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a(n) = floor(3/2 + sqrt(n)).
1

%I #25 Jan 10 2024 23:58:40

%S 1,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,

%T 7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,

%U 9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10

%N a(n) = floor(3/2 + sqrt(n)).

%C Burning number of the n-ladder (for n >= 1), n-Moebius ladder (for n >= 3), and n-prism (for n >= 3) graphs.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BurningNumber.html">Burning Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LadderGraph.html">Ladder Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusLadder.html">Moebius Ladder</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>.

%F a(n) = A000194(n) + 1. - _Andrew Howroyd_, Jan 10 2024

%F G.f.: x*(1 + QPochhammer(-x^2, x^4)*QPochhammer(x^8, x^8))/(1 - x).

%t Table[Floor[3/2 + Sqrt[n]], {n, 50}]

%t Floor[3/2 + Sqrt[Range[50]]]

%t CoefficientList[Series[(1 + QPochhammer[-x^2, x^4] QPochhammer[x^8, x^8])/(1 - x), {x, 0, 50}], x]

%Y Sequence agrees with the known terms of A155934.

%Y Cf. A000194, A003059.

%K nonn,easy

%O 0,2

%A _Eric W. Weisstein_, Jan 10 2024

%E Terms a(26) and beyond from _Andrew Howroyd_, Jan 10 2024