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.

A134225 A007436 + A134082 - A000012 as infinite lower triangular matrices; where A000012 = (1; 1,1; 1,1,1; ...).

Original entry on oeis.org

1, 3, 1, 2, 5, 1, 3, 2, 7, 1, 4, 3, 2, 9, 1, 5, 4, 3, 2, 11, 1, 6, 5, 4, 3, 2, 13, 1, 7, 6, 5, 4, 3, 2, 15, 1, 8, 7, 6, 5, 4, 3, 2, 17, 1, 9, 8, 7, 6, 5, 4, 3, 2, 19, 1
Offset: 1

Views

Author

Gary W. Adamson, Oct 14 2007

Keywords

Comments

Row sums = A034856: (1, 4, 8, 13, 19, 26, ...).

Examples

			First few rows of the triangle:
  1;
  3, 1;
  2, 5, 1;
  3, 2, 7, 1;
  4, 3, 2, 9,  1;
  5, 4, 3, 2, 11,  1;
  6, 5, 4, 3,  2, 13,  1;
  7, 6, 5, 4,  3,  2, 15, 1;
  ...

Crossrefs

Cf. A007436, A134082, A034856.

A130116 Inverse Moebius transform of a diagonalized matrix of A007436.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 0, 4, 1, 0, 1, 0, 0, 6, 1, 0, 0, 0, 0, 0, 12, 1, 0, 0, 2, 0, 0, 0, 18, 1, 0, 1, 0, 0, 0, 0, 0, 32, 1, 0, 0, 0, 4, 0, 0, 0, 0, 50
Offset: 1

Views

Author

Gary W. Adamson, May 09 2007

Keywords

Comments

Row sums = F(n); example: row 6 = F(6), 8 = (1 + 0 + 1 + 0 + 0 + 6).
Right border = A007436, (1, 0, 1, 2, 4, 6, 12, 18, 32, ...), the Moebius transform of the Fibonacci series.

Examples

			First few rows of the triangle:
  1;
  1,  0;
  1,  0,  1;
  1,  0,  0,  2;
  1,  0,  0,  0,  4;
  1,  0,  1,  0,  0,  6;
  1,  0,  0,  0,  0,  0, 12;
  1,  0,  0,  2,  0,  0,  0, 18;
  ...

Crossrefs

Cf. A007436, A051731, A054525.

Formula

A051731 * an infinite lower triangular matrix with A007436 in the main diagonal and the rest zeros.

A108700 Number of compositions of n into odd and relatively prime parts.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 12, 21, 32, 54, 88, 141, 232, 376, 604, 987, 1596, 2576, 4180, 6762, 10932, 17710, 28656, 46347, 75020, 121392, 196384, 317808, 514228, 831978, 1346268, 2178309, 3524488, 5702886, 9227448, 14930208, 24157816, 39088168
Offset: 1

Views

Author

Vladeta Jovovic, Jun 19 2005

Keywords

Links

Crossrefs

Cf. A000740, A007436, A000045, A008683.

Programs

  • Maple
    with(numtheory): seq(-add(mobius(2*d)*combinat[fibonacci](n/d), d in divisors(n)), n=1..80); # Ridouane Oudra, Apr 12 2025
  • Mathematica
    a[n_] := a[n] = Module[{d = Divisors[n]}, m = Plus @@ (MoebiusMu /@ (n/d)*Fibonacci /@ d); If[ OddQ[n], m, a[n/2] + m]]; Table[ a[n], {n, 38}] (* Robert G. Wilson v, Jun 21 2005 *)

Formula

a(n) = A007436(n) if n is odd, else a(n) = a(n/2) + A007436(n).
a(n) = - Sum_{d|n} mu(2*d)*Fibonacci(n/d). - Ridouane Oudra, Apr 12 2025

Extensions

More terms from Robert G. Wilson v, Jun 21 2005

A130095 Inverse Möbius transform of odd-indexed Fibonacci numbers.

Original entry on oeis.org

1, 3, 6, 16, 35, 97, 234, 626, 1603, 4218, 10947, 28767, 75026, 196654, 514269, 1346895, 3524579, 9229159, 24157818, 63250217, 165580380, 433505386, 1134903171, 2971244450, 7778742084, 20365086102, 53316292776, 139584059112, 365435296163, 956722544582
Offset: 1

Views

Author

Gary W. Adamson, May 06 2007

Keywords

Comments

Original name was: A051731 * A007436.
Conjecture: a(n)/a(n-1) tends to phi^2.

Examples

			The divisors of 6 are 1, 2, 3 and 6. Hence
a(6) = Fibonacci(1) + Fibonacci(3) + Fibonacci(5) + Fibonacci(11) = 97.

Crossrefs

Cf. A001519, A007435, A051731.

Programs

  • Maple
    #A130095
with(combinat): with(numtheory):
f := n -> fibonacci(2*n-1):
g := proc (n) local div; div := divisors(n):
add(f(div[j]), j = 1 .. tau(n)) end proc:
seq(g(n), n = 1 .. 30); # Peter Bala, Mar 26 2015

Formula

From Peter Bala, Mar 26 2015: (Start)
a(n) = sum {d | n} Fibonacci(2*d - 1).
O.g.f. Sum_{n >= 1} Fibonacci(2*n - 1)*x^n/(1 - x^n) = Sum_{n >= 1} x^n*(1 - x^n)/(1 - 3*x^n + x^(2*n)).
Sum_{n >= 1} a(n)*x^(2*n) = Sum_{n >= 1} x^n/( 1/(x^n - 1/x^n) - (x^n - 1/x^n) ).
For p prime, a(p) == k (mod p) where k = 3 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 0 if p = 5. (End)

Extensions

Incorrect original name removed and terms a(11) - a(30) added by Peter Bala, Mar 26 2015

A134545 A051731 * A004736.

Original entry on oeis.org

1, 3, 1, 4, 2, 1, 7, 4, 2, 1, 6, 4, 3, 2, 1, 12, 8, 5, 3, 2, 1, 8, 6, 5, 4, 3, 2, 1, 15, 11, 8, 6, 4, 3, 2, 1, 13, 10, 8, 6, 5, 4, 3, 2, 1, 18, 14, 11, 9, 7, 5, 4, 3, 2, 1
Offset: 1

Views

Author

Gary W. Adamson, Oct 31 2007

Keywords

Comments

Row sums = A007437: (1, 4, 7, 14, 16, 31, ...).
Left border = sigma(n), A000203: (1, 3, 4, 7, 6, 12, ...).

Examples

			First few rows of the triangle:
   1;
   3,  1;
   4,  2, 1;
   7,  4, 2, 1;
   6,  4, 3, 2, 1;
  12,  8, 5, 3, 2, 1;
   8,  6, 5, 4, 3, 2, 1;
  15, 11, 8, 6, 4, 3, 2, 1;
  ...

Crossrefs

Cf. A051731, A004736, A007436, A000203.

Formula

A051731 * A004736 as infinite lower triangular matrices.

A381936 Number of primitive binary words of length n that avoid 11, start with 1 and end with 0.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 8, 11, 20, 30, 55, 83, 144, 224, 373, 597, 987, 1572, 2584, 4146, 6756, 10890, 17711, 28557, 46365, 74880, 121372, 196184, 317811, 513818, 832040, 1345659, 2178253, 3523590, 5702876, 9225784, 14930352, 24155232, 39088024, 63241794, 102334155, 165573148, 267914296
Offset: 1

Views

Author

Aidan Diekmann, Mar 10 2025

Keywords

Comments

Here primitive means the word is not two or more repetitions of a smaller word.

Examples

			For n=5, the a(6) = 3 words are: 100000, 100010, 101000.
Notice 100100 is not included since it is repetitions of the smaller word 100 (from n=3).

Crossrefs

Cf. A000045, A007436, A008683, A056267, A113166.

Programs

  • PARI
    a(n) = sumdiv(n,d,moebius(d)*fibonacci(n/d-1)) \\ Andrew Howroyd, Mar 10 2025
  • Python
    from sympy import mobius, fibonacci, divisors
def A381936(n): return sum(mobius(n//d)*fibonacci(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Mar 18 2025

Formula

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

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

Content is available under The OEIS End-User License Agreement.