cp's OEIS Frontend

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

%I A194275 #60 Jun 29 2025 21:23:54
%S A194275 0,1,5,10,16,25,35,46,60,75,91,110,130,151,175,200,226,255,285,316,
%T A194275 350,385,421,460,500,541,585,630,676,725,775,826,880,935,991,1050,
%U A194275 1110,1171,1235,1300,1366,1435,1505,1576,1650,1725,1801,1880,1960,2041,2125
%N A194275 Concentric pentagonal numbers of the second kind: a(n) = floor(5*n*(n+1)/6).
%C A194275 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 A194275 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 A194275 Also partial sums of A008854.
%C A194275 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 A194275 For concentric pentagonal numbers see A032527. - _Omar E. Pol_, Sep 27 2011
%H A194275 Vincenzo Librandi, <a href="/A194275/b194275.txt">Table of n, a(n) for n = 0..10000</a>
%H A194275 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F A194275 G.f.: (-1 - 3*x - x^2)/((-1 + x)^3*(1 + x + x^2)). - _Alexander R. Povolotsky_, Aug 22 2011
%F A194275 a(n) = floor(5*n*(n+1)/6). - _Arkadiusz Wesolowski_, Aug 23 2011
%e A194275 Using the numbers A008706 we can write:
%e A194275 0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
%e A194275 0, 0, 0,  0,  1,  5, 10, 15, 20, 25, 30, ...
%e A194275 0, 0, 0,  0,  0,  0,  0,  1,  5, 10, 15, ...
%e A194275 0, 0, 0,  0,  0,  0,  0,  0,  0,  0,  1, ...
%e A194275 And so on.
%e A194275 ===========================================
%e A194275 The sums of the columns give this sequence:
%e A194275 0, 1, 5, 10, 16, 25, 35, 46, 60, 75, 91, ...
%e A194275 ...
%e A194275 Illustration of initial terms (in a precise representation the pentagons should appear strictly concentric):
%e A194275 .                                             o
%e A194275 .                                           o   o
%e A194275 .                            o            o       o
%e A194275 .                          o   o        o     o     o
%e A194275 .               o        o       o    o     o   o     o
%e A194275 .             o   o    o     o     o   o     o o     o
%e A194275 .      o    o       o   o         o     o           o
%e A194275 .    o   o   o     o     o       o       o         o
%e A194275 . o   o o     o o o       o o o o         o o o o o
%e A194275 .
%e A194275 . 1    5        10          16                25
%t A194275 Table[Floor[5 n (n + 1)/6], {n, 0, 50}] (* _Arkadiusz Wesolowski_, Oct 03 2011 *)
%o A194275 (PARI) a(n)=5*n*(n+1)\6 \\ _Charles R Greathouse IV_, Aug 23 2011
%o A194275 (Magma) [Floor(5*n*(n+1)/6): n in [0..60]]; // Vincenzo Librandi, Sep 27 2011
%Y A194275 Cf. A000326, A008706, A008854, A032528, A152734, A193273, A193274.
%Y A194275 Cf. similar sequences with the formula floor(k*n*(n+1)/(k+1)) listed in A281026.
%K A194275 nonn,easy
%O A194275 0,3
%A A194275 _Omar E. Pol_, Aug 20 2011
%E A194275 Name improved by _Arkadiusz Wesolowski_, Aug 23 2011
%E A194275 New name from _Omar E. Pol_, Sep 28 2011