A303834 Number of total dominating sets in the n-gear graph.

3, 21, 48, 161, 473, 1476, 4553, 14241, 44688, 141081, 447153, 1422596, 4539473, 14522361, 46556048, 149508801, 480810153, 1548053316, 4988972313, 16090635281, 51928966928, 167675418921, 541639730273, 1750245266436, 5657268819873, 18289912673001

Offset: 1

Author

Eric W. Weisstein, May 02 2018

Keywords

• nonn
• easy

Comments

Extended to a(1)-a(2) using the formula/recurrence.

Links

• Eric Weisstein's World of Mathematics, Gear Graph
• Eric Weisstein's World of Mathematics, Total Dominating Set
• Index entries for linear recurrences with constant coefficients, signature (4,2,-13,-6,4).

Programs

Mathematica

Table[2 (-1)^n + 2^n LucasL[n] + LucasL[2 n], {n, 20}]
LinearRecurrence[{4, 2, -13, -6, 4}, {3, 21, 48, 161, 473}, 20]
CoefficientList[Series[(-3 - 9 x + 42 x^2 + 34 x^3 - 24 x^4)/(-1 + 4 x + 2 x^2 - 13 x^3 - 6 x^4 + 4 x^5), {x, 0, 20}], x]

Formula

a(n) = 2*(-1)^n + 2^n*A000032(n) + A005248(n).

a(n) = 4*a(n-1) + 2*a(n-2) - 13*a(n-3) - 6*a(n-4) + 4*a(n-5).

G.f.: -x*(3+9*x-42*x^2-34*x^3+24*x^4) / ( (1+x)*(x^2-3*x+1)*(4*x^2+2*x-1) ).