

A196700


Number of n X 1 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 3,1,0,4,2 for x=0,1,2,3,4.


6



1, 2, 4, 6, 12, 22, 40, 74, 136, 250, 460, 846, 1556, 2862, 5264, 9682, 17808, 32754, 60244, 110806, 203804, 374854, 689464, 1268122, 2332440, 4290026, 7890588, 14513054, 26693668, 49097310, 90304032, 166095010, 305496352, 561895394
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OFFSET

1,2


COMMENTS

Every 0 is next to zero 3's, every 1 is next to one 1, every 2 is next to two 0's, every 3 is next to three 4's, every 4 is next to four 2's.
Column 1 of A196707.
The perimeter of cuboids with the dimensions of consecutive tribonacci numbers, signature (0,1,0).  Peter M. Chema, Feb 03 2017


LINKS

R. H. Hardin, Table of n, a(n) for n = 1..200


FORMULA

Empirical: a(n) = a(n1) + a(n2) + a(n3) for n > 4.
G.f.: 1  1/x  1/x^2 + 1/x^2/G(0), where G(k)= 1  (2*k+1)*x/(1  x/(x  (2*k+1)/G(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Jul 09 2013
Empirical: a(n) = 2*(A001590(n) + A001590(n1) + A001590(n2)) for n > 1.  Peter M. Chema, Feb 03 2017
From Gregory L. Simay, Jun 23 2017: (Start)
a(n) = A000073(n+2)  A000073(n2), the difference of two tribonacci numbers. The corresponding g.f. is (1  x^4)/(1  x  x^2  x^3). E.g.: a(10) = A000073(12)  A000073(8) = 274  24 = 250.
The tribonacci formula arises from considering the number of compositions of n where only the order of parts 1,2,3 matters (part of an upcoming paper), which may be denoted by C(n [4). We are convolving the number of partitions of n with parts >3 with the tribonacci numbers. The number of partitions of n with parts greater than 3 is P(n)  P(n1)  P(n2) + P(n4) + P(n5)  P(n6). (Derived from the corresponding gf which is (1x)(1x^2)(1x^3)gfP(x).) The rest is algebra. It looks like C(n, [4) = P(n) + Sum_{j=0..n3} P(n3j)*A196700(j+1). (End)


EXAMPLE

All solutions for n=4:
0 0 1 0 0 0
0 0 1 1 0 2
1 0 0 1 2 0
1 0 0 0 0 0


CROSSREFS

Cf. A000073, A001590, A196707.
Sequence in context: A005303 A292320 A057575 * A283834 A326114 A135231
Adjacent sequences: A196697 A196698 A196699 * A196701 A196702 A196703


KEYWORD

nonn


AUTHOR

R. H. Hardin, Oct 05 2011


STATUS

approved



