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A194273
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Concentric triangular numbers (see Comments lines for definition).
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3
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0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55, 63, 72, 81, 90, 99, 109, 120, 132, 144, 156, 168, 181, 195, 210, 225, 240, 255, 271, 288, 306, 324, 342, 360, 379, 399, 420, 441, 462, 483, 505, 528, 552, 576, 600, 624, 649, 675, 702, 729, 756, 783, 811
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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This can be interpreted as a cellular automaton on the infinite hexagonal net. The sequence gives the number of cells "ON" in the structure after n-th stage. A194272 gives the first differences. For a definition without words see the illustration of initial terms in the example section. For other concentric polygonal numbers see A194274, A194275 and A032528.
Also, row sums of an infinite square array T(n,k) in which column k lists 6*k-1 zeros followed by the numbers A008486 (see example).
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LINKS
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FORMULA
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G.f.: x/(1-3*x+3*x^2-3*x^4+3*x^5-x^6) = x/((1-x)^3*(1+x)*(1-x+x^2)).
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EXAMPLE
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Using the numbers A008486 we can write:
0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36,...
0, 0, 0, 0, 0, 0, 0, 1, 3, 6, 9, 12, 15, 18,...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,...
And so on.
=========================================================
The sums of the columns give this sequence:
0, 1, 3, 6, 9, 12, 15, 19, 24, 30, 36, 42, 48, 55,...
...
Illustration of initial terms:
. o
. o o o
. o o o o o
. o o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
.
. 1 3 6 9 12 15
.
. o
. o o o
. o o o o o
. o o o o o o
. o o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o
. o o o o o o
. o o o o o o o o o o o o o o o o o o o o o o o o
.
. 19 24 30
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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