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 A032528 Concentric hexagonal numbers: floor(3*n^2/2). 45
 0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, 181, 216, 253, 294, 337, 384, 433, 486, 541, 600, 661, 726, 793, 864, 937, 1014, 1093, 1176, 1261, 1350, 1441, 1536, 1633, 1734, 1837, 1944, 2053, 2166, 2281, 2400, 2521, 2646, 2773, 2904, 3037, 3174, 3313, 3456, 3601, 3750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS From Omar E. Pol, Aug 20 2011: (Start) 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. A033581 and A003154 interleaved. 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) 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1). Index entries for sequences related to cellular automata. FORMULA From Joerg Arndt, Aug 22 2011: (Start) 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). a(n) = +2*a(n-1) -2*a(n-3) +1*a(n-4). (End) a(n) = (6*n^2+(-1)^n-1)/4. - Bruno Berselli, Aug 22 2011 a(n) = A184533(n), n >= 2. - Clark Kimberling, Apr 20 2012 First differences of A011934: a(n) = A011934(n) - A011934(n-1) for n>0. - Franz Vrabec, Feb 17 2013 From Paul Curtz, Mar 31 2019: (Start) a(-n) = a(n). a(n) = a(n-2) + 6*(n-1) for n > 1. a(2*n) = A033581(n). a(2*n+1) = A003154(n+1). (End) E.g.f.: (3*x*(x + 1)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/2. - Stefano Spezia, Aug 19 2022 Sum_{n>=1} 1/a(n) = Pi^2/36 + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Jan 16 2023 EXAMPLE From Omar E. Pol, Aug 20 2011: (Start) Using the numbers A008458 we can write: 0, 1, 6, 12, 18, 24, 30, 36, 42, 48, 54, ... 0, 0, 0, 1, 6, 12, 18, 24, 30, 36, 42, ... 0, 0, 0, 0, 0, 1, 6, 12, 18, 24, 30, ... 0, 0, 0, 0, 0, 0, 0, 1, 6, 12, 18, ... 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, ... And so on. =========================================== The sums of the columns give this sequence: 0, 1, 6, 13, 24, 37, 54, 73, 96, 121, 150, ... ... Illustration of initial terms as concentric hexagons: . . 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 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 6 13 24 37 . (End) MATHEMATICA f[n_, m_] := Sum[Floor[n^2/k], {k, 1, m}]; t = Table[f[n, 2], {n, 1, 90}] (* Clark Kimberling, Apr 20 2012 *) PROG (Magma) [Floor(3*n^2/2): n in [0..50]]; // Vincenzo Librandi, Aug 21 2011 (Haskell) a032528 n = a032528_list !! n a032528_list = scanl (+) 0 a007310_list -- Reinhard Zumkeller, Jan 07 2012 (PARI) a(n)=3*n^2\2 \\ Charles R Greathouse IV, Sep 24 2015 CROSSREFS Cf. A003154, A007310, A008458, A033581, A083577, A000326, A001318, A005449, A045943, A032527, A195041. Column 6 of A195040. Cf. A004524, A004525, A033436, A212959, A238410, A227017, A282513. Cf. A033581, A003154, A011934, A184533. Sequence in context: A194126 A296310 A235450 * A058535 A131833 A101736 Adjacent sequences: A032525 A032526 A032527 * A032529 A032530 A032531 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS New name and more terms a(41)-a(50) from Omar E. Pol, Aug 20 2011 STATUS approved

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Last modified September 24 17:02 EDT 2023. Contains 365579 sequences. (Running on oeis4.)