login
A194275
Concentric pentagonal numbers of the second kind: a(n) = floor(5*n*(n+1)/6).
7
0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, 110, 130, 151, 175, 200, 226, 255, 285, 316, 350, 385, 421, 460, 500, 541, 585, 630, 676, 725, 775, 826, 880, 935, 991, 1050, 1110, 1171, 1235, 1300, 1366, 1435, 1505, 1576, 1650, 1725, 1801, 1880, 1960, 2041, 2125
OFFSET
0,3
COMMENTS
Quasipolynomial: trisections are (15*x^2 - 15*x + 2)/2, 5*(15*x^2 - 5*x)/2, and 5*(15*x^2 + 5*x)/2. - Charles R Greathouse IV, Aug 23 2011
Appears to be similar to cellular automaton. The sequence gives the number of elements in the structure after n-th stage. Positive integers of A008854 gives the first differences. For a definition without words see the illustration of initial terms in the example section.
Also partial sums of A008854.
Also row sums of an infinite square array T(n,k) in which column k lists 3*k-1 zeros followed by the numbers A008706 (see example).
For concentric pentagonal numbers see A032527. - Omar E. Pol, Sep 27 2011
FORMULA
G.f.: (-1 - 3*x - x^2)/((-1 + x)^3*(1 + x + x^2)). - Alexander R. Povolotsky, Aug 22, 2011
a(n) = floor(5*n*(n+1)/6). - Arkadiusz Wesolowski, Aug 23 2011
EXAMPLE
Using the numbers A008706 we can write:
0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
0, 0, 0, 0, 1, 5, 10, 15, 20, 25, 30, ...
0, 0, 0, 0, 0, 0, 0, 1, 5, 10, 15, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, ...
And so on.
===========================================
The sums of the columns give this sequence:
0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, ...
...
Illustration of initial terms (in a precise representation the pentagons should appear strictly concentric):
. o
. o o
. o o o
. o o o o o
. o o o o o o o
. o o o o o o o o o
. o o o o o o 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 5 10 16 25
MATHEMATICA
Table[Floor[5 n (n + 1)/6], {n, 0, 50}] (* Arkadiusz Wesolowski, Oct 03 2011 *)
PROG
(PARI) a(n)=5*n*(n+1)\6 \\ Charles R Greathouse IV, Aug 23 2011
(Magma) [Floor(5*n*(n+1)/6): n in [0..60]]; // Vincenzo Librandi, Sep 27 2011
CROSSREFS
Cf. similar sequences with the formula floor(k*n*(n+1)/(k+1)) listed in A281026.
Sequence in context: A306351 A215341 A345070 * A026059 A115002 A301290
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 20 2011
EXTENSIONS
Name improved by Arkadiusz Wesolowski, Aug 23 2011
New name from Omar E. Pol, Sep 28 2011
STATUS
approved