A194274 Concentric square numbers (see Comments lines for definition).

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

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

From R. J. Mathar, Aug 22 2011: (Start)

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

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

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(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

E.g.f.: (exp(x)*x*(3 + x) - sin(x))/2. - Stefano Spezia, Feb 26 2023

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), end=", ")
    curr = n*n - prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 03 2012
(Python)
def A194274(n): return (3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2) # Chai Wah Wu, Jul 15 2023
(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)
def A194274(n): return n if n<2 else n^2 - A194274(n-2)
[A194274(n) for n in range(41)] # G. C. Greubel, Jan 31 2024

Crossrefs

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

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

Sequence in context: A311553 A311554 A340266 * A098573 A092753 A376852

Adjacent sequences: A194271 A194272 A194273 * A194275 A194276 A194277

Keyword

nonn,easy

Author

Omar E. Pol, Aug 20 2011

Status

approved