A194273 -id:A194273 - cp's OEIS frontend

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

0, 3, 24, 63, 120, 195, 288, 399, 528, 675, 840, 1023, 1224, 1443, 1680, 1935, 2208, 2499, 2808, 3135, 3480, 3843, 4224, 4623, 5040, 5475, 5928, 6399, 6888, 7395, 7920, 8463, 9024, 9603, 10200, 10815, 11448, 12099, 12768, 13455, 14160, 14883, 15624, 16383, 17160

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) also can be represented as n concentric triangles (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric triangles:
.
.                                          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 o o o o o o o   o
.                              o                       o
.                             o o o o o o o o o o o o o o
.
.    3            24                       63
(End)

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567, A064201, A139267, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152759, A152767, A153783, A153448, A153875.
Cf. A033581, A085250, A152734, A194273. - Omar E. Pol, Aug 21 2011
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=18: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,18}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
3*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, May 08 2022 *)

a(n)=3*n*(3*n-2) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 9*n^2 - 6*n = 3*A000567(n) = A064201(n)/3.
a(n) = a(n-1) + 18*n - 15 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+5*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 3*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = n + A152995(n). (End)

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

Original entry on oeis.org

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

Omar E. Pol, Aug 20 2011

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).

			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

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).

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

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

[n le 2 select n-1 else (n-1)^2 - Self(n-2): n in [1..61]]; // G. C. Greubel, Jan 31 2024

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 *)

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

def A194274(n): return (3*n>>2)+(n*(n+2)+1>>1)-(3*n+1>>2) # Chai Wah Wu, Jul 15 2023

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

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

A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3n-2, 3n-1, 3n, 3n, 3n, 3n.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 36, 36, 36, 36

Offset: 1

Omar E. Pol, Aug 20 2011

Also first differences of A194273 which is also a sequence related to cellular automata.

			Array begins:
1,  2,  3,  3,  3,  3,
4,  5,  6,  6,  6,  6,
7,  8,  9,  9,  9,  9,
10, 11, 12, 12, 12, 12,
13, 14, 15, 15, 15, 15,
16, 17, 18, 18, 18, 18,
19, 20, 21, 21, 21, 21,
22, 23, 24, 24, 24, 24,
...
Sum of row n gives 18*n-3 = A008600(n) - 3.
G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Column 1: A016777. Column 2: A016789. Every column 3, 4, 5 and 6: positive integers of A008585.

Cf. A008600, A047926, A132868, A194273.

[Floor((n+3)/6) + Floor((n+4)/6) + Floor((n+5)/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2015

A194272:=n->floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6): seq(A194272(n), n=1..100); # Wesley Ivan Hurt, Apr 04 2015

Table[Floor[(n + 3)/6] + Floor[(n + 4)/6] + Floor[(n + 5)/6], {n, 100}] (* Wesley Ivan Hurt, Apr 04 2015 *)

x='x+O('x^60); Vec(x*(1-x^3)/((1-x)^2*(1-x^6))) \\ G. C. Greubel, Aug 13 2018

From Michael Somos, May 12 2014: (Start)
Euler transform of length 6 sequence [2, 0, -1, 0, 0, 1].
G.f.: x * (1-x^3) / ( (1-x)^2 * (1-x^6) ).
a(n-1) = A047926(n) - A132868(n). (End)
From Wesley Ivan Hurt, Apr 04 2015, Sep 08 2015: (Start)
a(n) = 2*a(n-1)-a(n-2)-a(n-3)+2*a(n-4)-a(n-5), n>5.
a(n) = floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6).
a(n) = Sum_{i=0..n-1} floor(i/6) - floor((i-3)/6). (End)

cp's OEIS Frontend

A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).

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

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A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3*n-2, 3*n-1, 3*n, 3*n, 3*n, 3*n.

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A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3n-2, 3n-1, 3n, 3n, 3n, 3n.