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 1-4 of 4 results.

A217873 a(n) = 4*n*(n^2 + 2)/3.

Original entry on oeis.org

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004
Offset: 0

Views

Author

M. F. Hasler, Oct 13 2012

Keywords

Comments

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

Links

Crossrefs

Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

Programs

  • Magma
    [4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012
  • Mathematica
    Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)
  • Maxima
    makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */
  • PARI
    a(n)=(n^2+2)*n/3*4

Formula

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138. (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.

Original entry on oeis.org

3, 5, 7, 10, 12, 15, 16, 18, 19, 23, 25, 26, 27, 28, 29, 30, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 55, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 95, 96, 97, 103, 104, 105, 106, 107, 109, 115, 119, 122, 123, 124, 125, 126
Offset: 1

Views

Author

James R. Buddenhagen, Feb 06 2012

Keywords

Comments

n>3 is in the sequence just in case the elliptic curve y^2 = 4*x^4 + (n^2+8)*x^2 + 4 has positive rank. Note that (0,2) is on that curve.
n is in the sequence just in case there are positive rational numbers x,y such that x*y>1 and x - 1/x + y - 1/y = n.
The triangle whose sides are [(4*k^6+8*k^5+8*k^4+4*k^3+2*k^2+2*k+1)/((k+1)*k*(2*k^2+2*k+1)), (4*k^6+16*k^5+28*k^4+28*k^3+18*k^2+6*k+1)/((k+1)*k*(2*k^2+2*k+1)), 4*k^2+4*k+4] has area equal to its third side. Hence, starting with the second term, A112087 is a subsequence of the present sequence.
The triangle whose sides are [(k^6+2*k^4+k^2+1)/(k*(k^2+1)), (k^4+3*k^2+1)/(k*(k^2+1)), (k^2+2)*k] has area equal to its third side. Hence, starting with the first positive term, A054602 is a subsequence of the present sequence. [This subsequence found by Dragan K, see second link, below.]
The triangle whose sides are [(k^8+6*k^6+13*k^4+13*k^2+4)/(k*(k^2+2)*(k^2+1)), (k^6+3*k^4+5*k^2+4)/(k*(k^2+2)*(k^2+1)), k*(k^2+4)] has area equal to its third side. Hence A155965 is a subsequence of the present sequence.

Examples

			5 is in the sequence because the triangle with sides (37/6, 13/6, 5) has area 5, one side 5, and the other two sides rational.

Links

Crossrefs

Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).

A280579 Square array read by antidiagonals downwards giving the first differences A261327(n+p) - A261327(n), with p >= 0.

Original entry on oeis.org

0, 0, 4, 0, -3, 1, 0, 11, 8, 12, 0, -8, 3, 0, 4, 0, 24, 16, 27, 24, 28, 0, -19, 5, -3, 8, 5, 9, 0, 43, 24, 48, 40, 51, 48, 52, 0, -36, 7, -12, 12, 4, 15, 12, 16, 0, 68, 32, 75, 56, 80, 72, 83, 80, 84, 0, -59, 9, -27
Offset: 0

Views

Author

Paul Curtz, Jan 05 2017

Keywords

Comments

Successive rows:
p
0: 0, 0, 0, 0, 0, 0, 0, ...
1: 4, -3, 11, -8, 24, -19, 43, ...
2: 1, 8, 3, 16, 5, 24, 7, ...
3: 12, 0, 27, -3, 48, -12, 75, ...
4: 4, 24, 8, 40, 12, 56, 16, ...
5: 28, 5, 51, 4, 80, -3, 115, ...
6: 9, 48, 15, 72, 21, 96, 27, ...
... .
Main diagonal: alternate 3*n^2, -3.
From p>0, the rows are multiples of 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, ... . Sequences appearing after division: shifted A144433 or A195161, A064680. For p=3, we have (n+2)^2, -n^2.
First column: alternate n^2, 4*(n^2 + n + 1). Its first differences (4, -3, 11, -8, 24, ...) is the sequence of the square array for p=1.
Third column: 0, 3, 8, 15, ... is A005563(n).
Fifth column: 5, 21, 45, 77, ... is a bisection of A061037(n).
Seventh column: 7, 16, 40, 55, 91, 112, ... is a subsequence of A061039(n).
Etc. From the Rydberg spectra of the hydrogen atom (mentioned in A261327).
Starting for instance from p=-3,at the main antidiagonal,yields:
-3: -12, 0, -27, 3, ... see p=3
-2: -1, -8, -3, -16, -5, ... p=2
-1: -4, 3, -11, 8, -24, 19, ... p=1.

Crossrefs

Cf. A000290, A002061, A005563, A061037, A061039, A064680, A112087, A144433, A195161, A261327.

A210239 Triangle, read by rows, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 2, 5, 3, 2, 9, 12, 5, 2, 13, 28, 25, 8, 2, 17, 52, 74, 50, 13, 2, 21, 84, 167, 177, 96, 21, 2, 25, 124, 320, 470, 397, 180, 34, 2, 29, 172, 549, 1041, 1211, 850, 331, 55, 2, 33, 228, 870, 2034, 3042, 2928, 1758, 600, 89
Offset: 0

Views

Author

Philippe Deléham, Mar 19 2012

Keywords

Examples

			Triangle begins :
1
2, 2
2, 5, 3
2, 9, 12, 5
2, 13, 28, 25, 8
2, 17, 52, 74, 50, 13
2, 21, 84, 167, 177, 96, 21
2, 25, 124, 320, 470, 397, 180, 34

Crossrefs

Cf. A000045, A026150, A112087 (3rd column, n>2).

Formula

G.f.: (1+x+y*x)/(1-x-y*x-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A122803(n), A000007(n), A040000(n), A026150(n+1) for x = -2, -1, 0, 1 respectively.
T(n,n) = Fibonacci(n+2) = A000045(n+2), T(n+1,n) = A067331(n).
Showing 1-4 of 4 results.

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