A194275 - OEIS

%I #58 Apr 20 2024 10:27:37

%S 0,1,5,10,16,25,35,46,60,75,91,110,130,151,175,200,226,255,285,316,

%T 350,385,421,460,500,541,585,630,676,725,775,826,880,935,991,1050,

%U 1110,1171,1235,1300,1366,1435,1505,1576,1650,1725,1801,1880,1960,2041,2125

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

%C 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

%C 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.

%C Also partial sums of A008854.

%C 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).

%C For concentric pentagonal numbers see A032527. - _Omar E. Pol_, Sep 27 2011

%H Vincenzo Librandi, <a href="/A194275/b194275.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F G.f.: (-1 - 3*x - x^2)/((-1 + x)^3*(1 + x + x^2)). - _Alexander R. Povolotsky_, Aug 22, 2011

%F a(n) = floor(5*n*(n+1)/6). - _Arkadiusz Wesolowski_, Aug 23 2011

%e Using the numbers A008706 we can write:

%e 0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, ...

%e 0, 0, 0,  0,  1,  5, 10, 15, 20, 25, 30, ...

%e 0, 0, 0,  0,  0,  0,  0,  1,  5, 10, 15, ...

%e 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  1, ...

%e And so on.

%e ===========================================

%e The sums of the columns give this sequence:

%e 0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, ...

%e ...

%e Illustration of initial terms (in a precise representation the pentagons should appear strictly concentric):

%e .                                             o

%e .                                           o   o

%e .                            o            o       o

%e .                          o   o        o     o     o

%e .               o        o       o    o     o   o     o

%e .             o   o    o     o     o   o     o o     o

%e .      o    o       o   o         o     o           o

%e .    o   o   o     o     o       o       o         o

%e . o   o o     o o o       o o o o         o o o o o

%e .

%e . 1    5        10          16                25

%t Table[Floor[5 n (n + 1)/6], {n, 0, 50}] (* _Arkadiusz Wesolowski_, Oct 03 2011 *)

%o (PARI) a(n)=5*n*(n+1)\6 \\ _Charles R Greathouse IV_, Aug 23 2011

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

%Y Cf. A000326, A008706, A008854, A032528, A152734, A193273, A193274.

%Y Cf. similar sequences with the formula floor(k*n*(n+1)/(k+1)) listed in A281026.

%K nonn,easy

%O 0,3

%A _Omar E. Pol_, Aug 20 2011

%E Name improved by _Arkadiusz Wesolowski_, Aug 23 2011

%E New name from _Omar E. Pol_, Sep 28 2011