%I #77 Mar 16 2024 17:37:13
%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
%C The number of Hamiltonian nonisomorphic unfoldings in an n-gonal Archimedean antiprism. See sequence A284647. - _Rick Mabry_, Apr 10 2021
%H Vincenzo Librandi, <a href="/A104249/b104249.txt">Table of n, a(n) for n = 0..3000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H Rick Mabry, <a href="https://doi.org/10.1080/00029890.2019.1644124">Fibonacci Numbers, Integer Compositions, and Nets of Antiprisms</a>, The American Mathematical Monthly, Vol. 126 (2019), no. 9, pp. 786-801.
%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: (n+3)*u(n+3) + (-5-n)*u(n+2)*(-2+2*n)*u(n+1) + (-2-2*n)*u(n) = 0 for n >= 0 with u(0) = 1, u(1) = 3, and u(2) = 8.
%F From _Paul Barry_, Nov 17 2005: (Start)
%F a(0) = 1, a(n) = a(n-1) + 3*n - 1 for n > 0;
%F a(n) = Sum_{k=0..n} C(n, k)*C(2, k)*J(k+1), where J(n) = A001045(n). (End)
%F Binomial transform of [1, 2, 3, 0, 0, 0, ...]. - _Gary W. Adamson_, Apr 23 2008
%F E.g.f.: exp(x)*(2 + 4*x + 3*x^2)/2. - _Stefano Spezia_, Apr 10 2021
%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, ... 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 *)
%t LinearRecurrence[{3,-3,1},{1,3,8},70] (* _Harvey P. Dale_, Jul 21 2023 *)
%o (Magma) [(3*n^2+n+2)/2: n in [0..50]]; // _Vincenzo Librandi_, May 09 2011
%o (Haskell)
%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.
%Y Counts special cases of A284647.
%K nonn,easy
%O 0,2
%A _Thomas Wieder_, Feb 26 2005