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.

A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A073570.
Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(n\k+4, 5));
  • Python
    from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+3,4) * floor(n/k).
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).
a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - 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

A366395 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+2,3).

Original entry on oeis.org

1, 3, 10, 16, 32, 49, 78, 100, 152, 194, 261, 318, 410, 489, 631, 717, 871, 1014, 1205, 1351, 1617, 1806, 2083, 2300, 2641, 2903, 3333, 3612, 4048, 4450, 4947, 5289, 5923, 6367, 7041, 7548, 8252, 8805, 9683, 10245, 11107, 11873, 12820, 13497, 14719, 15526, 16655
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Links

Crossrefs

Partial sums of A365007.
Cf. A078471, A366659, A366723.
Cf. A364970.

Programs

  • Mathematica
    Array[Sum[(-1)^(k - 1)*Binomial[Floor[#/k] + 2, 3], {k, #}] &, 56] (* Michael De Vlieger, Oct 25 2023 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+2, 3));

Formula

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

A366937 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+1,2) * floor(n/k).

Original entry on oeis.org

1, -1, 6, -6, 10, -7, 22, -26, 26, -16, 51, -54, 38, -41, 101, -83, 71, -72, 119, -143, 123, -66, 211, -230, 111, -151, 279, -216, 220, -182, 315, -397, 237, -207, 467, -430, 274, -279, 599, -519, 343, -423, 524, -665, 557, -250, 879, -874, 380, -612, 874, -776
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Partial sums of A320900.
Cf. A024919, A366938, A366939.
Cf. A364970, A366395.

Programs

  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+1, 2)*(n\k));
  • Python
    from math import isqrt
def A366937(n): return (((s:=isqrt(m:=n>>1))*(s+1)**2*((s<<2)+5)<<1)-(t:=isqrt(n))*(t+1)**2*(t+2)-sum((((q:=m//w)+1)*(q*((q<<2)+5)+6*w*((w<<1)+1))<<1) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)+3*w*(w+1)) for w in range(1,t+1)))//6 # Chai Wah Wu, Oct 29 2023

Formula

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).

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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