OFFSET
0,3
COMMENTS
From Omar E. Pol, Aug 20 2011: (Start)
Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.
Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example). (End)
Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - Omar E. Pol, Sep 08 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
FORMULA
From Joerg Arndt, Aug 22 2011: (Start)
G.f.: (x+4*x^2+x^3)/(1-2*x+2*x^3-x^4) = x*(1+4*x+x^2)/((1+x)*(1-x)^3).
a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End)
a(n) = (6*n^2+(-1)^n-1)/4. - Bruno Berselli, Aug 22 2011
a(n) = A184533(n), n >= 2. - Clark Kimberling, Apr 20 2012
From Paul Curtz, Mar 31 2019: (Start)
a(-n) = a(n).
a(n) = a(n-2) + 6*(n-1) for n > 1.
a(2*n) = A033581(n).
a(2*n+1) = A003154(n+1). (End)
E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Aug 19 2022
Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Jan 16 2023
EXAMPLE
From Omar E. Pol, Aug 20 2011: (Start)
Using the numbers A008458 we can write:
0, 1, 6, 12, 18, 24, 30, 36, 42, 48, 54, ...
0, 0, 0, 1, 6, 12, 18, 24, 30, 36, 42, ...
0, 0, 0, 0, 0, 1, 6, 12, 18, 24, 30, ...
0, 0, 0, 0, 0, 0, 0, 1, 6, 12, 18, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ...
...
Illustration of initial terms as concentric hexagons:
.
. 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
.
. 1 6 13 24 37
.
(End)
MATHEMATICA
f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* Clark Kimberling, Apr 20 2012 *)
PROG
(Magma) [Floor(3*n^2/2): n in [0..50]]; // Vincenzo Librandi, Aug 21 2011
(Haskell)
a032528 n = a032528_list !! n
a032528_list = scanl (+) 0 a007310_list
-- Reinhard Zumkeller, Jan 07 2012
(PARI) a(n)=3*n^2\2 \\ Charles R Greathouse IV, Sep 24 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011
STATUS
approved