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 1-6 of 6 results.

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472
Offset: 0

Views

Author

Labos Elemer, Apr 28 2003

Keywords

Comments

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

Examples

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Links

Crossrefs

Cf. A000010, A007283, A009195, A050339, A134059.
Essentially the same as A003945 (and perhaps also A058764).

Programs

  • Magma
    [0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009
  • Maple
    0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021
  • Mathematica
    {0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */
  • PARI
    a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012
  • Sage
    [0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

Formula

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

Extensions

More terms from Klaus Brockhaus, Dec 02 2009

A134061 Triangle read by rows: T(n,k) = A124928(n,k) + A134059(n,k) - A007318(n,k).

Original entry on oeis.org

1, 3, 5, 3, 10, 5, 3, 15, 15, 5, 3, 20, 30, 20, 5, 3, 25, 50, 50, 25, 5, 3, 30, 75, 100, 75, 30, 5, 3, 35, 105, 175, 175, 105, 35, 5, 3, 40, 140, 280, 350, 280, 140, 40, 5, 3, 45, 180, 420, 630, 630, 420, 180, 45, 5, 3, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 5
Offset: 0

Views

Author

Gary W. Adamson, Oct 05 2007

Keywords

Comments

Row sums = A134062: (1, 8, 18, 38, 78, 158, ...).

Examples

			First few rows of the triangle:
  1;
  3,  5;
  3, 10,  5;
  3, 15, 15,  5;
  3, 20, 30, 20,  5;
  3, 25, 50, 50, 25,  5;
  ...

Crossrefs

Cf. A007318, A124928, A134059, A134062 (row sums).

Programs

  • Mathematica
    Table[If[k > 0, 3*Binomial[n, k], 1] + 2*Binomial[n, k] - 2*Boole[n == 0], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 31 2025 *)

Formula

T(n,k) = 5*binomial(n,k) if k > 0, (0 <= k <= n), with leftmost column = (1,3,3,3,...).

Extensions

More terms from Amiram Eldar, May 31 2025

A134065 Triangle read by rows: A134059 + A124927 - A007318 as infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 4, 3, 8, 4, 3, 12, 12, 4, 3, 16, 24, 16, 4, 3, 20, 40, 40, 20, 4, 3, 24, 60, 80, 60, 24, 4, 3, 28, 84, 140, 140, 84, 28, 4, 3, 32, 112, 224, 280, 224, 112, 32, 4, 3, 36, 144, 336, 504, 504, 336, 144, 36, 4, 3, 40, 180, 480, 840, 1008, 840, 480, 180, 40, 4, 3, 44, 220, 660, 1320, 1848, 1848, 1320, 660, 220, 44, 4
Offset: 0

Views

Author

Gary W. Adamson, Oct 05 2007

Keywords

Comments

Row sums = A078485 starting (1, 7, 15, 31, 63, 127, ...).

Examples

			First few rows of the triangle:
  1;
  3,  4;
  3,  8,  4;
  3, 12, 12,  4;
  3, 16, 24, 16,  4;
  3, 20, 40, 40, 20,  4;
  ...

Crossrefs

Cf. A078485, A124927, A134059.

Extensions

a(15) = 3 inserted and more terms from Georg Fischer, Jun 07 2023

A134058 Triangle T(n, k) = 2*binomial(n, k) with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 6, 6, 2, 2, 8, 12, 8, 2, 2, 10, 20, 20, 10, 2, 2, 12, 30, 40, 30, 12, 2, 2, 14, 42, 70, 70, 42, 14, 2, 2, 16, 56, 112, 140, 112, 56, 16, 2, 2, 18, 72, 168, 252, 252, 168, 72, 18, 2
Offset: 0

Views

Author

Gary W. Adamson, Oct 05 2007

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, -1, 0, 0, 0, 0, 0, ...] DELTA [2, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007
Equals A028326 for all but the first term. - R. J. Mathar, Jun 08 2008
Warning: the row sums do not give A046055. - N. J. A. Sloane, Jul 08 2009

Examples

			First few rows of the triangle:
  1
  2,  2;
  2,  4,  2;
  2,  6,  6,  2;
  2,  8, 12,  8,  2;
  2, 10, 20, 20, 10,  2;
  ...

Links

Crossrefs

Cf. A007318, A028326, A046055, A084938, A134059, A151821, A173048, A173049.

Programs

  • Magma
    A134058:= func< n,k | n eq 0 select 1 else 2*Binomial(n,k) >;
[A134058(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021
  • Mathematica
    T[n_, k_]:= SeriesCoefficient[(1+x+y)/(1-x-y), {x, 0, n-k}, {y, 0, k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Jean-François Alcover, Apr 09 2015, after Vladimir Kruchinin *)
Table[2*Binomial[n,k] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)
  • Sage
    def A134058(n,k): return 2*binomial(n,k) - bool(n==0)
flatten([[A134058(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

Formula

Double Pascal's triangle and replace leftmost column with (1,2,2,2,...).
M*A007318, where M = an infinite lower triangular matrix with (1,2,2,2,...) in the main diagonal and the rest zeros.
Sum_{k=0..n} T(n,k) = A151821(n+1). - Philippe Deléham, Sep 17 2009
G.f.: (1+x+y)/(1-x-y). - Vladimir Kruchinin, Apr 09 2015
T(n, k) = 2*binomial(n, k) - [n=0]. - G. C. Greubel, Apr 26 2021
E.g.f.: 2*exp(x*(1+y)) - 1. - Stefano Spezia, Apr 03 2024

Extensions

Title changed by G. C. Greubel, Apr 26 2021

A132200 Numbers in (4,4)-Pascal triangle .

Original entry on oeis.org

1, 4, 4, 4, 8, 4, 4, 12, 12, 4, 4, 16, 24, 16, 4, 4, 20, 40, 40, 20, 4, 4, 24, 60, 80, 60, 24, 4, 4, 28, 84, 140, 140, 84, 28, 4, 4, 32, 112, 224, 280, 224, 112, 32, 4, 4, 36, 144, 336, 504, 504, 336, 144, 36, 4, 4, 40, 180, 480, 840, 1008, 840, 480, 180, 40, 4
Offset: 0

Views

Author

Philippe Deléham, Nov 19 2007

Keywords

Comments

This triangle belongs to the family of (x,y)-Pascal triangles ; other triangles arise by choosing different values for (x,y): (1,1) -> A007318 ; (1,0) -> A071919 ; (3,2) -> A029618 ; (2,2) -> A134058 ; (-1,1) -> A112467 ; (0,1) -> A097805 ; (5,5) -> A135089 ; etc..

Examples

			Triangle begins:
  1;
  4,  4;
  4,  8,  4;
  4, 12, 12,  4;
  4, 16, 24, 16,  4;
  4, 20, 40, 40, 20, 4;

Links

Crossrefs

Cf. A000079, A007318, A134058, A134059, A135089.

Programs

  • Magma
    [1] cat [4*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
  • Mathematica
    Table[4*Binomial[n,k] -3*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)
  • Sage
    def A132200(n,k): return 4*binomial(n,k) - 3*bool(n==0)
flatten([[A132200(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Formula

T(n,k) = 4*binomial(n,k), n>0 ; T(0,0)=1.
Sum_{k=0..n} T(n,k) = 2^(n+2) - 3*[n=0]. - G. C. Greubel, May 03 2021

A135089 Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.

Original entry on oeis.org

1, 5, 5, 5, 10, 5, 5, 15, 15, 5, 5, 20, 30, 20, 5, 5, 25, 50, 50, 25, 5, 5, 30, 75, 100, 75, 30, 5, 5, 35, 105, 175, 175, 105, 35, 5, 5, 40, 140, 280, 350, 280, 140, 40, 5, 5, 45, 180, 420, 630, 630, 420, 180, 45, 5, 5, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 5
Offset: 0

Views

Author

Gary W. Adamson, Nov 18 2007

Keywords

Comments

Row sums = A020714 (except for the first term).
Triangle T(n,k), 0 <= k <= n, read by rows given by (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 24 2013

Examples

			First few rows of the triangle:
  1;
  5,  5;
  5, 10,  5;
  5, 15, 15,   5;
  5, 20, 30   20,  5;
  5, 25, 50,  50, 25,  5;
  5, 30, 75, 100, 75, 30, 5.

Links

Crossrefs

Cf. A007318, A020714, A132200, A134058, A134059.

Programs

  • Magma
    [1] cat [5*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
  • Mathematica
    Table[5*Binomial[n,k] -4*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 22 2016; May 03 2021 *)
  • Sage
    def A135089(n,k): return 5*binomial(n,k) - 4*bool(n==0)
flatten([[A135089(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

Formula

T(n,k) = 5*binomial(n,k), n > 0, 0 <= k <= n.
Equals 2*A134059(n,k) - A007318(n,k).
G.f.: (1+4*x+4*x*y)/(1-x-x*y). - Philippe Deléham, Nov 24 2013
Sum_{k=0..n} T(n,k) = A020714(n) - 4*[n=0]. - G. C. Greubel, May 03 2021
Showing 1-6 of 6 results.

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

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