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A195971 Number of n X 1 0..4 arrays with each element x equal to the number its horizontal and vertical neighbors equal to 2,0,1,3,4 for x=0,1,2,3,4. 16
0, 1, 3, 4, 5, 9, 16, 25, 39, 64, 105, 169, 272, 441, 715, 1156, 1869, 3025, 4896, 7921, 12815, 20736, 33553, 54289, 87840, 142129, 229971, 372100, 602069, 974169, 1576240, 2550409, 4126647, 6677056, 10803705, 17480761, 28284464, 45765225 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Every 0 is next to 0 2's, every 1 is next to 1 0's, every 2 is next to 2 1's, every 3 is next to 3 3's, every 4 is next to 4 4's.

Column 1 of A195978.

a(n) is the number of total dominating sets in the (n+1)-path graph. - Eric W. Weisstein, Apr 10 2018

LINKS

R. H. Hardin, Table of n, a(n) for n = 0..200 (corrected by R. H. Hardin, Jan 19 2019)

Eric Weisstein's World of Mathematics, Path Graph

Eric Weisstein's World of Mathematics, Total Dominating Set

Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1).

FORMULA

a(n) = a(n-1) + a(n-3) + a(n-4).

G.f.: x*(1 + x)^2 / ((1 + x^2)*(1 - x - x^2)). - Colin Barker, Feb 17 2018

a(n) = (A000032(n + 3) - 2*sin(n*Pi/2) - 4*cos(n*Pi/2))/5. - Eric W. Weisstein, Apr 10 2018

a(n) = (Lucas(n+3) - (-1)^(floor(n/2))*(3+(-1)^n))/5. - G. C. Greubel, Apr 03 2019

EXAMPLE

All solutions for n=4:

  0   0   1   1   0

  0   0   0   0   1

  0   0   0   0   1

  1   0   1   0   0

MATHEMATICA

Table[(LucasL[n + 3] - 2 Sin[n Pi/2] - 4 Cos[n Pi/2])/5, {n, 0, 40}] (* Eric W. Weisstein, Apr 10 2018 *)

LinearRecurrence[{1, 0, 1, 1}, {0, 1, 3, 4, 5}, 40] (* Eric W. Weisstein, Apr 10 2018; amended for a(0) by Georg Fischer, Apr 03 2019 *)

CoefficientList[Series[x*(1+x)^2/(1-x-x^3-x^4), {x, 0, 40}], x] (* Eric W. Weisstein, Apr 10 2018 *)

PROG

(PARI) my(x='x+O('x^40)); concat([0], Vec(x*(1+x)^2/(1-x-x^3-x^4))) \\ G. C. Greubel, Apr 03 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1+x)^2/(1-x-x^3-x^4) )); // G. C. Greubel, Apr 03 2019

(Sage) (x*(1+x)^2/(1-x-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 03 2019

(GAP) a:=[1, 3, 4, 5];; for n in [5..40] do a[n]:=a[n-1]+a[n-3]+a[n-4]; od; Concatenation([0], a); # G. C. Greubel, Apr 03 2019

CROSSREFS

Cf. A195978.

Sequence in context: A195609 A117125 A000692 * A080552 A215176 A257041

Adjacent sequences:  A195968 A195969 A195970 * A195972 A195973 A195974

KEYWORD

nonn,easy

AUTHOR

R. H. Hardin, Sep 25 2011

STATUS

approved

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Last modified August 5 21:45 EDT 2020. Contains 336213 sequences. (Running on oeis4.)