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Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.
9

%I #24 Oct 20 2014 02:32:31

%S 8,24,48,80,112,160,200,264,328,408,464,560,624,728,832,960,1040,1184,

%T 1272,1432,1576,1728,1832,2024,2160,2336,2512,2736

%N Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

%C Partial sums of A244371.

%C If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.

%C If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

%F a(n) = 4*A244362(n) = 8*A244360(n).

%e Illustration of the structure after 16 stages (Contains 960 toothpicks):

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%Y Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

%K nonn,more

%O 1,1

%A _Omar E. Pol_, Jun 26 2014

%E a(8) corrected and more terms from _Omar E. Pol_, Oct 18 2014