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A194274 Concentric square numbers (see Comments lines for definition). 8
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; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

LINKS

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).

FORMULA

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

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(Python)

prpr = 0

prev = 1

for n in range(2, 777):

    print str(prpr)+', ',

    curr = n*n - prpr

    prpr = prev

    prev = curr

# from Alex Ratushnyak, Aug 03 2012

CROSSREFS

Cf. A000290, A085250, A008574, A032528, A046092, A077221, A098181, A194273, A194275.

Sequence in context: A311552 A311553 A311554 * A098573 A092753 A276338

Adjacent sequences:  A194271 A194272 A194273 * A194275 A194276 A194277

KEYWORD

nonn,easy

AUTHOR

Omar E. Pol, Aug 20 2011

STATUS

approved

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Last modified October 22 07:48 EDT 2019. Contains 328315 sequences. (Running on oeis4.)