A165271 - OEIS

A165271

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.

1, 2, 5, 9, 17, 27, 43, 64, 90, 124, 163, 211, 261, 321, 381, 446, 511, 576, 638, 694, 746, 786, 818, 836, 844, 836, 818, 786, 746, 694, 638, 576, 511, 446, 381, 321, 261, 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

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OFFSET

0,2

COMMENTS

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

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,

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

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

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

a(n) = 0 for n > 48;

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

A165272 and A165273 give first differences and partial sums.

LINKS

Table of n, a(n) for n=0..68.

Wikipedia, Ostomachion

Bibliotheca Augustana, Ostomachion

EXAMPLE

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

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

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

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.

CROSSREFS

A008585, A001651.

Sequence in context: A342854 A062492 A268346 * A308827 A139672 A308873

Adjacent sequences: A165268 A165269 A165270 * A165272 A165273 A165274

KEYWORD

nonn,look

AUTHOR

Reinhard Zumkeller, Sep 13 2009

STATUS

approved