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 1-5 of 5 results.

A084630 a(n) = floor(C(n+7,7)/C(n+5,5)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 22, 23, 25, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 56, 58, 60, 63, 65, 68, 70, 73, 76, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 115, 118, 121, 125, 128, 132, 135
Offset: 0

Views

Author

Paul Barry, Jun 01 2003

Keywords

Comments

The general Somos-6 sequence terms s(n), with general coefficients and initial values s(0)..s(5), are Laurent polynomials with denominators a product of initial values raised to powers being entries in this sequence. Thus, the denominator of s(n) = Product_{k=0..5} s(k)^a(n-k-6). - Michael Somos, Apr 10 2020

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ... - _Michael Somos_, Apr 10 2020

Links

Crossrefs

Cf. A084628, A084631.

Programs

  • Magma
    [Floor(Binomial(n+7,2)/21): n in [0..80]]; // G. C. Greubel, Mar 23 2023
  • Mathematica
    a[n_]:= Quotient[n(n+13), 42] + 1; (* Michael Somos, Apr 10 2020 *)
Floor[Binomial[Range[0,100]+7,2]/21] (* G. C. Greubel, Mar 23 2023 *)
  • PARI
    {a(n) = n*(n + 13)\42 + 1}; /* Michael Somos, Apr 10 2020 */
  • SageMath
    [binomial(n+7,2)//21 for n in range(81)] # G. C. Greubel, Mar 23 2023

Formula

a(n) = 1 + floor( n*(n+13)/42 ).
From Michael Somos, Apr 10 2020: (Start)
G.f.: (1-x+x^3-x^4+x^5-x^6+x^7-x^9+x^10)/((1-x)^2*(1-x^21)).
a(n) = a(-13-n).
a(n) = a(n-21) + n + 4 for all n in Z.
0 = +a(n)*(a(n+1) -a(n+3) -a(n+4) +a(n+6)) + a(n+1)*(-a(n+1) +a(n+3) +a(n+4) -a(n+5)) + a(n+2)*(-a(n+3) +a(n+4) +a(n+5) -a(n+6)) + a(n+3)*(+a(n+3) -a(n+5) +a(n+6) -a(n+6)) + a(n+5)*(-a(n+5) +a(n+6)) for all n in Z. (End)
a(n) = floor(binomial(n+7,2)/21). - G. C. Greubel, Mar 23 2023

A084626 a(n) = floor(C(n+6,6)/C(n+2,2)).

Original entry on oeis.org

1, 2, 4, 8, 14, 22, 33, 47, 66, 91, 121, 158, 204, 258, 323, 399, 487, 590, 708, 843, 996, 1170, 1365, 1583, 1827, 2097, 2397, 2728, 3091, 3490, 3927, 4403, 4921, 5483, 6092, 6751, 7462, 8227, 9050, 9933, 10879, 11891, 12972, 14125, 15353, 16660, 18048
Offset: 0

Views

Author

Paul Barry, Jun 01 2003

Keywords

Links

Crossrefs

Cf. A011865, A084624, A084627, A084628, A084630, A084631.

Programs

  • Magma
    [Floor(n*(n+9)*(n^2+9*n+38)/360)+1: n in [0..50]]; // Vincenzo Librandi, Aug 02 2013
  • Mathematica
    Table[Floor[n*(n+9)*(n^2+9*n+38)/360] +1, {n,0,50}] (* Vincenzo Librandi, Aug 02 2013 *)
Floor[Binomial[Range[6,76],4]/15] (* G. C. Greubel, Mar 24 2023 *)
  • SageMath
    [binomial(n+6,4)//15 for n in range(71)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = 1 + floor( n*(n+9)*(n^2 +9*n +38)/360 ).
a(n) = floor(binomial(n+6,4)/15). - G. C. Greubel, Mar 24 2023

A084627 a(n) = floor(C(n+6,6)/C(n+3,3)).

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 11, 14, 18, 22, 28, 34, 40, 48, 57, 66, 77, 88, 101, 115, 130, 146, 163, 182, 203, 224, 248, 272, 299, 327, 357, 388, 421, 456, 494, 533, 574, 617, 662, 709, 759, 810, 864, 921, 980, 1041, 1105, 1171, 1240, 1311, 1386, 1463, 1542, 1625, 1711
Offset: 0

Views

Author

Paul Barry, Jun 01 2003

Keywords

Links

Crossrefs

Cf. A084626, A084628, A084630, A084631.

Programs

  • Magma
    [Floor(Binomial(n+6,3)/20): n in [0..70]]; // G. C. Greubel, Mar 24 2023
  • Mathematica
    Table[Floor[Binomial[n+6,6]/Binomial[n+3,3]],{n,0,60}] (* Harvey P. Dale, Feb 07 2015 *)
Floor[Binomial[Range[6,76],3]/20] (* G. C. Greubel, Mar 24 2023 *)
  • SageMath
    [binomial(n+6,3)//20 for n in range(71)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = 1 + floor(n*(n^2 +15*n +74)/120).
a(n) = floor(binomial(n+6,3)/20). - G. C. Greubel, Mar 24 2023

A084624 a(n) = floor(C(n+5,5)/C(n+2,2)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 16, 22, 28, 36, 45, 56, 68, 81, 96, 114, 133, 154, 177, 202, 230, 260, 292, 327, 365, 406, 449, 496, 545, 598, 654, 714, 777, 843, 913, 988, 1066, 1148, 1234, 1324, 1419, 1518, 1621, 1729, 1842, 1960, 2082, 2210, 2342, 2480, 2623, 2772, 2926
Offset: 0

Views

Author

Paul Barry, Jun 01 2003

Keywords

Links

Crossrefs

Cf. A011865, A084625, A084626, A084627, A084628, A084630, A084631.

Programs

  • Magma
    [Floor(Binomial(n+5,3)/10): n in [0..60]]; // G. C. Greubel, Mar 24 2023
  • Mathematica
    LinearRecurrence[{3,-3,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-3,3,-1},{1,2,3, 5,8,12,16,22,28,36,45,56,68,81,96,114,133,154,177,202,230, 260,292},53] (* Ray Chandler, Jul 17 2015 *)
Table[Floor[Binomial[n+5,5]/Binomial[n+2,2]],{n,0,60}] (* or *) Table[ Floor[((3+n)(4+n)(5+n))/60],{n,0,60}] (* Harvey P. Dale, Sep 04 2017 *)
Floor[Binomial[Range[5,65],3]/10] (* G. C. Greubel, Mar 24 2023 *)
  • SageMath
    [(binomial(n+5,3)//10) for n in range(61)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = 1 + floor( n*(n^2 + 12*n + 47)/60 ).
From G. C. Greubel, Mar 24 2023: (Start)
a(n) = floor( binomial(n+5,3)/10 ).
G.f.: (1 -x +x^3 -x^6 +2*x^7 -2*x^8 +2*x^9 -x^10 +x^11 -x^12 +x^14 +x^15 -2*x^16 +x^17)/((1-x)^3*(1-x^20)). (End)

A084625 Binomial transform of A084624.

Original entry on oeis.org

1, 3, 8, 21, 55, 143, 366, 919, 2265, 5491, 13125, 31000, 72485, 168042, 386709, 884161, 2009742, 4543830, 10222264, 22891099, 51041560, 113359224, 250839510, 553173006, 1216070081, 2665518207, 5826533103, 12703217438, 27628250142
Offset: 0

Views

Author

Paul Barry, Jun 01 2003

Keywords

Links

Crossrefs

Cf. A084624, A084626, A084627, A084628, A084630, A084631.

Programs

  • Magma
    A084625:= func< n | (&+[Binomial(n,j)*Floor(Binomial(j+5,3)/10): j in [0..n]]) >;
[A084625(n): n in [0..50]]; // G. C. Greubel, Mar 24 2023
  • Mathematica
    a[n_]:= a[n]= 2^n +Sum[Binomial[n,j]*Floor[j*(j^2+12*j+47)/60], {j,0, n}];
Table[a[n], {n,0,50}] (* G. C. Greubel, Mar 24 2023 *)
  • SageMath
    def A084625(n): return sum(binomial(n,j)*(binomial(j+5,3)//10) for j in range(n+1))
[A084625(n) for n in range(51)] # G. C. Greubel, Mar 24 2023

Formula

a(n) = Sum_{k=0..n} C(n, k)*floor(C(k+5, 5)/C(k+2, 2)).
a(n) = 2^n + Sum_{k=0..n} binomial(n,k)*floor(k*(k^2 +12*k +47)/60). - G. C. Greubel, Mar 24 2023
Showing 1-5 of 5 results.

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