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
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).
Programs
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Magma
[ (2*n^4-(1-(-1)^n))/32: n in [0..50] ];
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Maple
seq(round((n/2)^4), n=0..40);
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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 *) -
PARI
a(n)=round((n/2)^4) \\ Charles R Greathouse IV, Jun 23 2011
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
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.
Comments