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.

A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

Original entry on oeis.org

1, 5, 12, 26, 42, 73, 102, 152, 204, 278, 345, 464, 556, 693, 835, 1021, 1175, 1422, 1613, 1907, 2173, 2496, 2773, 3228, 3569, 4015, 4445, 4998, 5434, 6120, 6617, 7331, 7965, 8717, 9391, 10392, 11096, 12031, 12909, 14059, 14921, 16219, 17166, 18489, 19711, 21072, 22201
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A007437.
Cf. A006218, A024916, A064602, A365409, A365439.

Programs

  • Mathematica
    Table[Sum[Binomial[Floor[n/k+2],3],{k,n}],{n,50}] (* Harvey P. Dale, Aug 04 2024 *)
  • PARI
    a(n) = sum(k=1, n, binomial(n\k+2, 3));
  • Python
    from math import isqrt
def A364970(n): return (-(s:=isqrt(n))**2*(s+1)*(s+2)+sum((q:=n//k)*(3*k*(k+1)+(q+1)*(q+2)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+1,2) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^3 = 1/(1-x) * Sum_{k>=1} binomial(k+1,2) * x^k/(1-x^k).
a(n) = (A064602(n)+A024916(n))/2. - Chai Wah Wu, Oct 26 2023

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A059358.
Cf. A006218, A024916, A064602, A064603, A364970, A365439.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(n\k+3, 4));
  • Python
    from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023

A366723 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 5, 21, 50, 121, 236, 447, 736, 1247, 1896, 2898, 4151, 5972, 8146, 11292, 14797, 19643, 25248, 32564, 40663, 51515, 63168, 78119, 94452, 114998, 136933, 164849, 193753, 229714, 268334, 314711, 362824, 422746, 483950, 558046, 635070, 726461, 820420, 934186, 1048245
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Crossrefs

Partial sums of A366814.
Cf. A078471, A366395, A366659.
Cf. A365439.

Programs

  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+4, 5));

Formula

a(n) = Sum_{k=1..n} binomial(k+3,4) * (floor(n/k) mod 2).
G.f.: -1/(1-x) * Sum_{k>=1} (-x)^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1+x^k).

A366939 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+3,4) * floor(n/k).

Original entry on oeis.org

1, -3, 13, -26, 45, -70, 141, -228, 283, -366, 636, -879, 942, -1232, 1914, -2331, 2515, -3090, 4226, -5313, 5539, -6114, 8837, -10558, 9988, -11947, 15969, -17705, 18256, -20364, 26013, -30592, 29330, -31874, 42222, -47034, 44357, -49602, 64164, -69115, 66637, -74017
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Cf. A024919, A366937, A366938.
Cf. A365439, A366723.

Programs

  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+3, 4)*(n\k));
  • Python
    from math import isqrt
from sympy import rf
def A366939(n): return ((rf(s:=isqrt(m:=n>>1),3)*(s+1)*((s**2<<2)+13*s+8)<<3)-rf(t:=isqrt(n),5)*(t+1)+sum((((q:=m//w)+1)*(-q*(q+2)*((q**2<<2)+13*q+8)-5*w*(w+1)*((r:=w<<1)+1)*(r+3))<<3) for w in range(1,s+1))+sum(rf(q:=n//w,5)+5*(q+1)*rf(w,4) for w in range(1,t+1)))//120 # Chai Wah Wu, Oct 29 2023

Formula

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1+x^k)^5 = -1/(1-x) * Sum_{k>=1} binomial(k+3,4) * (-x)^k/(1-x^k).

A366977 Array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} binomial(floor(n/j)+k,k+1).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 10, 1, 7, 17, 26, 21, 14, 1, 8, 23, 42, 42, 33, 16, 1, 9, 30, 64, 78, 73, 41, 20, 1, 10, 38, 93, 135, 149, 102, 56, 23, 1, 11, 47, 130, 220, 282, 234, 152, 69, 27, 1, 12, 57, 176, 341, 500, 493, 379, 204, 87, 29
Offset: 1

Views

Author

Chai Wah Wu, Oct 30 2023

Keywords

Examples

			Array begins:
   1,  1,   1,   1,   1,   1,    1,    1,    1,    1, ...
   3,  4,   5,   6,   7,   8,    9,   10,   11,   12, ...
   5,  8,  12,  17,  23,  30,   38,   47,   57,   68, ...
   8, 15,  26,  42,  64,  93,  130,  176,  232,  299, ...
  10, 21,  42,  78, 135, 220,  341,  507,  728, 1015, ...
  14, 33,  73, 149, 282, 500,  839, 1344, 2070, 3083, ...
  16, 41, 102, 234, 493, 963, 1764, 3061, 5074, 8089, ...

Crossrefs

First superdiagonal is A366978.
Columns k=0..4 give A006218, A024916, A364970, A365409, A365439.

Programs

  • Python
    from math import isqrt, comb
def A366977_T(n,k): return (-(s:=isqrt(n))**2*comb(s+k,k)+sum((q:=n//j)*((k+1)*comb(j+k-1,k)+comb(q+k,k)) for j in range(1,s+1)))//(k+1)
def A366977_gen(): # generator of terms
    return (A366977_T(k+1, n-k-1) for n in count(1) for k in range(n))
A366977_list = list(islice(A366977_gen(),30))

Formula

T(n,k) = Sum_{j=1..n} binomial(j+k-1,k)*floor(n/j) = (Sum_{j=1..floor(sqrt(n))} [floor(n/j)*((k+1)*binomial(j+k-1,k)+binomial(floor(n/j)+k,k))] - floor(sqrt(n))^2*binomial(floor(sqrt(n))+k,k))/(k+1).
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - x^j)^(k+1). - Seiichi Manyama, Oct 30 2023
Showing 1-5 of 5 results.

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