A131478 a(n) = ceiling(n^4/4).
0, 1, 4, 21, 64, 157, 324, 601, 1024, 1641, 2500, 3661, 5184, 7141, 9604, 12657, 16384, 20881, 26244, 32581, 40000, 48621, 58564, 69961, 82944, 97657, 114244, 132861, 153664, 176821, 202500, 230881, 262144, 296481, 334084, 375157, 419904, 468541, 521284
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
Programs
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Magma
[Ceiling(n^4/4) : n in [0..50]]; // Vincenzo Librandi, Oct 01 2011
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Mathematica
Ceiling[Range[0,40]^4/4] (* Harvey P. Dale, May 17 2019 *) CoefficientList[Series[(x(x^3 + 6x^2 + 7x + 1)Cosh[x]+ (x^4 + 6x^3 + 7x^2 + x + 3)Sinh[x])/4,{x,0,35}],x]Table[n!,{n,0,35}] (* Stefano Spezia, Feb 19 2023 *)
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PARI
vector(50, n, n--;ceil(n^4/4)) \\ Michel Marcus, Jun 16 2015
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Python
def A131478(n): return n**4+3>>2 # Chai Wah Wu, Jan 30 2023
Formula
From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x*(1 + 10*x^2 + x^4)/((1 - x)^5*(1 + x)).
a(n) + a(n+1) = A058919(n+1). (End)
a(n) = floor(n^4/4 + 3/4). - Bruno Berselli, Dec 21 2017
E.g.f.: (x*(x^3 + 6*x^2 + 7*x + 1)*cosh(x) + (x^4 + 6*x^3 + 7*x^2 + x + 3)*sinh(x))/4. - Stefano Spezia, Feb 18 2023