1,2

a(n) counts the partitions of {1...n} with the property that all elements of the partition are congruent, modulo translation and reflection, to the same tile.

Two tilings that are reflections of each other are considered distinct. E.g. {{1,2,6},{3,7,8},{4,5,9}} and {{1,5,6},{2,3,7},{4,8,9}} are both included in the count for a(9). The first tile that allows more than one tiling for the same set without one being a reflection of the other is {1,2,7} on the span {1...12}.

a(n)-4 often seems to be a power of 2. - Don Reble

a(8)=13 because the following are the 13 tilings of {1...8}:

{{1},{2},{3},{4},{5},{6},{7},{8}} tile: {1}

{{1,2},{3,4},{5,6},{7,8}} tile: {1,2}

{{1,3},{2,4},{5,7},{6,8}} tile: {1,3}

{{1,5},{2,6},{3,7},{4,8}} tile: {1,5}

{{1,2,3,4},{5,6,7,8}} tile: {1,2,3,4}

{{1,2,3,5},{4,6,7,8}} tile: {1,2,3,5}

{{1,5,6,7},{2,3,4,8}} tile: {1,2,3,7}

{{1,2,4,6},{3,5,7,8}} tile: {1,2,4,6}

{{1,4,6,7},{2,3,5,8}} tile: {1,2,4,7}

{{1,2,5,6},{3,4,7,8}} tile: {1,2,5,6}

{{1,3,4,7},{2,5,6,8}} tile: {1,3,4,7}

{{1,3,5,7},{2,4,6,8}} tile: {1,3,5,7}

{{1,2,3,4,5,6,7,8}} tile: {1,2,3,4,5,6,7,8}

Cf. A096202, A096203.

Sequence in context: A100577 A018818 A067538 this_sequence A084540 A113416 A131999

Adjacent sequences: A096151 A096152 A096153 this_sequence A096155 A096156 A096157

nonn

Jon Wild (wild(AT)music.mcgill.ca), Jul 26 2004

More terms from Don Reble (djr(AT)nk.ca), Jul 04 2004