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A194274
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Concentric square numbers (see Comments lines for definition).
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8
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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
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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, union of A046092 and A077221, the bisections of this sequence.
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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Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
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FORMULA
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a(n) = n^2 - a(n-2), with a(0)=0, a(1)=1. - Alex Ratushnyak, Aug 03 2012
G.f.: x*(1+x) / ((1+x^2)*(1-x)^3). a(n) = (A005563(n)-A056594(n-1))/2. - R. J. Mathar, Aug 22 2011
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) = floor(3*n/4) + floor((n*(n+2)+1)/2) - floor((3*n+1)/4). - Arkadiusz Wesolowski, Nov 08 2011
a(0)=0, a(1)=1, a(2)=4, a(3)=8, a(4)=12, a(n)=3*a(n-1)-4*a(n-2)+ 4*a(n-3)- 3*a(n-4)+a(n-5). - 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
# Alex Ratushnyak, Aug 03 2012
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CROSSREFS
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Cf. A000290, A085250, A008574, A032528, A046092, A077221, A098181, A194273, A194275.
Sequence in context: A311553 A311554 A340266 * A098573 A092753 A276338
Adjacent sequences: A194271 A194272 A194273 * A194275 A194276 A194277
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KEYWORD
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nonn,easy
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AUTHOR
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Omar E. Pol, Aug 20 2011
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STATUS
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approved
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