, 0, 0, 1089, 3645, 15129, 3645, 1089, 0, 0, 3969, 31329, 281961, 281961, 31329, 3969, 0, 0, 16641, 266805, 6487209, 14082255, 6487209, 266805, 16641, 0, 0, 65025, 2277081, 137945025

Offset: 1

R. H. Hardin, Oct 08 2013

Table starts
.0.....0.......0..........0.............0................0...................0
.0.....9.......9.........81...........225.............1089................3969
.0.....9......45........441..........3645............31329..............266805
.0....81.....441......15129........281961..........6487209...........137945025
.0...225....3645.....281961......14082255........793041921.........43142680647
.0..1089...31329....6487209.....793041921.....114228452529......15615593045649
.0..3969..266805..137945025...43142680647...15615593045649....5392746297039195
.0.16641.2277081.3020271849.2369158188849.2168295406671681.1888668174848027049

			Some solutions for n=4 k=4
..0..1..2..0....1..1..1..0....1..2..1..2....0..0..0..2....0..1..0..1
..1..2..2..0....1..1..2..1....1..1..0..0....2..2..0..2....2..2..1..2
..1..2..0..1....2..2..0..1....1..1..2..2....0..0..2..2....0..0..1..1
..0..1..1..2....0..0..1..2....1..0..1..0....2..2..0..0....2..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..312

Column 2 is A208556(n-2)

Empirical for column k:
k=1: a(n) = a(n-1)
k=2: a(n) = 3*a(n-1) +6*a(n-2) -8*a(n-3)
k=3: a(n) = 7*a(n-1) +14*a(n-2) -8*a(n-3)
k=4: [order 10] for n>11
k=5: [order 16] for n>19

A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.

Original entry on oeis.org

1, 9, 9, 81, 225, 1089, 3969, 16641, 65025, 263169, 1046529, 4198401, 16769025, 67125249, 268402689, 1073807361, 4294836225, 17180131329, 68718952449, 274878955521, 1099509530625, 4398050705409, 17592177655809, 70368760954881, 281474943156225, 1125899973951489

Offset: 0

Peter Luschny, Jan 09 2019

Colin Barker conjectures that A208556 is a shifted version of this sequence.

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Cf. A001045, A062510, A139818, A208556, A322942, A323232.

gf := (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)):
ser := series(gf,x,32): seq(coeff(ser,x,n), n=0..25);

LinearRecurrence[{3, 6, -8}, {1, 9, 9, 81}, 25]

# Demonstrates the product formula.
CC = ComplexField(200)
def t(n,k): return CC(3)*cos(CC(pi*k/n)) - CC(i)*sin(CC(pi*k/n))
def T(n,k): return t(n,k)*(t(n,k).conjugate())
def a(n): return prod(T(n,k) for k in (1..n))
print([a(n).real().round() for n in (0..29)])

a(n) = Product_{k=1..n} T(n, k) where T(n, k) = t(n,k)*conjugate(t(n,k)) and t(n,k) = 3*cos(Pi*k/n) - i*sin(Pi*k/n), i is the imaginary unit.
a(n) = [x^n] (8*x^3 - 24*x^2 + 6*x + 1)/((4*x - 1)*(2*x + 1)*(x - 1)).
a(n) = n! [x^n] (1 + exp(x) - 2*exp(-2*x) + exp(4*x)).
a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) for n >= 4.
A062510(n) = sqrt(a(n)) for n > 0.
a(n) = 4^n-2*(-2)^n+1, n>0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A230068 T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element diagonally or antidiagonally next to at least one element with value 2-x(i,j).

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A323210 a(n) = 9*J(n)^2 where J(n) are the Jacobsthal numbers A001045 with J(0) = 1.

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