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.

A034729 a(n) = Sum_{ k, k|n } 2^(k-1).

Original entry on oeis.org

1, 3, 5, 11, 17, 39, 65, 139, 261, 531, 1025, 2095, 4097, 8259, 16405, 32907, 65537, 131367, 262145, 524827, 1048645, 2098179, 4194305, 8390831, 16777233, 33558531, 67109125, 134225995, 268435457, 536887863, 1073741825, 2147516555, 4294968325, 8590000131
Offset: 1

Views

Author

Erich Friedman

Keywords

Comments

Dirichlet convolution of b_n=1 with c_n = 2^(n-1).
Equals row sums of triangle A143425, & inverse Möbius transform (A051731) of [1, 2, 4, 8, ...]. - Gary W. Adamson, Aug 14 2008
Number of constant multiset partitions of normal multisets of size n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Sep 16 2018

Examples

			From _Gus Wiseman_, Sep 16 2018: (Start)
The a(4) = 11 constant multiset partitions:
  (1)(1)(1)(1)
    (11)(11)
    (12)(12)
     (1111)
     (1222)
     (1122)
     (1112)
     (1233)
     (1223)
     (1123)
     (1234)
(End)

Links

Crossrefs

Cf. A000005, A002033, A003238, A008480, A018783, A047968, A051731.
Cf. A052409, A055895, A078392, A143425, A215366, A245282, A248906.
Cf. A289508.
Sums of the form Sum_{d|n} q^(d-1): this sequence (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

Programs

  • Magma
    A034729:= func< n | (&+[2^(d-1): d in Divisors(n)]) >;
[A034729(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024
  • Maple
    seq(add(2^(k-1),k=numtheory:-divisors(n)), n = 1 .. 100); # Robert Israel, Aug 22 2014
  • Mathematica
    Rest[CoefficientList[Series[Sum[x^k/(1-2*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 08 2014 *)
  • PARI
    A034729(n) = sumdiv(n,k,2^(k-1)) \\ Michael B. Porter, Mar 11 2010
  • PARI
    {a(n)=polcoeff(sum(m=1,n,2^(m-1)*x^m/(1-x^m +x*O(x^n))),n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014
  • PARI
    {a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,1/(1 - x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,40,print1(a(n),", ")) \\ Paul D. Hanna, Aug 21 2014
  • Python
    from sympy import divisors
def A034729(n): return sum(1<<(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 15 2022
  • SageMath
    def A034729(n): return sum(2^(k-1) for k in (1..n) if (k).divides(n))
[A034729(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

Formula

G.f.: Sum_{n>0} x^n/(1-2*x^n). - Vladeta Jovovic, Nov 14 2002
a(n) = 1/2 * A055895(n). - Joerg Arndt, Aug 14 2012
G.f.: Sum_{n>=1} 2^(n-1) * x^n / (1 - x^n). - Paul D. Hanna, Aug 21 2014
G.f.: Sum_{n>=1} x^n * Sum_{d|n} 1/(1 - x^d)^(n/d). - Paul D. Hanna, Aug 21 2014
a(n) ~ 2^(n-1). - Vaclav Kotesovec, Sep 09 2014
a(n) = Sum_{k in row n of A215366} A008480(k) * A000005(A289508(k)). - Gus Wiseman, Sep 16 2018
a(n) = Sum_{c is a composition of n} A000005(gcd(c)). - Gus Wiseman, Sep 16 2018

A007435 Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...

Original entry on oeis.org

1, 2, 3, 5, 6, 12, 14, 26, 37, 62, 90, 159, 234, 392, 618, 1013, 1598, 2630, 4182, 6830, 10962, 17802, 28658, 46548, 75031, 121628, 196455, 318206, 514230, 832722, 1346270, 2179322, 3524670, 5704486, 9227484, 14933129, 24157818, 39092352, 63246222, 102341006
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

For p prime, a(p) == k (mod p) where k = 0 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 1 if p = 5. - Michael Somos, Apr 15 2012

Examples

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 12*x^6 + 14*x^7 + 26*x^8 + 37*x^9 + 62*x^10 + ...

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000045, A051731, A127647, A245282, A108046, A034772, A100107, A108031, A008683.

Programs

  • Maple
    with(numtheory): seq(add(combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 12 2025
  • Mathematica
    Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */

Formula

Row sums of A051731 * A127647. - Gary W. Adamson, Jan 22 2007
G.f.: Sum_{k>0} Fibonacci(k)*x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k-x^(2*k)). - Vladeta Jovovic, Dec 17 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(Fibonacci(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018
a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2018
From Ridouane Oudra, Apr 12 2025 : (Start)
a(n) = Sum_{d|n} Fibonacci(d).
a(n) = Sum_{d|n} mu(d)*A034772(n/d).
a(n) = A245282(n) - A108046(n).
a(n) = 2*A245282(n) - A100107(n).
a(n) = (A108031(n) + A108046(n))/2. (End)

Extensions

More terms from Joerg Arndt, Aug 14 2012

A108046 Inverse Moebius transform of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, ...

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 16, 22, 38, 55, 98, 144, 242, 381, 626, 987, 1625, 2584, 4221, 6774, 11002, 17711, 28768, 46371, 75170, 121415, 196662, 317811, 514650, 832040, 1346895, 2178365, 3525566, 5702898, 9229181, 14930352, 24160402, 39088314, 63250220, 102334155
Offset: 1

Views

Author

Emeric Deutsch, Jun 01 2005

Keywords

Examples

			a(4)=3 because the divisors of 4 are 1,2,4 and the first, second and fourth Fibonacci numbers are 0, 1 and 2, respectively, having sum 3.

Links

Crossrefs

Cf. A000045, A007435, A245282.

Programs

  • Maple
    with(combinat): with(numtheory): f:=n->fibonacci(n-1): g:=proc(n) local div: div:=divisors(n): sum(f(div[j]),j=1..tau(n)) end: seq(g(n),n=1..45);
  • Mathematica
    a[n_] := DivisorSum[n, Fibonacci[#-1]&]; Array[a, 40] (* Jean-François Alcover, Dec 17 2015 *)
  • PARI
    a(n)=if(n<1,1,sumdiv(n,d,fibonacci(d-1))); /* Joerg Arndt, Aug 14 2012 */
  • Python
    from sympy import fibonacci, divisors
def a(n): return 1 if n<1 else sum([fibonacci(d - 1) for d in divisors(n)]) # Indranil Ghosh, May 23 2017

Formula

G.f.: Sum_{k>=1} Fibonacci(k-1)*x^k/(1 - x^k). - Ilya Gutkovskiy, May 23 2017
a(n) = Sum_{d|n} Fibonacci(d-1). - Ridouane Oudra, Apr 11 2025

A256272 G.f.: Sum_{n>=1} Pell(n+1) * x^n / (1 - x^n), where Pell(n) = A000129(n).

Original entry on oeis.org

2, 7, 14, 36, 72, 188, 410, 1021, 2392, 5818, 13862, 33678, 80784, 195440, 470916, 1137710, 2744212, 6627675, 15994430, 38619812, 93222780, 225072548, 543339722, 1311772784, 3166816034, 7645450834, 18457558444, 44560677618, 107578520352, 259717999680, 627013566050, 1513745792655, 3654502889812
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2015

Keywords

Examples

			G.f.: A(x) = 2*x + 7*x^2 + 14*x^3 + 36*x^4 + 72*x^5 + 188*x^6 +...
where by definition
A(x) = 2*x/(1-x) + 5*x^2/(1-x^2) + 12*x^3/(1-x^3) + 29*x^4/(1-x^4) + 70*x^5/(1-x^5) + 169*x^6/(1-x^6) + 408*x^7/(1-x^7) + 985*x^8/(1-x^8) + 2378*x^9/(1-x^9) + 5741*x^10/(1-x^10) +...+ Pell(n+1)*x^n/(1-x^n) +...
The g.f. is also given by the series identity:
A(x) = x*(2+x)/(1-2*x-x^2) + x^2*(2+x^2)/(1-2*x^2-x^4) + x^3*(2+x^3)/(1-2*x^3-x^6) + x^4*(2+x^4)/(1-2*x^4-x^8) + x^5*(2+x^5)/(1-2*x^5-x^10) + x^6*(2+x^6)/(1-2*x^6-x^12) + x^7*(2+x^7)/(1-2*x^7-x^14) +...+ x^n*(1+x^n)/(1-x^n-x^(2*n)) +...
And also we have the series:
A(x) = x*(2 + x) + x^2*((2+x)^2 + (2+x^2)) + x^3*((2+x)^3 + (2+x^3))
+ x^4*((2+x)^4 + (2+x^2)^2 + (2+x^4)) + x^5*((2+x)^5 + (2+x^5))
+ x^6*((2+x)^6 + (2+x^2)^3 + (2+x^3)^2 + (2+x^6))
+ x^7*((2+x)^7 + (2+x^7))
+ x^8*((2+x)^8 + (2+x^2)^4 + (2+x^4)^2 + (2+x^8))
+ x^9*((2+x)^9 + (2+x^3)^3 + (2+x^9))
+ x^10*((2+x)^10 + (2+x^2)^5 + (2+x^5)^2 + (2+x^10))
+ x^11*((2+x)^11 + (2+x^11))
+ x^12*((2+x)^12 + (2+x^2)^6 + (2+x^3)^4 + (2+x^4)^3 + (2+x^6)^2 + (2+x^12))
+...+ x^n * Sum_{d|n} (2 + x^d)^(n/d) +...
or, more explicitly,
A(x) = x*(2 + x) + x^2*(6 + 4*x + 2*x^2)
+ x^3*(10 + 12*x + 6*x^2 + 2*x^3)
+ x^4*(22 + 32*x + 28*x^2 + 8*x^3 + 3*x^4)
+ x^5*(34 + 80*x + 80*x^2 + 40*x^3 + 10*x^4 + 2*x^5)
+ x^6*(78 + 192*x + 252*x^2 + 164*x^3 + 66*x^4 + 12*x^5 + 4*x^6)
+ x^7*(130 + 448*x + 672*x^2 + 560*x^3 + 280*x^4 + 84*x^5 + 14*x^6 + 2*x^7)
+ x^8*(278 + 1024*x + 1824*x^2 + 1792*x^3 + 1148*x^4 + 448*x^5 + 120*x^6 + 16*x^7 + 4*x^8)
+ x^9*(522 + 2304*x + 4608*x^2 + 5388*x^3 + 4032*x^4 + 2016*x^5 + 678*x^6 + 144*x^7 + 18*x^8 + 3*x^9) +...

Links

Crossrefs

Cf. A256281, A245282, A000129.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Sum[(x^k*(2+x^k))/(1-2*x^k-x^(2*k)), {k, 1, n}], {x, 0, n}]; Array[a, 40] (* Jean-François Alcover, Dec 19 2015 *)
  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(sum(m=1, n, Pell(m+1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1, 40, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=1, n, x^m*(2+x^m)/(1-2*x^m-x^(2*m) +x*O(x^n)) ), n)}
for(n=1, 40, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=x+x^2); A=sum(m=1, n, x^m*sumdiv(m, d, (2 + x^(m/d) +x*O(x^n))^d) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=1} x^n * (2 + x^n) / (1 - 2*x^n - x^(2*n)).
G.f.: Sum_{n>=1} x^n * Sum_{d|n} (2 + x^d)^(n/d).
a(2*n^2) == 1 (mod 2), with a(n) == 0 (mod 2) elsewhere.
a(n) ~ (1+sqrt(2))^(n+1) / (2*sqrt(2)). - Vaclav Kotesovec, Jun 02 2015
Showing 1-4 of 4 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.