A194275 Concentric pentagonal numbers of the second kind: a(n) = floor(5*n*(n+1)/6).

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

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. A000326, A008706, A008854, A032528, A152734, A193273, A193274.

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

Adjacent sequences: A194272 A194273 A194274 * A194276 A194277 A194278

Keyword

nonn,easy,changed

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