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 1-6 of 6 results.

A317843 Dirichlet inverse of Stern's diatomic sequence (A002487).

Original entry on oeis.org

1, -1, -2, 0, -3, 2, -3, 0, 0, 3, -5, 0, -5, 3, 8, 0, -5, 0, -7, 0, 4, 5, -7, 0, 2, 5, 0, 0, -7, -8, -5, 0, 14, 5, 9, 0, -11, 7, 10, 0, -11, -4, -13, 0, -8, 7, -9, 0, 0, -2, 8, 0, -13, 0, 19, 0, 18, 7, -11, 0, -9, 5, 14, 0, 23, -14, -11, 0, 14, -9, -13, 0, -15, 11, -20, 0, 13, -10, -13, 0, 2, 11, -19, 0, 9, 13, 10, 0, -17, 8, 11, 0
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Links

Crossrefs

Cf. A002487, A317839, A317841, A317842.

Programs

  • Mathematica
    s[0] = 0; s[1] = 1;
s[n_] := s[n] = If[EvenQ[n], s[n/2], s[(n-1)/2] + s[(n+1)/2]];
a[n_] := a[n] = If[n == 1, 1, -Sum[s[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)
  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317843(n) = if(1==n,1,-sumdiv(n,d,if(dA002487(n/d)*A317843(d),0)));

Formula

a(1) = 1; for n > 1, a(n) = -Sum_{d|n, dA002487(n/d)*a(d).

A317839 Möbius transform of A002487, Stern's Diatomic sequence.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 6, 0, 4, 0, 6, 0, 4, 0, 4, 0, 6, 0, 4, 0, 0, 0, 4, 0, 10, 0, 4, 0, 10, 0, 12, 0, 6, 0, 8, 0, 6, 0, 6, 0, 12, 0, 4, 0, 2, 0, 10, 0, 8, 0, -4, 0, 0, 0, 10, 0, 6, 0, 12, 0, 14, 0, 10, 0, 10, 0, 12, 0, 6, 0, 18, 0, 14, 0, 10, 0, 16, 0, 12, 0, 10, 0, 2, 0, 10, 0, 8, 0, 18, 0, 16, 0, 4
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Links

Crossrefs

Cf. A002487, A008683, A317837, A317838, A317840, A317841, A317843.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317839(n) = sumdiv(n,d,moebius(n/d)*A002487(d));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A002487(d).
a(n) = A000010(n) - A317841(n).

A317837 a(n) = Sum_{d|n, dA002487(d).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 3, 5, 1, 7, 1, 5, 6, 4, 1, 10, 1, 9, 6, 7, 1, 10, 4, 7, 7, 9, 1, 16, 1, 5, 8, 7, 7, 17, 1, 9, 8, 13, 1, 20, 1, 13, 14, 9, 1, 13, 4, 15, 8, 13, 1, 22, 9, 13, 10, 9, 1, 26, 1, 7, 18, 6, 9, 22, 1, 13, 10, 23, 1, 24, 1, 13, 17, 17, 9, 26, 1, 17, 15, 13, 1, 34, 9, 15, 10, 19, 1, 40, 9, 17, 8, 11, 11
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Links

Crossrefs

Cf. A002487, A293216, A317838, A317839, A317840, A317841, A317843.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317837(n) = sumdiv(n,d,(dA002487(d));

Formula

a(n) = Sum_{d|n, dA002487(d).
a(n) = A317838(n) - A002487(n).
a(n) = A001222(A293216(n)).

A317838 a(n) = Sum_{d|n} A002487(d).

Original entry on oeis.org

1, 2, 3, 3, 4, 6, 4, 4, 7, 8, 6, 9, 6, 8, 10, 5, 6, 14, 8, 12, 14, 12, 8, 12, 11, 12, 15, 12, 8, 20, 6, 6, 14, 12, 16, 21, 12, 16, 18, 16, 12, 28, 14, 18, 26, 16, 10, 15, 13, 22, 20, 18, 14, 30, 20, 16, 20, 16, 12, 30, 10, 12, 24, 7, 16, 28, 12, 18, 24, 32, 14, 28, 16, 24, 35, 24, 26, 36, 14, 20, 29, 24, 20, 42, 30, 28, 28, 24
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Comments

Inverse Möbius transform of A002487, Stern's Diatomic sequence.

Links

Crossrefs

Cf. A002487, A317837, A317839, A317840, A317841, A317843.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317838(n) = sumdiv(n,d,A002487(d));

Formula

a(n) = Sum_{d|n} A002487(d).
a(n) = A317837(n) + A002487(n).

A317840 Difference between Stern's Diatomic sequence (A002487) and its Möbius transform (A317839).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 4, 1, 1, 4, 1, 3, 4, 5, 1, 2, 3, 5, 4, 3, 1, 4, 1, 1, 6, 5, 5, 4, 1, 7, 6, 3, 1, 8, 1, 5, 6, 7, 1, 2, 3, 7, 6, 5, 1, 8, 7, 3, 8, 7, 1, 4, 1, 5, 10, 1, 7, 6, 1, 5, 8, 9, 1, 4, 1, 11, 8, 7, 7, 10, 1, 3, 8, 11, 1, 8, 7, 13, 8, 5, 1, 12, 7, 7, 6, 9, 9, 2, 1, 9, 8, 7, 1, 12, 1, 5, 14
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Links

Crossrefs

Cf. A002487, A008683, A317837, A317838, A317839, A317841.

Programs

  • PARI
    A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A317840(n) = -sumdiv(n,d,(dA002487(d));

Formula

a(n) = -Sum_{d|n, dA008683(n/d)*A002487(d).
a(n) = A002487(n) - A317839(n).

A317842 Dirichlet convolution of A284013 (= n - A002487(n)) with itself, where A002487 is Stern's Diatomic sequence.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 6, 1, 4, 0, 14, 0, 8, 4, 23, 0, 18, 0, 26, 8, 12, 0, 58, 4, 16, 10, 46, 0, 52, 0, 72, 12, 24, 16, 94, 0, 24, 16, 104, 0, 80, 0, 70, 42, 32, 0, 190, 16, 64, 24, 90, 0, 106, 24, 172, 24, 44, 0, 248, 0, 52, 66, 201, 32, 136, 0, 130, 32, 152, 0, 342, 0, 52, 80, 134, 48, 164, 0, 334, 63, 60, 0, 364, 48, 60, 44
Offset: 1

Views

Author

Antti Karttunen, Aug 09 2018

Keywords

Links

Crossrefs

Cf. A002487, A284013, A317839, A317841, A317843.

Programs

Formula

a(n) = Sum_{d|n} A284013(n/d)*A284013(d).
Showing 1-6 of 6 results.

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