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A194274
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Concentric square numbers (see Comments lines for definition).
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11
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0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, 72, 84, 97, 112, 128, 144, 161, 180, 200, 220, 241, 264, 288, 312, 337, 364, 392, 420, 449, 480, 512, 544, 577, 612, 648, 684, 721, 760, 800, 840, 881, 924, 968, 1012, 1057, 1104, 1152, 1200, 1249, 1300, 1352, 1404
(list;
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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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Cellular automaton on the first quadrant of the square grid. The sequence gives the number of cells "ON" in the structure after n-th stage. A098181 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 A194273, A194275 and A032528.
Also row sums of an infinite square array T(n,k) in which column k lists 4*k-1 zeros followed by the numbers A008574 (see example).
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LINKS
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FORMULA
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G.f.: x*(1 + x)/((1 + x^2)*(1 - x)^3).
a(n) = a(-n-2) = (2*n*(n+2) + (1-(-1)^n)*i^(n+1))/4, where i=sqrt(-1). - Bruno Berselli, Sep 22 2011
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5), with a(0)=0, a(1)=1, a(2)=4, a(3)=8, a(4)=12. - Harvey P. Dale, Sep 11 2013
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EXAMPLE
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Using the numbers A008574 we can write:
0, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
0, 0, 0, 0, 0, 1, 4, 8, 12, 16, 20, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, ...
...
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 o o o o o o o o o o o o o o o o o o o o
.
. 1 4 8 12 17 24
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MATHEMATICA
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Table[Floor[3*n/4] + Floor[(n*(n + 2) + 1)/2] - Floor[(3*n + 1)/4], {n, 0, 52}] (* Arkadiusz Wesolowski, Nov 08 2011 *)
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==n^2-a[n-2]}, a, {n, 60}] (* or *) LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 4, 8, 12}, 60] (* Harvey P. Dale, Sep 11 2013 *)
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PROG
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(Python)
prpr = 0
prev = 1
for n in range(2, 777):
print(str(prpr), end=", ")
curr = n*n - prpr
prpr = prev
prev = curr
(Python)
(Magma) [n le 2 select n-1 else (n-1)^2 - Self(n-2): n in [1..61]]; // G. C. Greubel, Jan 31 2024
(SageMath)
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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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