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URL: https://oeis.org/A302654

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A302654
Number of minimum total dominating sets in the n-path graph.
4
0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1, 3, 16, 5, 1, 4, 25, 6, 1, 5, 36, 7, 1, 6, 49, 8, 1, 7, 64, 9, 1, 8, 81, 10, 1, 9, 100, 11, 1, 10, 121, 12, 1, 11, 144, 13, 1, 12, 169, 14, 1, 13, 196, 15, 1, 14, 225, 16, 1, 15, 256, 17, 1, 16, 289, 18, 1, 17, 324, 19, 1, 18, 361, 20, 1
OFFSET
1,3
LINKS
Eric Weisstein's World of Mathematics, Minimum Total Dominating Set.
Eric Weisstein's World of Mathematics, Path Graph.
FORMULA
a(n) = ((-1)^n*(n - 2)^2 + (6 + n)^2 - 2*(n - 2)*(n + 6)*cos(n*Pi/2) - 48*sin(n*Pi/2))/6.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
G.f.: x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3). - Colin Barker, Dec 25 2019
E.g.f.: ((12 + x^2)*cos(x) + (20 + 8*x + x^2)*cosh(x) + (5*x - 24)*sin(x) + (16 + 5*x)*sinh(x) - 32)/32. - Stefano Spezia, Jun 10 2025
MATHEMATICA
Table[Piecewise[{{1, Mod[n, 4] == 0}, {((n + 2)/4)^2, Mod[n, 4] == 2}, {(n - 1)/4, Mod[n, 4] == 1}, {(n + 5)/4, Mod[n, 4] == 3}}], {n, 20}]
Table[((-1)^n (n - 2)^2 + (6 + n)^2 - 2 (n - 2) (n + 6) Cos[n Pi/2] - 48 Sin[n Pi/2])/64, {n, 20}]
LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 2, 1, 1, 4, 3, 1, 2, 9, 4, 1}, 20]
PROG
(PARI) concat(0, Vec(x^2*(1 + 2*x + x^2 + x^3 + x^4 - 3*x^5 - 2*x^6 - x^7 + x^9 + x^10) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3) + O(x^70))) \\ Colin Barker, Dec 25 2019
CROSSREFS
Row 1 of A303293.
Sequence in context: A370005 A294082 A129705 * A264831 A264728 A306790
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Apr 11 2018
STATUS
approved