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.

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).

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).

A366813 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+2,3).

Original entry on oeis.org

1, 3, 11, 15, 36, 49, 85, 95, 176, 188, 287, 313, 456, 479, 726, 671, 970, 1024, 1331, 1300, 1866, 1741, 2301, 2265, 2961, 2824, 3830, 3431, 4496, 4514, 5457, 5023, 6842, 6174, 7890, 7444, 9140, 8553, 11126, 9780, 12342, 11998, 14191, 12885, 17106, 14999, 18425
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Links

Crossrefs

Partial sums give A366659.
Cf. A000593, A365007, A366814.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(n/# - 1)*Binomial[# + 2, 3] &], {n, 56}] (* Michael De Vlieger, Oct 25 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+2, 3));

Formula

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

A366938 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(k+2,3) * floor(n/k).

Original entry on oeis.org

1, -2, 9, -14, 22, -27, 58, -85, 91, -97, 190, -243, 213, -266, 460, -499, 471, -553, 778, -970, 896, -845, 1456, -1697, 1264, -1560, 2270, -2289, 2207, -2307, 3150, -3793, 3049, -3125, 4765, -5079, 4061, -4492, 6634, -6714, 5628, -6370, 7821, -9120, 7986, -7013
Offset: 1

Views

Author

Seiichi Manyama, Oct 29 2023

Keywords

Crossrefs

Partial sums of A320901.
Cf. A024919, A366937, A366939.
Cf. A365409, A366659.

Programs

  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(k+2, 3)*(n\k));
  • Python
    from math import isqrt
def A366938(n): return (((s:=isqrt(m:=n>>1))*(s+1)**3*(s+2)<<4)-(t:=isqrt(n))*(t+1)**2*(t+2)*(t+3)-sum((((q:=m//w)+1)*(q*(q+1)*(q+2)+(w*(w+1)*((w<<1)+1)<<1))<<4) for w in range(1,s+1))+sum(((q:=n//w)+1)*(q*(q+2)*(q+3)+(w*(w+1)*(w+2)<<2)) for w in range(1,t+1)))//24 # Chai Wah Wu, Oct 29 2023

Formula

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).
Showing 1-4 of 4 results.

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