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

Author

Omar E. Pol, Aug 20 2011

Keywords

• nonn
• easy

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

Examples

			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
		

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).

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.

Programs

Magma

[Floor(5*n*(n+1)/6): n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

Mathematica

Table[Floor[5 n (n + 1)/6], {n, 0, 50}] (* Arkadiusz Wesolowski, Oct 03 2011 *)

PARI

a(n)=5*n*(n+1)\6 \\ Charles R Greathouse IV, Aug 23 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

Extensions

Name improved by Arkadiusz Wesolowski, Aug 23 2011

New name from Omar E. Pol, Sep 28 2011