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.

A155724 Triangle read by rows: T(n, k) = 2*n*k + n + k - 4.

Original entry on oeis.org

0, 3, 8, 6, 13, 20, 9, 18, 27, 36, 12, 23, 34, 45, 56, 15, 28, 41, 54, 67, 80, 18, 33, 48, 63, 78, 93, 108, 21, 38, 55, 72, 89, 106, 123, 140, 24, 43, 62, 81, 100, 119, 138, 157, 176, 27, 48, 69, 90, 111, 132, 153, 174, 195, 216, 30, 53, 76, 99, 122, 145, 168, 191, 214, 237, 260
Offset: 1

Views

Author

Vincenzo Librandi, Jan 25 2009

Keywords

Examples

			Triangle begins:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27, 36;
  12, 23, 34, 45,  56;
  15, 28, 41, 54,  67,  80;
  18, 33, 48, 63,  78,  93, 108;
  21, 38, 55, 72,  89, 106, 123, 140;
  24, 43, 62, 81, 100, 119, 138, 157, 176;
  27, 48, 69, 90, 111, 132, 153, 174, 195, 216;

Links

Crossrefs

All terms are in A155723.
Cf. A008585, A016885, A017053, A020705, A154685, A155722.
Cf. A162261 (row sums).
Columns k: A008585 (k=1), A016885 (k=2), A017053 (k=3), 9*A020705 (k=4).
Diagonals include: A139570, A181510, A271625.

Programs

  • Magma
    /* Triangle: */ [[2*m*n+m+n-4: m in [1..n]]: n in [1..10]]; // Bruno Berselli, Aug 31 2012
  • Mathematica
    Flatten[Table[2 n m + m + n - 4, {n, 10}, {m, n}]] (* Vincenzo Librandi, Mar 01 2012 *)
  • Python
    def A155724(n,k): return 2*n*k+n+k-4
print(flatten([[A155724(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

Formula

T(n, k) = A154685(n, k) - 8. - L. Edson Jeffery, Oct 12 2012
2*T(n, k) + 9 = (2*k+1)*(2*n+1). - Vincenzo Librandi, Nov 18 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n - 5 (main diagonal).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*( 4*(-1)^(n+1)*n^2 + 2*(2-3*(-1)^n)*n - 7*(1-(-1)^n)).
G.f.: x*y*(3*x + 3*y - 4*x*y)/((1-x)*(1-y))^2. (End)

Extensions

Edited by N. J. A. Sloane, Jun 23 2010

A163657 Triangle T(m,n) = 2*m*n + m + n + 8 read by rows.

Original entry on oeis.org

12, 15, 20, 18, 25, 32, 21, 30, 39, 48, 24, 35, 46, 57, 68, 27, 40, 53, 66, 79, 92, 30, 45, 60, 75, 90, 105, 120, 33, 50, 67, 84, 101, 118, 135, 152, 36, 55, 74, 93, 112, 131, 150, 169, 188, 39, 60, 81, 102, 123, 144, 165, 186, 207, 228, 42, 65, 88, 111, 134, 157, 180
Offset: 1

Views

Author

Vincenzo Librandi, Aug 02 2009

Keywords

Comments

If p=2*n+1 is a prime number, then T(n,n) = (p^2+15)/2.
First column: 3*A020705; second column: 5*A020705; third column: A017029. - Vincenzo Librandi, Nov 18 2012

Examples

			Triangle begins:
12;
15, 20;
18, 25, 32;
21, 30, 39, 48;
24, 35, 46, 57, 68;
27, 40, 53, 66, 79, 92;
30, 45, 60, 75, 90, 105, 120;
33, 50, 67, 84, 101, 118, 135, 152; etc.

Links

Crossrefs

Cf. A153047, A020705, A017029.

Programs

  • Magma
    [2*n*k+n+k+8: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012
  • Mathematica
    Flatten[Table[2nm + m + n + 8, {n, 10}, {m, n}]] (* Vincenzo Librandi, Nov 18 2012 *)

Formula

T(n,m) = A163672(n,m)+1.

Extensions

Edited by R. J. Mathar, Oct 12 2009

A020744 Pisot sequences P(8,10), T(8,10).

Original entry on oeis.org

8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

Conjecturally, even sums of four primes. - Charles R Greathouse IV, Feb 16 2012

Links

Crossrefs

Subsequence of A005843, A020739. See A008776 for definitions of Pisot sequences.
Cf. A020705, A028563.

Programs

  • Mathematica
    LinearRecurrence[{2,-1},{8,10},70] (* Harvey P. Dale, Jul 19 2015 *)
P[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Ceiling[a[n - 1]^2/a[n - 2] - 1/2]; Table[a[n], {n, 0, z}]]; P[8, 10, 65] (* or *)
T[x_, y_, z_] := Block[{a}, a[0] = x; a[1] = y; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2]]; Table[a[n], {n, 0, z}]]; T[8, 10, 65] (* Michael De Vlieger, Aug 08 2016 *)
  • PARI
    a(n)=2*n+8 \\ Charles R Greathouse IV, Feb 16 2012
  • PARI
    pisotP(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = ceil(a[n-1]^2/a[n-2]-1/2));
  a
}
pisotP(50, 8, 10) \\ Colin Barker, Aug 08 2016

Formula

a(n) = 2*n + 8.
a(n) = 2*a(n-1) - a(n-2).
From Elmo R. Oliveira, Oct 30 2024: (Start)
G.f.: 2*(4 - 3*x)/(1 - x)^2.
E.g.f.: 2*(4 + x)*exp(x).
a(n) = 2*A020705(n) = A028563(n+1) - A028563(n). (End)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.