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%I #71 Dec 22 2023 12:11:25
%S 1,2,6,15,31,56,92,141,205,286,386,507,651,820,1016,1241,1497,1786,
%T 2110,2471,2871,3312,3796,4325,4901,5526,6202,6931,7715,8556,9456,
%U 10417,11441,12530,13686,14911,16207,17576,19020,20541,22141,23822
%N a(n) = (n + 2)*(2*n^2 - n + 3)/6.
%C Hankel transform of A030238. - _Paul Barry_, Oct 16 2007
%C Equals (1, 2, 3, 4, 5, ...) convolved with (1, 0, 3, 5, 7, 9, ...). - _Gary W. Adamson_, Jul 31 2010
%C a(n) equals n!^2 times the determinant of the n X n matrix whose (i,j)-entry is 1 + KroneckerDelta[i, j] (-1 + (1 + i^2)/i^2). - _John M. Campbell_, May 20 2011
%C Positions of ones in A253903 (with offset 1). - _Harvey P. Dale_, Mar 05 2015
%H Vincenzo Librandi, <a href="/A056520/b056520.txt">Table of n, a(n) for n = 0..1000</a>
%H Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, <a href="https://arxiv.org/abs/2312.05145">Cyclic permutations avoiding patterns in both one-line and cycle forms</a>, arXiv:2312.05145 [math.CO], 2023. See p. 2.
%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 Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, <a href="http://www.ece.nus.edu.sg/stfpage/eleak/pdf/akumar_todaes_2012.pdf">Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs</a>, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 25 2012
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = a(n-1) + n^2.
%F a(n) = A000330(n) + 1.
%F G.f.: (1 - 2*x + 4*x^2 - x^3)/(1 - x)^4. - _Paul Barry_, Apr 14 2010
%F Let b(0) = b(1) = 1, b(n) = max(b(n-1) + (n - 1)^2, b(n-2) + (n - 2)^2) for n >= 2; then a(n) = b(n+1). - _Yalcin Aktar_, Jul 28 2011
%t a[n_] := (n+2)*(2*n^2-n+3)/6; Table[a[n], {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 17 2008 *)
%t s = 1; lst = {s}; Do[s += n^2; AppendTo[lst, s], {n, 1, 41, 1}]; lst (* _Zerinvary Lajos_, Jul 12 2009 *)
%t Table[n!^2*Det[Array[KroneckerDelta[#1,#2](((#1^2+1)/(#1^2))-1)+1&,{n,n}]],{n,1,20}] (* _John M. Campbell_, May 20 2011 *)
%t FoldList[#1 + #2^2 &, 1, Range@ 40] (* _Robert G. Wilson v_, Oct 28 2011 *)
%o (Magma) [(n+2)*(2*n^2-n+3)/6: n in [0..40]]; // _Vincenzo Librandi_, May 24 2011
%o (PARI) a(n)=(n+2)*(2*n^2-n+3)/6 \\ _Charles R Greathouse IV_, Jul 02 2013
%Y Cf. A000330, A153056, A153057, A153058, A179904.
%Y Cf. A030238, A253903.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Laura Kasavan (maui12129(AT)cswebmail.com), Aug 26 2000
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