cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A331987 a(n) = ((n + 1) - 9*(n + 1)^2 + 8*(n + 1)^3)/6.

Original entry on oeis.org

0, 5, 23, 62, 130, 235, 385, 588, 852, 1185, 1595, 2090, 2678, 3367, 4165, 5080, 6120, 7293, 8607, 10070, 11690, 13475, 15433, 17572, 19900, 22425, 25155, 28098, 31262, 34655, 38285, 42160, 46288, 50677, 55335, 60270, 65490, 71003, 76817, 82940, 89380, 96145
Offset: 0

Views

Author

Peter Luschny, Feb 19 2020

Keywords

Comments

The start values of the partial rows on the main diagonal of A332662 in the representation in the example section.
Apparently the sum of the hook lengths over the partitions of 2*n + 1 with exactly 2 parts (cf. A180681).

Links

Crossrefs

Apparently a bisection of A049779 and of A024862.
Cf. A180681, A332662.

Programs

  • Magma
    [n*(n+1)*(8*n+7)/6: n in [0..50]]; // G. C. Greubel, Apr 19 2023
  • Maple
    a := n -> ((n+1) - 9*(n+1)^2 + 8*(n+1)^3)/6: seq(a(n), n=0..41);
gf := (x*(3*x + 5))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);
  • Mathematica
    LinearRecurrence[{4,-6,4,-1}, {0,5,23,62}, 42]
Table[(n-9n^2+8n^3)/6,{n,50}] (* Harvey P. Dale, Apr 11 2024 *)
  • SageMath
    def A331987(n): return n*(n+1)*(8*n+7)/6
[A331987(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

Formula

a(n) = [x^n] (x*(5 + 3*x)/(1 - x)^4).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2).
From G. C. Greubel, Apr 19 2023: (Start)
a(n) = 3*binomial(n+1,1) - 11*binomial(n+2,2) + 8*binomial(n+3,3).
a(n) = n*binomial(8*n+8,2)/24.
a(n) = n*(n+1)*(8*n+7)/6.
E.g.f.: (1/6)*x*(30 + 39*x + 8*x^2)*exp(x). (End)

A285188 a(n) = Sum_{k=1..n} (k^2*floor(k/2)).

Original entry on oeis.org

0, 4, 13, 45, 95, 203, 350, 606, 930, 1430, 2035, 2899, 3913, 5285, 6860, 8908, 11220, 14136, 17385, 21385, 25795, 31119, 36938, 43850, 51350, 60138, 69615, 80591, 92365, 105865, 120280, 136664, 154088, 173740, 194565, 217893, 242535, 269971
Offset: 1

Views

Author

Néstor Jofré, Apr 24 2017

Keywords

Examples

			For n = 4, a(4) = 1^2*floor(1/2)  + 2^2*floor(2/2) + 3^2*floor(3/2) + 4^2*floor(4/2) =  0 + 4 + 9 + 32 = 45.

Links

Crossrefs

Cf. A049779.
Partial sums of A265645.

Programs

  • MATLAB
    s = @(n) sum((1:n).^2.*floor((1:n)/2)); %summation handle function
         s_cf = @(n) 1/8*n^2*(n+1)^2 - 2/3*floor((n+1)/2)^3 + 1/6*floor((n+1)/2); %faster closed-form handle function
  • Maple
    seq( n*(n+1)*(3*n^2+n-1+3*(-1)^n)/24, n=1..100); # Robert Israel, Apr 26 2017
  • PARI
    a(n) = sum(k=1, n, k^2*(k\2)); \\ Michel Marcus, Apr 24 2017

Formula

Theorem: a(n) = (1/8)*n^2*(n+1)^2 - (2/3)*floor((n+1)/2)^3 + (1/6)*floor((n+1)/2).
From Chai Wah Wu, Apr 24 2017: (Start)
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 8.
G.f.: x^2*(x^4 + 3*x^3 + 11*x^2 + 5*x + 4)/((1 - x)^5*(1 + x)^3). (End)
a(n) = n*(n+1)*(3*n^2+n-1+3*(-1)^n)/24. - Robert Israel, Apr 26 2017
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