|
|
A032528
|
|
Concentric hexagonal numbers: floor(3*n^2/2).
|
|
45
|
|
|
0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601, 3750
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
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
|
|
|
FORMULA
|
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) = a(n).
a(n) = a(n-2) + 6*(n-1) for n > 1.
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
|
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
|
(Haskell)
a032528 n = a032528_list !! n
a032528_list = scanl (+) 0 a007310_list
|
|
CROSSREFS
|
Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011
|
|
STATUS
|
approved
|
|
|
|