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.

A131246 Row sums of triangle A131245.

Original entry on oeis.org

1, 3, 6, 13, 27, 57, 119, 250, 523, 1097, 2297, 4815, 10086, 21137, 44283, 92793, 194419, 407378, 853559, 1788481, 3747361, 7851867, 16451910, 34471669, 72228171, 151339401, 317100335, 664418698, 1392152131
Offset: 0

Views

Author

Gary W. Adamson, Jun 22 2007

Keywords

Comments

A131244 is a companion sequence.

Examples

			a(3) = 13 = sum of row 3 terms of triangle A131245: (5 + 5 + 2 + 1)

Links

Crossrefs

Cf. A131243, A131244, A131245, A065941, A046854.

Programs

  • Maple
    A046854 := proc(n,k) binomial(floor((n+k)/2),k) ; end proc:
A131245 := proc(n,k) local a,j ; a := 0 ; for j from k to n do a := a+ A046854(n,j)*A046854(j,k) ;  end do: a ; end proc:
A131246 := proc(n) add(A131245(n,k),k=0..n) ; end proc:
seq(A131246(n),n=0..50) ; # R. J. Mathar, Jan 29 2011
  • Mathematica
    CoefficientList[Series[-(1+x)(x^2-x-1)/(1-x-3x^2+x^3+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,-1,-1},{1,3,6,13},30] (* Harvey P. Dale, Sep 07 2013 *)

Formula

G.f. -(1+x)*(x^2-x-1)/ ( 1-x-3*x^2+x^3+x^4 ). - R. J. Mathar, Jan 29 2011
a(0)=1, a(1)=3, a(2)=6, a(3)=13, a(n)=a(n-1)+3*a(n-2)-a(n-3)-a(n-4). - Harvey P. Dale, Sep 07 2013

A131243 A065941^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 4, 1, 8, 7, 10, 4, 1, 13, 12, 23, 12, 6, 1, 21, 20, 48, 29, 21, 6, 1, 34, 33, 96, 64, 62, 24, 8, 1, 55, 54, 185, 132, 160, 74, 36, 8, 1, 89, 88, 348, 261, 382, 200, 130, 40, 10, 1, 144, 143, 642, 500, 859, 492, 400, 150, 55, 10, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 22 2007

Keywords

Comments

Left border, Fibonacci numbers; next border to the right, (Fibonacci numbers - 1).

Examples

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,   1;
   5,  4,   4,   1;
   8,  7,  10,   4,   1;
  13, 12,  23,  12,   6,   1;
  21, 20,  48,  29,  21,   6,   1;
  34, 33,  96,  64,  62,  24,   8,  1;
  55, 54, 185, 132, 160,  74,  36,  8,  1;
  89, 88, 348, 261, 382, 200, 130, 40, 10, 1;

Links

Crossrefs

Cf. A065941, A131244 (row sums), A131245, A131246.

Extensions

Data and Example corrected by Jon E. Schoenfield, Feb 26 2022
More terms from Hakan Icoz, Jan 23 2023

A131245 A046854^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 8, 9, 7, 2, 1, 13, 19, 13, 9, 2, 1, 21, 33, 34, 17, 11, 2, 1, 34, 65, 61, 53, 21, 13, 2, 1, 55, 111, 141, 97, 76, 25, 15, 2, 1, 89, 210, 248, 257, 141, 103, 29, 17, 2, 1, 144, 355, 534, 461, 421, 193, 134, 33, 19, 2, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 22 2007

Keywords

Comments

Left border = Fibonacci numbers.
Row sums = A131246.
A131243 is the square of the reflection triangle to A046854: A065941.
Row sums of A131243 = (1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, ...).

Examples

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  1;
   5,  5,  2,  1;
   8,  9,  7,  2,  1;
  13, 19, 13,  9,  2,  1;
  21, 33, 34, 17, 11,  2,  1;
  ...

Crossrefs

Cf. A131243, A131244, A131246, A046854, A065941.

Programs

  • PARI
    T(n, k) = binomial((n+k)\2, k);
row(n) = my(m=matrix(n+1, n+1, i, j, T(i-1,j-1))); vector(n+1, i, (m^2)[n+1,i]);
lista(nn) = for (n=0, nn, my(v=row(n)); for (i=1, #v, print1(v[i], ", "));); \\ Michel Marcus, Feb 28 2022

Extensions

More terms from Michel Marcus, Feb 28 2022

A131345 Triangle read by rows: A065941 * A046854 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 3, 1, 8, 10, 8, 3, 1, 13, 20, 19, 10, 4, 1, 21, 38, 42, 26, 14, 4, 1, 34, 71, 89, 65, 41, 16, 5, 1, 55, 130, 182, 151, 110, 50, 21, 5, 1, 89, 235, 363, 338, 276, 146, 72, 23, 6, 1, 144, 420, 709, 730, 659, 392, 223, 83, 29, 6, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 30 2007

Keywords

Comments

Left border = Fibonacci numbers starting with F(2). Row sums = A131244: (1, 3, 6, 14, 30, 67, 146,...). A131344 = A046854 * A065941.

Examples

			First few rows of the triangle are:
1;
2, 1;
3, 2, 1;
5, 5, 3, 1;
8, 10, 8, 3, 1;
13, 20, 19, 10, 4, 1;
21, 38, 42, 26, 14, 4, 1;
...

Crossrefs

Cf. A046854, A065941, A131244, A131344.

Formula

A065941 * A046854 as infinite lower triangular matrics.

Extensions

a(49) split and more terms from Georg Fischer, May 29 2023
Showing 1-4 of 4 results.

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