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A011863 Nearest integer to (n/2)^4.

%I A011863 #101 Nov 03 2023 22:34:45
%S A011863 0,0,1,5,16,39,81,150,256,410,625,915,1296,1785,2401,3164,4096,5220,
%T A011863 6561,8145,10000,12155,14641,17490,20736,24414,28561,33215,38416,
%U A011863 44205,50625,57720,65536,74120,83521,93789,104976,117135,130321,144590
%N A011863 Nearest integer to (n/2)^4.
%C A011863 First differences are in A019298.
%C A011863 The bisections are A000583 and A219086.
%C A011863 Number of ways to put n-1 copies of 1,2,3 into sets. [Zeilberger?]
%C A011863 s(n) is the number of 4-tuples (w,x,y,z) with all terms in {1,...,n} and |w-x| >= w + |y-z|; see A186707. - _Clark Kimberling_, May 24 2012
%H A011863 N. J. A. Sloane, <a href="/A011863/b011863.txt">Table of n, a(n) for n = 0..10000</a> (first 2000 terms from Vicenzo Librandi)
%H A011863 A. J. Guttmann, <a href="https://doi.org/10.1016/S0012-365X(99)00262-9">Indicators of solvability for lattice models</a>, Discrete Math., 217 (2000), 167-189 (H_2 for square lattice of Section 6).
%H A011863 Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/babushkas.html">In How Many Ways Can You Reassemble Several Russian Dolls?</a>, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger (2009); <a href="/A011863/a011863.pdf">Local copy</a>. [PDF file only, no active links]
%H A011863 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F A011863 G.f.: x^2*(1 + x + x^2)/((1 - x)^5*(1+x)).
%F A011863 a(n) = +4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6). - _R. J. Mathar_, Dec 07 2010
%F A011863 a(n)+a(n+1) = A002817(n). - _R. J. Mathar_, Dec 19 2008
%F A011863 a(n) = n^4/16 - 1/32 + (-1)^n/32 - _R. J. Mathar_, Dec 07 2010, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019
%F A011863 a(n) = (2*A000583(n) + (-1)^n - 1)/32. - _Bruno Berselli_, Dec 07 2010, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019
%F A011863 n*(n^2+n+2)*a(n+1) = 4*(n^2+2*n+2)*a(n)+(n+2)*(n^2+3*n+4)*a(n-1). Holonomic Ansatz with smallest order of recurrence. - _Thotsaporn Thanatipanonda_, Dec 12 2010
%F A011863 a(n) = floor(n^4/8)/2. - _Gary Detlefs_, Feb 19 2011, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019
%F A011863 a(n) = A212714(n)/2, n >= 0. - _Wolfdieter Lang_, Oct 03 2016, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019
%F A011863 E.g.f.: (1/32)*exp(-x)*(1 + exp(2*x)*(-1 + 2*x + 14*x^2 + 12*x^3 + 2*x^4)). - _Stefano Spezia_, Dec 29 2019
%F A011863 Sum_{n>=2} 1/a(n) = 6 + Pi^4/90 - 2*Pi*tanh(Pi/2). - _Amiram Eldar_, Aug 13 2022
%p A011863 seq(round((n/2)^4), n=0..40);
%t A011863 Round[(Range[40]/2)^4] (* or *) LinearRecurrence[{4,-5,0,5,-4,1},{0,1,5,16,39,81},40] (* _Harvey P. Dale_, Feb 07 2015 *)
%o A011863 (Magma) [ (2*n^4-(1-(-1)^n))/32: n in [0..50] ];
%o A011863 (PARI) a(n)=round((n/2)^4) \\ _Charles R Greathouse IV_, Jun 23 2011
%Y A011863 Cf. A000583, A002817, A019298, A106707, A212714, A219086.
%K A011863 nonn,easy
%O A011863 0,4
%A A011863 _R. K. Guy_
%E A011863 Missing a(0) added by _N. J. A. Sloane_, Dec 29 2019. As a result some of the comments and formulas will need to be adjusted.