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Nearest integer to (n/2)^4.
15

%I #101 Nov 03 2023 22:34:45

%S 0,0,1,5,16,39,81,150,256,410,625,915,1296,1785,2401,3164,4096,5220,

%T 6561,8145,10000,12155,14641,17490,20736,24414,28561,33215,38416,

%U 44205,50625,57720,65536,74120,83521,93789,104976,117135,130321,144590

%N Nearest integer to (n/2)^4.

%C First differences are in A019298.

%C The bisections are A000583 and A219086.

%C Number of ways to put n-1 copies of 1,2,3 into sets. [Zeilberger?]

%C 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 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 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 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 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).

%F G.f.: x^2*(1 + x + x^2)/((1 - x)^5*(1+x)).

%F 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 a(n)+a(n+1) = A002817(n). - _R. J. Mathar_, Dec 19 2008

%F 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 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 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 a(n) = floor(n^4/8)/2. - _Gary Detlefs_, Feb 19 2011, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019

%F a(n) = A212714(n)/2, n >= 0. - _Wolfdieter Lang_, Oct 03 2016, adapted to added a(0) by _Hugo Pfoertner_, Dec 29 2019

%F 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 Sum_{n>=2} 1/a(n) = 6 + Pi^4/90 - 2*Pi*tanh(Pi/2). - _Amiram Eldar_, Aug 13 2022

%p seq(round((n/2)^4), n=0..40);

%t 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 (Magma) [ (2*n^4-(1-(-1)^n))/32: n in [0..50] ];

%o (PARI) a(n)=round((n/2)^4) \\ _Charles R Greathouse IV_, Jun 23 2011

%Y Cf. A000583, A002817, A019298, A106707, A212714, A219086.

%K nonn,easy

%O 0,4

%A _R. K. Guy_

%E 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.