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.

A100206 Row sums of Clark's triangle A046902.

Original entry on oeis.org

0, 7, 20, 46, 98, 202, 410, 826, 1658, 3322, 6650, 13306, 26618, 53242, 106490, 212986, 425978, 851962, 1703930, 3407866, 6815738, 13631482, 27262970, 54525946, 109051898, 218103802, 436207610, 872415226, 1744830458, 3489660922
Offset: 0

Views

Author

Jorge Coveiro, Dec 28 2004

Keywords

Examples

			a(0) =  0.
a(1) =  6 +  1.
a(2) = 12 +  7 +  1.
a(3) = 18 + 19 +  8 + 1.
a(4) = 24 + 37 + 27 + 9 + 1.

Links

Crossrefs

Cf. A046902.

Programs

  • Magma
    [0] cat [13*2^(n-1)-6: n in [1..40]]; // Vincenzo Librandi, May 29 2016
  • Mathematica
    Join[{0}, Table[13 2^(n-1) - 6, {n, 1, 40}]] (* Vincenzo Librandi, May 29 2016 *)
LinearRecurrence[{3,-2},{0,7,20},30] (* Harvey P. Dale, Jul 07 2024 *)
  • PARI
    {a(n) = if(n,13*2^(n-1)-6,0)} \\ Max Alekseyev, May 12 2005
  • SageMath
    [(13*2^n - 12 - int(n==0))/2 for n in range(41)] # G. C. Greubel, Apr 02 2024

Formula

a(0)=0; for n>0, a(n) = 13*2^(n-1) - 6. - Max Alekseyev, May 12 2005
From Chai Wah Wu, May 28 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
G.f.: x*(7 - x)/((1 - x )*(1 - 2*x)). (End)
E.g.f.: (1/2)*(13*exp(2*x) - 12*exp(x) - 1). - G. C. Greubel, Apr 02 2024

Extensions

More terms from Max Alekseyev, May 12 2005

A257448 a(n) = 13*(2^n - 1) - 3*n^2 - 9*n.

Original entry on oeis.org

1, 9, 37, 111, 283, 657, 1441, 3051, 6319, 12909, 26149, 52695, 105859, 212265, 425161, 851043, 1702903, 3406725, 6814477, 13630095, 27261451, 54524289, 109050097, 218101851, 436205503, 872412957, 1744828021, 3489658311, 6979319059, 13958640729
Offset: 1

Views

Author

Luciano Ancora, Apr 23 2015

Keywords

Comments

These numbers belong to a family of sequences obtained as follows:
. A000225: 1*(2^n-1);
. A050488: 3*(2^n-1) - 2*n;
. a(n): 13*(2^n-1) - 3*n^2 - 9*n;
. A257449: 75*(2^n-1) - 4*n^3 - 18*n^2 - 52*n;
. A257450: 541*(2^n-1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n,
where the sequence 1, 3, 13, 75, 541, ... is A000670 (after the first term), and A208744 gives the triangle of coefficients:
2;
3, 9;
4, 18, 52;
5, 30, 130, 375;
6, 45, 260, 1125, 3246;
7, 63, 455, 2625, 11361, 32781, etc.
Also, the antidiagonal sums in the array are given by the formula (6*n^2 + 6*k*n + (k-1)*k)*(k+n)!/((k+3)!*(n-1)!) for k = 0, 1, 2, 3, 4, ... (see Example field).

Examples

			By the second comment, the array begins (antidiagonals in A046902):
k=0: 1,  8, 27,  64,  125,  216, ...  A000578
k=1: 1,  9, 36, 100,  225,  441, ...  A000537
k=2: 1, 10, 46, 146,  371,  812, ...  A024166
k=3: 1, 11, 57, 203,  574, 1386, ...  A101094
k=4: 1, 12, 69, 272,  846, 2232, ...  A101097
k=5: 1, 13, 82, 354, 1200, 3432, ...  A101102
k=6: 1, 14, 96, 450, 1650, 5082, ...  A254469
...
See also A254469 (Example field).

Links

Crossrefs

Cf. A000225, A000670, A050488, A208744, A257449, A257450.

Programs

  • Magma
    [13*(2^n-1)-3*n^2-9*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015
  • Mathematica
    Table[13 (2^n - 1) - 3 n^2 - 9n, {n, 30}]
CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^3*(1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Nov 14 2016 *)

Formula

G.f.: x*(1+4*x+x^2)/((1-x)^3*(1-2*x)).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Ray Chandler, Jul 25 2015

Extensions

Edited by Bruno Berselli, Apr 28 2015

A185080 a(n) = 6 * binomial(2*n,n-1) + binomial(2*n-1,n).

Original entry on oeis.org

7, 27, 100, 371, 1386, 5214, 19734, 75075, 286858, 1100138, 4232592, 16328942, 63146500, 244711260, 950094810, 3694876515, 14390571690, 56122547250, 219140635560, 856617714810, 3351878581740, 13127747882340, 51458942047500, 201869999056206, 792497263436676
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 26 2012

Keywords

Links

Crossrefs

Cf. A000108, A007318, A024482, A046224, A046902.

Programs

  • Haskell
    a185080 n = 6 * a007318 (2 * n) (n - 1) + a007318 (2 * n - 1) n
  • Magma
    [(13*n+1)*Catalan(n)/2: n in [1..40]]; // G. C. Greubel, Apr 03 2024
  • Mathematica
    Table[6Binomial[2n,n-1]+Binomial[2n-1,n],{n,30}] (* Harvey P. Dale, Dec 28 2012 *)
  • SageMath
    [(13*n+1)*binomial(2*n,n)/(2*n+2) for n in range(1,41)] # G. C. Greubel, Apr 03 2024

Formula

a(n) = A046902(2*n,n) (Central terms of Clark's triangle).
a(n) = 6 * A007318(2*n,n-1) + A007318(2*n-1,n).
From G. C. Greubel, Apr 03 2024: (Start)
a(n) = (13*n+1)*A000108(n)/2.
a(n) = (2 + 22*n - 52*n^2)*a(n-1)/(12 - n - 13*n^2).
G.f.: ((6 - 11*x)*sqrt(1-4*x) - (1-4*x)*(6+x))/(2*x*(1-4*x)).
E.g.f.: (1/2)*(-1 + exp(2*x)*(BesselI(0, 2*x) + 12*BesselI(1, 2*x))).(End)

A090850 Clark's triangle with f=6 read by row.

Original entry on oeis.org

0, 6, 1, 12, 7, 1, 18, 19, 8, 1, 24, 37, 27, 9, 1, 30, 61, 64, 36, 10, 1, 36, 91, 125, 100, 46, 11, 1, 42, 127, 216, 225, 146, 57, 12, 1, 48, 169, 343, 441, 371, 203, 69, 13, 1, 54, 217, 512, 784, 812, 574, 272, 82, 14, 1, 60, 271, 729, 1296, 1596, 1386, 846, 354, 96, 15, 1
Offset: 0

Views

Author

Eric W. Weisstein, Dec 09 2003

Keywords

Examples

			Triangle starts
   0;
   6,  1;
  12,  7,  1;
  18, 19,  8,  1;
  ...

Links

Crossrefs

Cf. A046902, A100206.

Programs

  • Mathematica
    Join[{0},Rest[Flatten[Table[6*Binomial[n,k+1]+Binomial[n-1,k-1],{n,0,10},{k,0,n}]]]] (* Harvey P. Dale, Mar 29 2014 *)
  • Python
    from operator import add
f = 6
A090850_list = blist = [0]
for _ in range(20):
    blist = [blist[0]+f]+list(map(add,blist[:-1],blist[1:]))+[1]
    A090850_list.extend(blist) # Chai Wah Wu, Sep 18 2014

Formula

c(n, k) = 6*binomial(n, k+1) + binomial(n-1, k-1). - Max Alekseyev, Nov 06 2005
Showing 1-4 of 4 results.

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