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A011863 Nearest integer to (n/2)^4. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of ways to put (n-1) copies of 1,2,3 into sets.

First differences are in A019298.

Second bisection preceded by zero is A000583.

Let s(0)=0 and s(n)=a(n-1) for n>0.  Then 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

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

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).

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

FORMULA

G.f.: H_2(x) = x*(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).

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

a(n) = n^4/16+n^3/4+3*n^2/8+n/4+1/32-(-1)^n/32. - R. J. Mathar, Dec 07 2010]

a(n) = (2*A000583(n+1)-(1+(-1)^n))/32. - Bruno Berselli, Dec 07 2010

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+1)^4/8)/2. - Gary Detlefs, Feb 19 2011

a(n) = A212714(n+1)/2, n >= 0. - Wolfdieter Lang, Oct 03 2016

MAPLE

seq(round((n/2)^4), n=1..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+1)^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, A212714.

Sequence in context: A155965 A216173 A269747 * A027085 A099452 A006007

Adjacent sequences:  A011860 A011861 A011862 * A011864 A011865 A011866

KEYWORD

nonn,easy

AUTHOR

R. K. Guy

STATUS

approved

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Last modified August 20 14:06 EDT 2019. Contains 326152 sequences. (Running on oeis4.)