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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. 17
 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 Equivalently, a(n) is the number of 0-1 sequences (every term is "0" or "1") of length n+1 whose every term is adjacent to a term "1". - Yuda Chen, Apr 06 2022 From Wajdi Maaloul, Jun 23 2022: (Start) For n > 1, a(n) is the number of ways to tile the figure below using squares and dominoes: a horizontal strip of length n-1 that contains a central vertical strip of length 3). Below are figures for a(2) through a(5): _ _ _ _ |_| _|_| _|_|_ _ _|_|_ |_| |_|_| |_|_|_| |_|_|_|_| |_| |_| |_| |_| (End) 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 From Wajdi Maaloul, Jun 23 2022: (Start) a(2n) = A226205(n+1) = - A121646(n+1) = Fibonacci(n+1)^2 - Fibonacci(n)^2 = Fibonacci(n+2)*Fibonacci(n-1); a(2n+1) = Fibonacci(n+2)^2 = A007598(n+2). (End) 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:=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 A063781 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 July 13 12:04 EDT 2024. Contains 374282 sequences. (Running on oeis4.)