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.

Showing 1-3 of 3 results.

A098077 a(n) = n^2*(n+1)*(2*n+1)/3.

Original entry on oeis.org

2, 20, 84, 240, 550, 1092, 1960, 3264, 5130, 7700, 11132, 15600, 21294, 28420, 37200, 47872, 60690, 75924, 93860, 114800, 139062, 166980, 198904, 235200, 276250, 322452, 374220, 431984, 496190, 567300, 645792, 732160, 826914, 930580, 1043700
Offset: 1

Views

Author

Alexander Adamchuk, Oct 24 2004

Keywords

Comments

Sum of all matrix elements M(i,j) = i^2 + j^2 (i,j = 1,...,n).
From Torlach Rush, Jan 05 2020: (Start)
a(n) = n * A006331(n).
tr(M(n)) = A006331(n).
The sum of the antidiagonal of M(n) equals tr(M(n)).
M(n) = M(n)' (Symmetric).
M(1,) = M(,1) = A002522(n), n > 0.
M(2,) = M(,2) = A087475(n), n > 0.
M(3,) = M(,3) = A189834(n), n > 0.
M(4,) = M(,4) = A241751(n), n > 0.
(End)
Consider the partitions of 2n into two parts (p,q) where p <= q. Then a(n) is the total volume of the family of rectangular prisms with dimensions p, p and p+q. - Wesley Ivan Hurt, Apr 15 2018

Examples

			a(2) = (1^2 + 1^2) + (1^2 + 2^2) + (2^2 + 1^2) + (2^2 + 2^2) = 2 + 5 + 5 + 8 = 20.

Links

Crossrefs

Cf. A002522, A006331, A087475, A100431, A108678, A189834, A241751.

Programs

  • Magma
    [n^2*(n+1)*(2*n+1)/3: n in [1..40]]; // G. C. Greubel, Apr 09 2023
  • Mathematica
    Table[ Sum[i^2 + j^2, {i, n}, {j, n}], {n, 35}]
LinearRecurrence[{5, -10, 10, -5, 1}, {2, 20, 84, 240, 550}, 40] (* Vincenzo Librandi, Apr 16 2018 *)
  • PARI
    a(n)=n^2*(n+1)*(2*n+1)/3 \\ Charles R Greathouse IV, Oct 07 2015
  • SageMath
    [n^2*(n+1)*(2*n+1)/3 for n in range(1,41)] # G. C. Greubel, Apr 09 2023

Formula

a(n) = Sum_{j=1..n} Sum_{i=1..n} (i^2 + j^2).
G.f.: 2*x*(1 + 5*x + 2*x^2)/(1-x)^5. - Colin Barker, May 04 2012
E.g.f.: (1/3)*exp(x)*x*(6 + 24*x + 15*x^2 + 2*x^3) . - Stefano Spezia, Jan 06 2020
a(n) = a(n-1) + (8*n^3 - 3*n^2 + n)/3. - Torlach Rush, Jan 07 2020
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/2 + 24*log(2) - 21.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/4 - 6*Pi - 6*log(2) + 21. (End)
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (1/4)*A100431(n-1).
a(n) = 2*A108678(n-1). (End)

Extensions

More terms from Robert G. Wilson v, Nov 01 2004
New definition from Ralf Stephan, Dec 01 2004

A108678 a(n) = (n+1)^2*(n+2)*(2*n+3)/6.

Original entry on oeis.org

1, 10, 42, 120, 275, 546, 980, 1632, 2565, 3850, 5566, 7800, 10647, 14210, 18600, 23936, 30345, 37962, 46930, 57400, 69531, 83490, 99452, 117600, 138125, 161226, 187110, 215992, 248095, 283650, 322896, 366080, 413457, 465290, 521850, 583416, 650275, 722722
Offset: 0

Views

Author

Emeric Deutsch, Jun 17 2005

Keywords

Comments

Kekulé numbers for certain benzenoids.

References

  • S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 232, # 44).

Links

Crossrefs

Cf. A000217, A014105, A098077, A100431.

Programs

  • Magma
    [(n+1)^2*(n+2)*(2*n+3)/6: n in [0..60]]; // G. C. Greubel, Apr 09 2023
  • Maple
    a:=n->(n+1)^2*(n+2)*(2*n+3)/6: seq(a(n),n=0..42);
a:=n->sum(n*j^2, j=1..n): seq(a(n), n=1..36); # Zerinvary Lajos, Apr 29 2007
  • Mathematica
    Table[(n+1)^2*(n+2)(2n+3)/6,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
  • SageMath
    [(n+1)^2*(n+2)*(2*n+3)/6 for n in range(61)] # G. C. Greubel, Apr 09 2023

Formula

G.f.: (1 + 5*x + 2*x^2)/(1-x)^5.
a(n) = A098077(n+1)/2. - Alexander Adamchuk, Apr 12 2006
From Amiram Eldar, May 31 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi^2 + 48*log(2) - 42.
Sum_{n>=0} (-1)^n/a(n) = Pi^2/2 - 12*Pi - 12*log(2) + 42. (End)
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (1/3)*binomial(n+2, 2)*binomial(2*n+3, 2).
a(n) = (1/3)*A000217(n+1)*A014105(n+1)
a(n) = (1/8)*A100431(n).
E.g.f.: (1/6)*(6 + 54*x + 69*x^2 + 23*x^3 + 2*x^4)*exp(x). (End)
a(n) = (n+1)*A000330(n+1). - Olivier Gérard, Jan 13 2024

A100430 Bisection of A002417.

Original entry on oeis.org

1, 30, 175, 588, 1485, 3146, 5915, 10200, 16473, 25270, 37191, 52900, 73125, 98658, 130355, 169136, 215985, 271950, 338143, 415740, 505981, 610170, 729675, 865928, 1020425, 1194726, 1390455, 1609300, 1853013, 2123410, 2422371, 2751840
Offset: 0

Views

Author

N. J. A. Sloane, Nov 20 2004

Keywords

Links

Crossrefs

Cf. A000447, A002417, A100431.

Programs

  • Magma
    [(2*n+1)*Binomial(2*n+3,3): n in [0..50]]; // G. C. Greubel, Apr 09 2023
  • Mathematica
    Table[(2*n+1)*Binomial[2*n+3,3], {n,0,50}] (* G. C. Greubel, Apr 09 2023 *)
  • SageMath
    [(2*n+1)*binomial(2*n+3,3) for n in range(51)] # G. C. Greubel, Apr 09 2023

Formula

a(n) = (8*n^4 +28*n^3 +34*n^2 +17*n+3)/3. - Ralf Stephan, May 15 2007
From G. C. Greubel, Apr 09 2023: (Start)
a(n) = (2*n+1)*binomial(2*n+3, 3).
a(n) = (2*n+1)*A000447(n+1).
G.f.: (1 + 25*x + 35*x^2 + 3*x^3)/(1-x)^5.
E.g.f.: (1/3)*(3 + 87*x + 174*x^2 + 76*x^3 + 8*x^4)*exp(x). (End)

Extensions

More terms from Hugo Pfoertner, Nov 26 2004
Showing 1-3 of 3 results.

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