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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A131241 3*A046854 - 2*I.

Original entry on oeis.org

1, 3, 1, 3, 3, 1, 3, 6, 3, 1, 3, 6, 9, 3, 1, 3, 9, 9, 12, 3, 1, 3, 9, 18, 12, 15, 3, 1, 3, 12, 18, 30, 15, 18, 3, 1, 3, 12, 30, 30, 45, 18, 21, 3, 1, 3, 15, 30, 60, 45, 63, 21, 24, 3, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 21 2007

Keywords

Comments

Row sums = A111314: (1, 4, 7, 13, 22, 37, ...). A131240 = 2*A046854 - I.

Examples

			First few rows of the triangle:
  1;
  3,  1;
  3,  3,  1;
  3,  6,  3,  1;
  3,  6,  9,  3,  1;
  3,  9,  9, 12,  3,  1
  3,  9, 18, 12, 15,  3,  1;
  ...

Crossrefs

Cf. A046854, A111314, A131240.

Formula

3*A046854 - 2*I, where A046854 = Pascal's triangle with repeats by columns and I = Identity matrix.

A131344 A046854 * A065941.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 8, 7, 8, 3, 1, 13, 12, 18, 9, 4, 1, 21, 20, 38, 21, 14, 4, 1, 34, 33, 76, 47, 39, 15, 5, 1, 55, 54, 147, 97, 100, 43, 21, 5, 1, 89, 88, 277, 194, 236, 115, 69, 22, 6, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 30 2007

Keywords

Comments

Left border = Fibonacci numbers starting with F(2). Row sums = A131246: (1, 3, 6, 13, 27, 57,...). A131345 = A065941 * A046854.

Examples

			First few rows of the triangle are:
1;
2, 1;
3, 2, 1;
5, 4, 3, 1;
8, 7, 8, 3, 1;
13, 12, 18, 9, 4, 1;
...

Crossrefs

Cf. A046854, A065941, A131246, A131345.

Formula

A046854 * A065941 as infinite lower triangular matrices.

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

A131400 A046854 + A065941 - I (Identity matrix).

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 3, 3, 1, 2, 3, 6, 3, 1, 2, 4, 7, 7, 4, 1, 2, 4, 11, 8, 11, 4, 1, 2, 5, 12, 15, 15, 12, 5, 1, 2, 5, 17, 16, 30, 16, 17, 5, 1, 2, 6, 18, 27, 36, 36, 27, 18, 6, 1, 2, 6, 24, 28, 63, 42, 63, 28, 24, 6, 1, 2, 7, 25, 44, 71, 84, 84, 71, 44, 25, 7, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 06 2007

Keywords

Comments

Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67,...).

Examples

			First few rows of the triangle are:
  1;
  2, 1;
  2, 2,  1;
  2, 3,  3, 1;
  2, 3,  6, 3,  1;
  2, 4,  7, 7,  4, 1;
  2, 4, 11, 8, 11, 4, 1; ...

Links

Crossrefs

Cf. A046854, A065941, A001595.

Programs

  • GAP
    B:=Binomial;;
T:= function(n,k)
    if k=n then return 1;
    else return B(Int((n+k)/2), k) + B(n - Int((k+1)/2), Int(k/2));
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Jul 13 2019
  • Magma
    B:=Binomial; [k eq n select 1 else B(Floor((n+k)/2), k) + B(n - Floor((k+1)/2), Floor(k/2)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 13 2019
  • Mathematica
    With[{B = Binomial}, Table[If[k==n, 1, B[Floor[(n+k)/2], k] + B[n - Floor[(k+1)/2], Floor[k/2]]], {n,0,12}, {k,0,n}]]//Flatten (* G. C. Greubel, Jul 13 2019 *)
  • PARI
    b=binomial; T(n,k) = if(k==n, 1, b((n+k)\2, k) + b(n - (k+1)\2, k\2));
for(n=0,12, for(k=0,n, print1(T(n,k), ", ", ))) \\ G. C. Greubel, Jul 13 2019
  • Sage
    def T(n, k):
    b=binomial;
    if (k==n): return 1
    else: return b(floor((n+k)/2), k) + b(n - floor((k+1)/2), floor(k/2))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 13 2019

Extensions

More terms added by G. C. Greubel, Jul 13 2019

A191586 Binomial row sums of the Riordan matrix (1/(1-x),x/(1-x^2)) (A046854).

Original entry on oeis.org

1, 2, 4, 11, 32, 92, 271, 814, 2464, 7508, 23024, 70952, 219503, 681358, 2121116, 6619571, 20703040, 64873328, 203625604, 640109128, 2014951552, 6350490808, 20037015200, 63284778256, 200063948527, 633007850942, 2004431426716, 6351693835169, 20141013776384
Offset: 0

Views

Author

Emanuele Munarini, Jun 07 2011

Keywords

Links

Crossrefs

Cf. A046854.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]Binomial[Floor[(n+k)/2],k],{k,0,n}],{n,0,100}]
  • Maxima
    makelist(sum(binomial(n,k)*binomial(floor((n+k)/2),k),k,0,n),n,0,12);

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*binomial(floor((n+k)/2),k).
(8*n^2+88*n+240)*a(n+6) - (72*n^2+636*n+1380)*a(n+5) + (180*n^2+1300*n+2232)*a(n+4) - (180*n^2+1170*n+1842)*a(n+3) + (326*n^2+2074*n+3164)*a(n+2) - (228*n^2+948*n+984)*a(n+1) + (35*n^2+105*n+70)*a(n) = 0. - Emanuele Munarini, Aug 31 2017

A131331 A046854 * A000012(signed).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, -1, 2, 0, 1, 2, -1, 3, 0, 1, -3, 4, -1, 4, 0, 1, 5, -4, 7, -1, 5, 0, 1, -8, 9, -5, 11, -1, 6, 0, 1, 13, -12, 16, -6, 16, -1, 7, 0, 1, -21, 22, -17, 27, -7, 22, -1, 8, 0, 1
Offset: 0

Views

Author

Gary W. Adamson, Jun 29 2007

Keywords

Comments

Row sums = A094967: (1, 1, 2, 2, 5, 5, 13, 13, 34, 34...).

Examples

			First few rows of the triangle are:
1;
0, 1;
1, 0, 1;
-1, 2, 0, 1;
2, -1, 3, 0, 1;
-3, 4, -1, 4, 0, 1;
5, -4, 7, -1, 5, 0, 1;
-8, 9, -5, 11, -1, 6, 0, 1;
...

Crossrefs

Cf. A046854, A000012, A094967.

Formula

A046854 * A000012(signed by columns, + - + -...).

A131373 A046854 + A065941 - A000012.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 5, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 10, 7, 10, 3, 1, 1, 4, 11, 14, 14, 11, 4, 1, 1, 4, 16, 15, 29, 15, 16, 4, 1, 1, 5, 17, 26, 35, 35, 26, 17, 5, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 03 2007

Keywords

Comments

Row sums = A131269: (1, 2, 3, 6, 11, 20, 35, 60,...).

Examples

			First few rows of the triangle are:
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 5, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 10, 7, 10, 3, 1;
1, 4, 11, 14, 14, 11, 4, 1;
1, 4, 16, 15, 29, 15, 16, 4, 1;
...

Crossrefs

Cf. A056854, A065941, A000012, A131269.

Formula

A046854 + A065941 - A000012 as infinite lower triangular matrices.

A131374 A046854 + A065941 - A049310.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 3, 3, 1, 2, 1, 7, 3, 4, 1, 1, 4, 5, 8, 6, 4, 1, 2, 1, 12, 5, 15, 6, 5, 1, 1, 5, 7, 16, 15, 16, 10, 5, 1, 2, 1, 18, 7, 36, 15, 27, 10, 6, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 03 2007

Keywords

Comments

Row sums = the Lucas numbers, A000032 starting (1, 3, 4, 7, 11, 18,...).

Examples

			First few rows of the triangle are:
1;
2, 1;
1, 2, 1;
2, 1, 3, 1;
1, 3, 3, 3, 1
2, 1, 7, 3, 4, 1;
1, 4, 5, 8, 6, 4, 1;
...

Crossrefs

Cf. A046854, A065941, A049310, A000032.

Formula

A046854 + A065941 - A049310 as infinite lower triangular matrices.

A131399 3*A007318 - A046854 - A065941.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 9, 12, 9, 1, 1, 11, 23, 23, 11, 1, 1, 14, 34, 52, 34, 14, 1, 1, 16, 51, 90, 90, 51, 16, 1, 1, 19, 67, 152, 180, 152, 67, 19, 1, 1, 21, 90, 225, 342, 342, 225, 90, 21, 1
Offset: 0

Views

Author

Gary W. Adamson, Jul 05 2007

Keywords

Comments

Row sums = A074878: (1, 2, 6, 14, 32, 70, 150,...).

Examples

			First few rows of the triangle are:
1;
1, 1;
1, 4, 1;
1, 6, 6, 1;
1, 9, 12, 9, 1;
1, 11, 23, 23, 11, 1;
1, 14, 34, 52, 34, 14, 1;
1, 16, 51, 90, 90, 51, 16, 1;
...

Crossrefs

Cf. A007318, A046854, A065941, A074878.

Formula

3*A007318 - A046854 - A065941 as infinite lower triangular matrices.

A131781 2*A046854 - A000012 (signed by columns + - + -, ...).

Original entry on oeis.org

1, 3, 1, 1, 3, 1, 3, 3, 3, 1, 1, 5, 5, 3, 1, 3, 5, 7, 7, 3, 1, 1, 7, 11, 9, 9, 3, 1, 3, 7, 13, 19, 11, 11, 3, 1, 1, 9, 19, 21, 29, 13, 13, 3, 1, 3, 9, 21, 39, 31, 41, 15, 15, 3, 1
Offset: 1

Views

Author

Gary W. Adamson, Jul 14 2007

Keywords

Comments

Row sums = A131780: (1, 4, 5, 10, 15, 26, 41, ...), a sequence with a(n)/a(n-1) tending to phi.
A131779 is a companion triangle, same row sums.

Examples

			First few rows of the triangle:
  1;
  3,  1;
  1,  3,  1;
  3,  3,  3,  1;
  1,  5,  5,  3,  1;
  3,  5,  7,  7,  3,  1;
  1,  7, 11,  9,  9,  3,  1;
  3,  7, 13, 19, 11, 11,  3,  1;
  ...

Crossrefs

Cf. A131779, A131780, A046854, A000012.
Previous Showing 11-20 of 58 results. Next

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

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