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 A104249 a(n) = (3*n^2+n+2)/2. 11

%I

%S 1,3,8,16,27,41,58,78,101,127,156,188,223,261,302,346,393,443,496,552,

%T 611,673,738,806,877,951,1028,1108,1191,1277,1366,1458,1553,1651,1752,

%U 1856,1963,2073,2186,2302,2421,2543,2668,2796,2927,3061,3198,3338,3481

%N a(n) = (3*n^2+n+2)/2.

%C Second differences are all 3.

%C Related to the sequence of odd numbers A005408 since for these numbers the first differences are all 2.

%C Column 2 of A114202. - _Paul Barry_, Nov 17 2005

%C Equals third row of A167560 divided by 2. - _Johannes W. Meijer_, Nov 12 2009

%C A242357(a(n)) = n + 1. - _Reinhard Zumkeller_, May 11 2014

%C Also, this sequence is related to A011379, for n>0, by A011379(n) = n*a(n) - Sum_{i=0..n-1} a(i). - _Bruno Berselli_, Jul 08 2015

%H Vincenzo Librandi, <a href="/A104249/b104249.txt">Table of n, a(n) for n = 0..3000</a>

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: (1+2*x^2)/(1-x)^3.

%F Recurrence: {u(1) = 3, u(2) = 8, (n+3)*u(n+3)+(-5-n)*u(n+2)*(-2+2*n)*u(n+1) +(-2-2*n)*u(n), u(0) = 1}.

%F a(0)=1, a(n) = a(n-1)+3n-1, n>0; a(n) = Sum_{k=0..n} C(n, k)C(2, k)J(k+1), J(n) = A001045(n). - _Paul Barry_, Nov 17 2005

%F Binomial transform of [1,2,3,0,...]. - _Gary W. Adamson_, Apr 23 2008

%e The sequence of first differences delta_a(n) = a(n+1) - a(n) is: 2,5,8,11,14,17,20,23,26,...

%e The sequence of second differences delta_delta_a(n) = a(n+2) - 2*a(n+1) + a(n) is: 3,3,3,3,3,3,3,3,3,... E.g. 78 - 2*58 + 41 = 3.

%p a := proc (n) local i, u; option remember; u[0] := 1; u[1] := 3; u[2] := 8; for i from 3 to n do u[i] := -(4*u[i-3]-8*u[i-2]-2*u[i-1]+(-2*u[i-3]+2*u[i-2]-u[i-1])*i)/i end do; [seq(u[i],i = 0 .. n)] end proc;

%t A104249[n_] := (3*n^2 + n + 2)/2; Table[A104249[n], {n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)

%o (MAGMA) [(3*n^2+n+2)/2: n in [0..50]]; // _Vincenzo Librandi_, May 09 2011

%o a104249 n = n*(3*n+1) `div` 2 + 1 -- _Reinhard Zumkeller_, May 11 2014

%o (PARI) a(n)=n*(3*n+1)/2+1 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A001399, A002597, A005408, A011379, A016777, A143689.

%K nonn,easy

%O 0,2

%A _Thomas Wieder_, Feb 26 2005

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