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.

A366659 a(n) = Sum_{k=1..n} (-1)^(k-1) * binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 4, 15, 30, 66, 115, 200, 295, 471, 659, 946, 1259, 1715, 2194, 2920, 3591, 4561, 5585, 6916, 8216, 10082, 11823, 14124, 16389, 19350, 22174, 26004, 29435, 33931, 38445, 43902, 48925, 55767, 61941, 69831, 77275, 86415, 94968, 106094, 115874, 128216, 140214, 154405
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Links

Crossrefs

Partial sums of A366813.
Cf. A078471, A366395, A366723.

Programs

  • Mathematica
    Array[Sum[(-1)^(k - 1)*Binomial[Floor[#/k] + 3, 4], {k, #}] &, 56] (* Michael De Vlieger, Oct 25 2023 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k-1)*binomial(n\k+3, 4));

Formula

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

A365007 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+1,2).

Original entry on oeis.org

1, 2, 7, 6, 16, 17, 29, 22, 52, 42, 67, 57, 92, 79, 142, 86, 154, 143, 191, 146, 266, 189, 277, 217, 341, 262, 430, 279, 436, 402, 497, 342, 634, 444, 674, 507, 704, 553, 878, 562, 862, 766, 947, 677, 1222, 807, 1129, 857, 1254, 992, 1486, 942, 1432, 1250, 1622, 1079
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Links

Crossrefs

Partial sums give A366395.
Cf. A000593, A366813, A366814.
Cf. A007437.

Programs

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

Formula

G.f.: -Sum_{k>=1} (-x)^k/(1-x^k)^3 = 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).

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).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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