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.

A307776 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+1)*a(d).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 0, 1, -2, -1, 0, 1, 0, 1, 1, 1, -3, -2, 0, 1, 2, 3, 3, 4, 1, 1, 1, 0, -2, -1, -2, -1, -6, -5, -2, -2, 2, 3, 3, 4, 3, 4, 3, 4, 0, -1, -4, -3, -11, -10, -11, -13, -15, -14, -15, -15, -18, -17, -15, -14, -11, -10, -8, -7, -11, -11, -2, -1, 6, 10, 13, 14, 21, 22, 20
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307777, A307778, A307779.

Programs

  • Maple
    with(numtheory): P:=proc(q) local a,d,n; a:=[1]:
for n from 1 to q do a:=[op(a),add((-1)^(n/d+1)*a[d],d=divisors(n))]:
od; op(a); end: P(74); # Paolo P. Lava, Apr 30 2019
  • Mathematica
    a[n_] := a[n] = Sum[(-1)^((n - 1)/d + 1) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 75}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[a[k] x^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 75}]

Formula

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 + x^n)).
L.g.f.: log(Product_{n>=1} (1 + x^n)^(a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A307779 a(1) = 1; a(n+1) = Sum_{d|n, n/d odd} a(d).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 11, 12, 13, 17, 17, 18, 22, 23, 25, 31, 32, 33, 38, 41, 42, 49, 51, 52, 63, 64, 64, 74, 75, 82, 93, 94, 95, 108, 113, 114, 130, 131, 133, 155, 156, 157, 174, 179, 187, 206, 208, 209, 231, 242, 247, 271, 272, 273, 307, 308, 309, 345, 345
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A002033, A003238, A307776, A307777, A307778.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[Boole[OddQ[(n - 1)/d]] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 65}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[a[k] x^k/(1 - x^(2 k)), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 65}]

Formula

G.f.: x * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^(2*n))).
L.g.f.: log(Product_{n>=1} ((1 + x^n)/(1 - x^n))^(a(n)/(2*n))) = Sum_{n>=1} a(n+1)*x^n/n.

A307777 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d+d)*a(d).

Original entry on oeis.org

1, 1, -2, -1, 1, 2, -2, -1, 0, -1, -2, -1, 6, 7, -7, -7, 5, 6, -8, -7, 6, 3, -3, -2, 5, 7, -15, -16, 26, 27, -22, -21, 12, 9, -16, -16, 28, 29, -23, -18, 9, 10, -23, -22, 28, 21, -20, -19, 25, 24, -31, -27, 29, 30, -23, -23, 16, 7, -35, -34, 79, 80, -60, -57, 27, 35, -50, -49, 54, 50, -41
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 28 2019

Keywords

Crossrefs

Cf. A003238, A281487, A307776, A307778, A307779.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[(-1)^((n - 1)/d + d) a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 71}]
a[n_] := a[n] = SeriesCoefficient[x (1 - Sum[a[k] (-x)^k/(1 + x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 71}]

Formula

G.f.: x * (1 - Sum_{n>=1} a(n)*(-x)^n/(1 + x^n)).
L.g.f.: log(Product_{n>=1} (1 + x^n)^((-1)^(n+1)*a(n)/n)) = Sum_{n>=1} a(n+1)*x^n/n.

A343370 a(1) = 1; a(n) = Sum_{d|n, d < n} (-1)^d * a(d).

Original entry on oeis.org

1, -1, -1, -2, -1, -1, -1, -4, 0, -1, -1, -4, -1, -1, 1, -8, -1, -2, -1, -4, 1, -1, -1, -12, 0, -1, 0, -4, -1, -3, -1, -16, 1, -1, 1, -10, -1, -1, 1, -12, -1, -3, -1, -4, 0, -1, -1, -32, 0, -2, 1, -4, -1, -4, 1, -12, 1, -1, -1, -16, -1, -1, 0, -32, 1, -3, -1, -4, 1, -3, -1, -36, -1, -1, 0, -4, 1, -3, -1, -32, 0
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 12 2021

Keywords

Links

Crossrefs

Cf. A008683, A053850 (positions of 0's), A056913 (positions of 1's), A067856, A074206, A307778, A308077, A325144.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<2, 1,
      add((-1)^d*a(d), d=numtheory[divisors](n) minus {n}))
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 12 2021
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[d < n, (-1)^d a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 70}]
  • PARI
    memoA343370 = Map();
A343370(n) = if(1==n,1,my(v); if(mapisdefined(memoA343370,n,&v), v, v = sumdiv(n,d,if(dA343370(d),0)); mapput(memoA343370,n,v); (v))); \\ Antti Karttunen, Jan 02 2023

Formula

G.f.: x + Sum_{n>=1} (-1)^n * a(n) * x^(2*n) / (1 - x^n).

Extensions

Data section extended up to a(81) by Antti Karttunen, Jan 02 2023

A351407 a(1) = 1; a(n+1) = Sum_{d|n} (-1)^(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -3, 3, -4, 8, -9, 6, -9, 14, -15, 16, -17, 27, -33, 21, -22, 36, -37, 34, -45, 61, -62, 51, -55, 73, -82, 76, -77, 124, -125, 80, -97, 120, -132, 132, -133, 171, -190, 153, -154, 221, -222, 194, -233, 296, -297, 239, -248, 313, -337, 301, -302
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 10 2022

Keywords

Crossrefs

Cf. A003238, A067856, A281487, A307776, A307778, A308077, A325144, A343370.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1,
      add((-1)^((n-1)/d)*a(d), d=numtheory[divisors](n-1)))
    end:
seq(a(n), n=1..54);  # Alois P. Heinz, Feb 10 2022
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[(-1)^((n - 1)/d) a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 54}]
nmax = 54; A[] = 0; Do[A[x] = x (1 + Sum[(-1)^k A[x^k], {k, 1, nmax}]) + O[x]^(nmax + 1) //Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

Formula

G.f. A(x) satisfies: A(x) = x * ( 1 - A(x) + A(x^2) - A(x^3) + A(x^4) - A(x^5) + ... ).
G.f.: x * ( 1 - Sum_{n>=1} a(n) * x^n / (1 + x^n) ).
Showing 1-5 of 5 results.

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