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a(n) is the minimum number of square tiles needed for constructing a figure whose corresponding graph has n cycles.
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%I #14 Jul 02 2022 14:50:53

%S 0,4,7,6,9,12,8,11,14,13,10,13,16,15,18,12,15,18,17,20,23,14,17,20,19,

%T 22,25,21,16,19,22,21,24,27,23,26,18,21,24,23,26,29,25,28,31,20,23,26,

%U 25,28,31,27,30,33,32,22,25,28,27,30,33,29,32,35,34,31,24

%N a(n) is the minimum number of square tiles needed for constructing a figure whose corresponding graph has n cycles.

%C The square tiles are connected only at corners.

%H Stefano Spezia, <a href="/A354763/a354763_2.jpg">Illustrations for n = 1..10</a>

%F a(n) = 1 + Sum_{i=1..A057945(n)} sqrt(1 + 8*A354762(n, i)) for n > 0.

%F a(A000217(n)) = A163300(n+1).

%t r[n_]:=First[IntegerPartitions[n,All,Table[k(k+1)/2,{k,Sqrt[1+8n]}]]]; (* A354762 *)

%t Join[{0}, Table[1+Sum[Sqrt[1+8Part[r[n],i]],{i,Length[r[n]]}],{n,66}]]

%Y Cf. A000217, A057945, A163300, A354762.

%K nonn,easy

%O 0,2

%A _Stefano Spezia_, Jun 06 2022