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a(n) = p(3*n), where p(n) = number of partitions of n into parts that correspond to areas of distinct pieces of Archimedes' ostomachion.
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%I #4 Jan 28 2014 03:48:17

%S 1,2,5,9,17,27,43,64,90,124,163,211,261,321,381,446,511,576,638,694,

%T 746,786,818,836,844,836,818,786,746,694,638,576,511,446,381,321,261,

%U 211,163,124,90,64,43,27,17,9,5,2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N a(n) = p(3*n), where p(n) = number of partitions of n into parts that correspond to areas of distinct pieces of Archimedes' ostomachion.

%C The 14 ostomachion pieces and their areas are (following the notation as published in the Bibliotheca Augustana website):

%C 11 triangles: NCO=3, KHT=3, EFQ=6, GCQ=6, BKH=6, ZLF=6, GNC=9, EQG=12, ALZ=12, ABM=12, and BML=12,

%C 2 tetragons: FQCZ=12 and DOCZ=24, and 1 pentagon: LFEHT=21,

%C the sum of these areas is 144 = 12*12;

%C p(3*n+1) = p(3*n+2) = 0, as the areas of all pieces are multiples of 3;

%C a(n) = 0 for n > 48;

%C a(n) <= a(24)=844 and a(24 - k) = a(24 + k), 0 < k <= 24;

%C A165272 and A165273 give first differences and partial sums.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ostomachion">Ostomachion</a>

%H Bibliotheca Augustana, <a href="http://www.hs-augsburg.de/~harsch/graeca/Chronologia/S_ante03/Archimedes/arc_ost2.html">Ostomachion</a>

%e p(3) = a(1) = #{NCO, KHT} = 2;

%e p(6) = a(2) = #{EFQ, GCQ, BKH, ZLF, NCO+KHT} = 5;

%e p(9) = a(3) = #{GNC, EFQ+NCO, EFQ+KHT, GCQ+NCO, GCQ+KHT, BKH+NCO, BKH+KHT, ZLF+NCO, ZLF+KHT} = 9;

%e p(12) = a(4) = #{EQG, ALZ, ABM, BML, FQCZ, GNC+NCO, GNC+KHT, EFQ+GCQ, EFQ+BKH, EFQ+ZLF, EFQ+NCO+KHT, GCQ+BKH, GCQ+ZLF, GCQ+NCO+KHT, BKH+ZLF, BKH+NCO+KHT, ZLF+NCO+KHT} = 17.

%Y A008585, A001651.

%K nonn,look

%O 0,2

%A _Reinhard Zumkeller_, Sep 13 2009