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Concentric square numbers (see Comments lines for definition).
11

%I #66 Jan 31 2024 13:11:42

%S 0,1,4,8,12,17,24,32,40,49,60,72,84,97,112,128,144,161,180,200,220,

%T 241,264,288,312,337,364,392,420,449,480,512,544,577,612,648,684,721,

%U 760,800,840,881,924,968,1012,1057,1104,1152,1200,1249,1300,1352,1404

%N Concentric square numbers (see Comments lines for definition).

%C 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.

%C Also, union of A046092 and A077221, the bisections of this sequence.

%C 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).

%H Arkadiusz Wesolowski, <a href="/A194274/b194274.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).

%F a(n) = n^2 - a(n-2), with a(0)=0, a(1)=1. - _Alex Ratushnyak_, Aug 03 2012

%F From _R. J. Mathar_, Aug 22 2011: (Start)

%F G.f.: x*(1 + x)/((1 + x^2)*(1 - x)^3).

%F a(n) = (A005563(n) - A056594(n-1))/2. (End)

%F 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

%F a(n) = floor(3*n/4) + floor((n*(n+2)+1)/2) - floor((3*n+1)/4). - _Arkadiusz Wesolowski_, Nov 08 2011

%F 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

%F E.g.f.: (exp(x)*x*(3 + x) - sin(x))/2. - _Stefano Spezia_, Feb 26 2023

%e Using the numbers A008574 we can write:

%e 0, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

%e 0, 0, 0, 0, 0, 1, 4, 8, 12, 16, 20, ...

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, ...

%e And so on.

%e ===========================================

%e The sums of the columns give this sequence:

%e 0, 1, 4, 8, 12, 17, 24, 32, 40, 49, 60, ...

%e ...

%e Illustration of initial terms:

%e . o o o o o o

%e . o o o o o o o

%e . o o o o o o o o o o

%e . o o o o o o o o o o o o

%e . o o o o o o o o o o

%e . o o o o o o o o o o o o o o o o o o o o o

%e .

%e . 1 4 8 12 17 24

%t Table[Floor[3*n/4] + Floor[(n*(n + 2) + 1)/2] - Floor[(3*n + 1)/4], {n, 0, 52}] (* _Arkadiusz Wesolowski_, Nov 08 2011 *)

%t 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 *)

%o (Python)

%o prpr = 0

%o prev = 1

%o for n in range(2,777):

%o print(str(prpr), end=", ")

%o curr = n*n - prpr

%o prpr = prev

%o prev = curr

%o # _Alex Ratushnyak_, Aug 03 2012

%o (Python)

%o def A194274(n): return (3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2) # _Chai Wah Wu_, Jul 15 2023

%o (Magma) [n le 2 select n-1 else (n-1)^2 - Self(n-2): n in [1..61]]; // _G. C. Greubel_, Jan 31 2024

%o (SageMath)

%o def A194274(n): return n if n<2 else n^2 - A194274(n-2)

%o [A194274(n) for n in range(41)] # _G. C. Greubel_, Jan 31 2024

%Y Cf. A000290, A005563, A008574, A056594, A032528, A046092.

%Y Cf. A077221, A085250, A098181, A194273, A194275.

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Aug 20 2011