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

0, 0, 1, 5, 16, 39, 81, 150, 256, 410, 625, 915, 1296, 1785, 2401, 3164, 4096, 5220, 6561, 8145, 10000, 12155, 14641, 17490, 20736, 24414, 28561, 33215, 38416, 44205, 50625, 57720, 65536, 74120, 83521, 93789, 104976, 117135, 130321, 144590

Offset

0,4

Comments

First differences are in A019298.

The bisections are A000583 and A219086.

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

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

Links

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 2000 terms from Vicenzo Librandi)

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189 (H_2 for square lattice of Section 6).

Doron Zeilberger, In How Many Ways Can You Reassemble Several Russian Dolls?, The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger (2009); Local copy. [PDF file only, no active links]

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Formula

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

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

a(n)+a(n+1) = A002817(n). - R. J. Mathar, Dec 19 2008

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

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

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

a(n) = floor(n^4/8)/2. - Gary Detlefs, Feb 19 2011, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019

a(n) = A212714(n)/2, n >= 0. - Wolfdieter Lang, Oct 03 2016, adapted to added a(0) by Hugo Pfoertner, Dec 29 2019

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

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

Maple

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

Mathematica

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 *)

Prog

(Magma) [ (2*n^4-(1-(-1)^n))/32: n in [0..50] ];
(PARI) a(n)=round((n/2)^4) \\ Charles R Greathouse IV, Jun 23 2011

Crossrefs

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

Sequence in context: A155965 A216173 A269747 * A027085 A099452 A006007

Adjacent sequences: A011860 A011861 A011862 * A011864 A011865 A011866

Keyword

nonn,easy

Author

R. K. Guy

Extensions

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.

Status

approved