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%I #110 Jan 16 2023 08:52:39
%S 0,1,6,13,24,37,54,73,96,121,150,181,216,253,294,337,384,433,486,541,
%T 600,661,726,793,864,937,1014,1093,1176,1261,1350,1441,1536,1633,1734,
%U 1837,1944,2053,2166,2281,2400,2521,2646,2773,2904,3037,3174,3313,3456,3601,3750
%N Concentric hexagonal numbers: floor(3*n^2/2).
%C From _Omar E. Pol_, Aug 20 2011: (Start)
%C Cellular automaton on the hexagonal net. The sequence gives the number of "ON" cells in the structure after n-th stage. A007310 gives the first differences. For a definition without words see the illustration of initial terms in the example section. Note that the cells become intermittent. A083577 gives the primes of this sequences.
%C A033581 and A003154 interleaved.
%C Row sums of an infinite square array T(n,k) in which column k lists 2*k-1 zeros followed by the numbers A008458 (see example). (End)
%C Sequence found by reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Main axis perpendicular to A045943 in the same spiral. - _Omar E. Pol_, Sep 08 2011
%H Vincenzo Librandi, <a href="/A032528/b032528.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.
%F From _Joerg Arndt_, Aug 22 2011: (Start)
%F G.f.: (x+4*x^2+x^3)/(1-2*x+2*x^3-x^4) = x*(1+4*x+x^2)/((1+x)*(1-x)^3).
%F a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End)
%F a(n) = (6*n^2+(-1)^n-1)/4. - _Bruno Berselli_, Aug 22 2011
%F a(n) = A184533(n), n >= 2. - _Clark Kimberling_, Apr 20 2012
%F First differences of A011934: a(n) = A011934(n) - A011934(n-1) for n>0. - _Franz Vrabec_, Feb 17 2013
%F From _Paul Curtz_, Mar 31 2019: (Start)
%F a(-n) = a(n).
%F a(n) = a(n-2) + 6*(n-1) for n > 1.
%F a(2*n) = A033581(n).
%F a(2*n+1) = A003154(n+1). (End)
%F E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - _Stefano Spezia_, Aug 19 2022
%F Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - _Amiram Eldar_, Jan 16 2023
%e From _Omar E. Pol_, Aug 20 2011: (Start)
%e Using the numbers A008458 we can write:
%e 0, 1, 6, 12, 18, 24, 30, 36, 42, 48, 54, ...
%e 0, 0, 0, 1, 6, 12, 18, 24, 30, 36, 42, ...
%e 0, 0, 0, 0, 0, 1, 6, 12, 18, 24, 30, ...
%e 0, 0, 0, 0, 0, 0, 0, 1, 6, 12, 18, ...
%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, ...
%e And so on.
%e ===========================================
%e The sums of the columns give this sequence:
%e 0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ...
%e ...
%e Illustration of initial terms as concentric hexagons:
%e .
%e . o o o o o
%e . o o o o o o
%e . o o o o o o o o o o
%e . o o o o 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 . o o o o o o o o o o o o
%e . o o o o o o o o o o
%e . o o o o o o
%e . o o o o o
%e .
%e . 1 6 13 24 37
%e .
%e (End)
%t f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* _Clark Kimberling_, Apr 20 2012 *)
%o (Magma) [Floor(3*n^2/2): n in [0..50]]; // _Vincenzo Librandi_, Aug 21 2011
%o (Haskell)
%o a032528 n = a032528_list !! n
%o a032528_list = scanl (+) 0 a007310_list
%o -- _Reinhard Zumkeller_, Jan 07 2012
%o (PARI) a(n)=3*n^2\2 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040.
%Y Cf. A004524, A004525, A033436, A212959, A238410, A227017, A282513.
%Y Cf. A033581, A003154, A011934, A184533.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E New name and more terms a(41)-a(50) from _Omar E. Pol_, Aug 20 2011
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