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 1-10 of 13 results. Next

A100107 Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..

Original entry on oeis.org

1, 4, 5, 11, 12, 26, 30, 58, 81, 138, 200, 355, 522, 876, 1380, 2265, 3572, 5880, 9350, 15272, 24510, 39806, 64080, 104084, 167773, 271968, 439285, 711530, 1149852, 1862022, 3010350, 4873112, 7881400, 12755618, 20633280, 33391491, 54018522, 87413156
Offset: 1

Views

Author

Jonathan Vos Post, Dec 26 2004

Keywords

Examples

			a(2) = 4 because the prime 2 is divisible only by 1 and 2, so L(1) + L(2) = 1 + 3 = 4.
a(3) = 5 because the prime 3 is divisible only by 1 and 3, so L(1) + L(3) = 1 + 4 = 5.
a(4) = 11 because the semiprime 4 is divisible only by 1, 2, 4, so L(1) + L(2) + L(4) = 1 + 3 + 7 = 11.

Links

Crossrefs

Cf. A000032, A007435, A100279.

Programs

  • Maple
    with(numtheory): with(combinat): a:=proc(n) local div: div:=divisors(n): sum(2*fibonacci(div[j]+1)-fibonacci(div[j]),j=1..tau(n)) end: seq(a(n),n=1..42); # Emeric Deutsch, Jul 31 2005
  • Mathematica
    Table[Plus @@ Map[Function[d, LucasL[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)

Formula

a(n) = Sum_{d|n} Lucas(d) = Sum_{d|n} A000032(d).
G.f.: Sum_{k>=1} Lucas(k) * x^k/(1 - x^k) = Sum_{k>=1} x^k * (1 + 2*x^k)/(1 - x^k - x^(2*k)). - Ilya Gutkovskiy, Aug 14 2019
a(n) ~ phi^n, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 10 2021

Extensions

More terms from Emeric Deutsch, Jul 31 2005

A111075 a(n) = F(n) * Sum_{k|n} 1/F(k), where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

1, 2, 3, 7, 6, 21, 14, 50, 52, 122, 90, 427, 234, 784, 1038, 2351, 1598, 6860, 4182, 17262, 17262, 35622, 28658, 139703, 90031, 243308, 300405, 766850, 514230, 2367006, 1346270, 5188658, 5326470, 11409346, 11782764, 44717548, 24157818
Offset: 1

Views

Author

Leroy Quet, Oct 10 2005

Keywords

Comments

a(n) = a(n+1) for n = 20, but for no other n < 25000. - Klaus Brockhaus, Oct 11 2005
If k|n then F(k)|F(n). Therefore A111075(n) = F(n) * sum{k|n} 1/F(k) = sum{k|n} F(n)/F(k) is a sum of integers. - Max Alekseyev, Oct 22 2005

Examples

			a(6) = F(6) sum{k|6} 1/F(k) = F(6) * (1/F(1) + 1/F(2) + 1/F(3) + 1/F(6)) = 8 * (1/1 + 1/1 + 1/2 + 1/8) = 21.

Links

Crossrefs

Cf. A000045, A007435, A111159, A000204 (Lucas), A203318.

Programs

  • Maple
    with(combinat): with(numtheory): a:=proc(n) local div: div:=divisors(n): fibonacci(n)*sum(1/fibonacci(div[j]),j=1..tau(n)) end: seq(a(n),n=1..40); # Emeric Deutsch, Oct 11 2005
# second Maple program:
a:= n-> (F-> F(n)*add(1/F(d),d=numtheory[divisors(n)))(
         combinat[fibonacci]):
seq(a(n), n=1..42);  # Alois P. Heinz, Aug 20 2019
  • Mathematica
    f[n_] := Fibonacci[n]*Plus @@ (1/Fibonacci /@ Divisors[n]); Table[ f[n], {n, 37}] (* Robert G. Wilson v, Oct 11 2005 *)
  • PARI
    {for(n=1,37,d=divisors(n);print1(fibonacci(n)*sum(j=1,length(d), 1/fibonacci(d[j])),","))}
  • PARI
    {a(n)=fibonacci(n) * sumdiv(n,d, 1/fibonacci(d))} /* Paul D. Hanna, Oct 11 2005 */
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n, x^m/(1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))),n)} /* Paul D. Hanna, Oct 11 2005 */

Formula

G.f.: Sum_{n>=1} x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000204(n). [Paul D. Hanna, Jan 09 2012]

Extensions

More terms from Robert G. Wilson v, Emeric Deutsch, Paul D. Hanna and Klaus Brockhaus, Oct 11 2005

A245282 G.f.: Sum_{n>=1} Fibonacci(n+1) * x^n / (1 - x^n).

Original entry on oeis.org

1, 3, 4, 8, 9, 19, 22, 42, 59, 100, 145, 257, 378, 634, 999, 1639, 2585, 4255, 6766, 11051, 17736, 28804, 46369, 75316, 121402, 196798, 317870, 514868, 832041, 1347372, 2178310, 3526217, 5703035, 9230052, 14930382, 24162310, 39088170, 63252754, 102334536, 165591226, 267914297
Offset: 1

Views

Author

Paul D. Hanna, Aug 21 2014

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 4*x^3 + 8*x^4 + 9*x^5 + 19*x^6 + 22*x^7 +...
where by definition
A(x) = 1*x/(1-x) + 2*x^2/(1-x^2) + 3*x^3/(1-x^3) + 5*x^4/(1-x^4) + 8*x^5/(1-x^5) + 13*x^6/(1-x^6) + 21*x^7/(1-x^7) + 34*x^8/(1-x^8) + 55*x^9/(1-x^9) + 89*x^10/(1-x^10) + 144*x^11/(1-x^11) +...+ Fibonacci(n+1)*x^n/(1-x^n) +...
The g.f. is also given by the series identity:
A(x) = x*(1+x)/(1-x-x^2) + x^2*(1+x^2)/(1-x^2-x^4) + x^3*(1+x^3)/(1-x^3-x^6) + x^4*(1+x^4)/(1-x^4-x^8) + x^5*(1+x^5)/(1-x^5-x^10) + x^6*(1+x^6)/(1-x^6-x^12) + x^7*(1+x^7)/(1-x^7-x^14) +...+ x^n*(1+x^n)/(1-x^n-x^(2*n)) +...
And also we have the series:
A(x) = x*(1 + x) + x^2*((1+x)^2 + (1+x^2)) + x^3*((1+x)^3 + (1+x^3))
+ x^4*((1+x)^4 + (1+x^2)^2 + (1+x^4)) + x^5*((1+x)^5 + (1+x^5))
+ x^6*((1+x)^6 + (1+x^2)^3 + (1+x^3)^2 + (1+x^6))
+ x^7*((1+x)^7 + (1+x^7))
+ x^8*((1+x)^8 + (1+x^2)^4 + (1+x^4)^2 + (1+x^8))
+ x^9*((1+x)^9 + (1+x^3)^3 + (1+x^9))
+ x^10*((1+x)^10 + (1+x^2)^5 + (1+x^5)^2 + (1+x^10))
+ x^11*((1+x)^11 + (1+x^11))
+ x^12*((1+x)^12 + (1+x^2)^6 + (1+x^3)^4 + (1+x^4)^3 + (1+x^6)^2 + (1+x^12))
+...+ x^n * Sum_{d|n} (1 + x^d)^(n/d) +...
or, more explicitly,
A(x) = x*(1 + x) + x^2*(2 + 2*x + 2*x^2) + x^3*(2 + 3*x + 3*x^2 + 2*x^3)
+ x^4*(3 + 4*x + 8*x^2 + 4*x^3 + 3*x^4)
+ x^5*(2 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + 2*x^5)
+ x^6*(4 + 6*x + 18*x^2 + 22*x^3 + 18*x^4 + 6*x^5 + 4*x^6)
+ x^7*(2 + 7*x + 21*x^2 + 35*x^3 + 35*x^4 + 21*x^5 + 7*x^6 + 2*x^7)
+ x^8*(4 + 8*x + 32*x^2 + 56*x^3 + 78*x^4 + 56*x^5 + 32*x^6 + 8*x^7 + 4*x^8)
+ x^9*(3 + 9*x + 36*x^2 + 87*x^3 + 126*x^4 + 126*x^5 + 87*x^6 + 36*x^7 + 9*x^8 + 3*x^9)
+ x^10*(4 + 10*x + 50*x^2 + 120*x^3 + 220*x^4 + 254*x^5 + 220*x^6 + 120*x^7 + 50*x^8 + 10*x^9 + 4*x^10)
+ x^11*(2 + 11*x + 55*x^2 + 165*x^3 + 330*x^4 + 462*x^5 + 462*x^6 + 330*x^7 + 165*x^8 + 55*x^9 + 11*x^10 + 2*x^11)
+ x^12*(6 + 12*x + 72*x^2 + 224*x^3 + 513*x^4 + 792*x^5 + 952*x^6 + 792*x^7 + 513*x^8 + 224*x^9 + 72*x^10 + 12*x^11 + 6*x^12) + ...

Links

Crossrefs

Cf. A034729, A000045, A007435.

Programs

  • Maple
    with(numtheory): seq(add(combinat[fibonacci](d+1), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 10 2025
  • PARI
    {a(n)=polcoeff(sum(m=1, n, fibonacci(m+1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1,50, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=1, n, x^m*(1+x^m)/(1-x^m-x^(2*m) +x*O(x^n)) ), n)}
for(n=1, 50, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=x+x^2);A=sum(m=1,n,x^m*sumdiv(m,d,(1 + x^(m/d) +x*O(x^n))^d) );polcoeff(A,n)}
for(n=1,50,print1(a(n),", "))

Formula

G.f.: Sum_{n>=1} x^n * (1 + x^n) / (1 - x^n - x^(2*n)).
G.f.: Sum_{n>=1} x^n * Sum_{d|n} (1 + x^d)^(n/d).
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n+1). - Vaclav Kotesovec, Aug 22 2014
a(n) = Sum_{d|n} Fibonacci(d+1). - Ridouane Oudra, Apr 10 2025

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

A368687 Expansion of g.f. Sum_{n>=1} q^n/(1-q^n-q^(2*n)-q^(3*n)).

Original entry on oeis.org

1, 2, 3, 6, 8, 17, 25, 50, 84, 158, 275, 525, 928, 1731, 3146, 5818, 10610, 19611, 35891, 66174, 121442, 223593, 410745, 756045, 1389545, 2556686, 4700854, 8647799, 15902592, 29252734, 53798081, 98955914, 181997878, 334756388, 615693506, 1132456971, 2082876104, 3831042321, 7046320314, 12960268134
Offset: 1

Views

Author

Joerg Arndt, Jan 03 2024

Keywords

Crossrefs

Cf. A007435 (g.f. Sum{n>=1} q^n/(1-q^n-q^(2*n))), A368689 (g.f. Sum_{n>=1} q^n/(1-q^(2*n)-q^(3*n))), A368688 (g.f. Sum_{n>=1} q^n/(1-q^n-q^(3*n))).

Programs

  • PARI
    my(N=55,q='q+O('q^N)); Vec(sum(n=1,N,q^n/(1-q^n-q^(2*n)-q^(3*n))))

A368688 Expansion of g.f. Sum_{n>=1} q^n/(1-q^n-q^(3*n)).

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 7, 13, 15, 24, 29, 50, 61, 96, 134, 202, 278, 426, 596, 898, 1286, 1903, 2746, 4082, 5900, 8703, 12679, 18658, 27202, 40023, 58426, 85828, 125521, 184195, 269552, 395502, 578950, 849088, 1243586, 1823380, 2670965, 3915867, 5736962, 8409830, 12322560, 18062121, 26467300, 38793983, 56849093
Offset: 1

Views

Author

Joerg Arndt, Jan 03 2024

Keywords

Crossrefs

Cf. A007435 (g.f. Sum{n>=1} q^n/(1-q^n-q^(2*n))), A368689 (g.f. Sum_{n>=1} q^n/(1-q^(2*n)-q^(3*n))), A368687 (g.f. Sum_{n>=1} q^n/(1-q^n-q^(2*n)-q^(3*n))).

Programs

  • PARI
    my(N=55, q='q+O('q^N)); Vec(sum(n=1,N,q^n/(1-q^n-q^(3*n))))

A368689 Expansion of g.f. Sum_{n>=1} q^n/(1-q^(2*n)-q^(3*n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 3, 5, 6, 7, 8, 14, 13, 19, 24, 33, 38, 57, 66, 94, 118, 159, 201, 282, 353, 478, 622, 836, 1082, 1463, 1898, 2546, 3338, 4448, 5846, 7806, 10253, 13647, 18005, 23930, 31573, 41960, 55406, 73556, 97257, 129002, 170626, 226340, 299429, 397013, 525495, 696560, 922112, 1222210, 1618201, 2144487
Offset: 1

Views

Author

Joerg Arndt, Jan 03 2024

Keywords

Crossrefs

Cf. A007435 (g.f. Sum{n>=1} q^n/(1-q^n-q^(2*n))), A368688 (g.f. Sum_{n>=1} q^n/(1-q^n-q^(3*n))), A368687 (g.f. Sum_{n>=1} q^n/(1-q^n-q^(2*n)-q^(3*n))).

Programs

  • PARI
    my(N=55, q='q+O('q^N)); Vec(sum(n=1,N,q^n/(1-q^(2*n)-q^(3*n))))

A066769 a(n) = Sum_{d|n} d*Fibonacci(n/d).

Original entry on oeis.org

1, 3, 5, 9, 10, 21, 20, 39, 49, 80, 100, 195, 246, 424, 650, 1065, 1614, 2715, 4200, 6940, 11020, 17922, 28680, 46821, 75075, 121898, 196565, 318680, 514258, 833560, 1346300, 2180439, 3524900, 5706132, 9227600, 14936241, 24157854, 39096588
Offset: 1

Views

Author

Vladeta Jovovic, Jan 17 2002

Keywords

Comments

Dirichlet convolution of f(n)=n with the Fibonacci numbers F(n)=A000045. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.

Links

Crossrefs

Cf. A000045, A007435.

Programs

  • Maple
    N:= 100:
A:= Vector(N):
for k from 1 to N do
  f:= combinat:-fibonacci(k);
  ds:= [$1..floor(N/k)];
  A[k*ds] := A[k*ds] + f*Vector(ds);
od:
convert(A,list); # Robert Israel, Feb 08 2016
  • Mathematica
    a[n_] := DivisorSum[n, # * Fibonacci[n/#] &]; Array[a, 38] (* Amiram Eldar, Sep 16 2020 *)
  • PARI
    a(n) = sumdiv(n, d, d*fibonacci(n/d)); \\ Michel Marcus, Sep 16 2020

Formula

G.f.: Sum_{i>0} i*x^i/(1-x^i-x^(2*i)). - Vladeta Jovovic, Oct 06 2003

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

A309729 Expansion of Sum_{k>=1} x^k/(1 - x^k - 2*x^(2*k)).

Original entry on oeis.org

1, 2, 4, 7, 12, 26, 44, 92, 175, 354, 684, 1396, 2732, 5506, 10938, 21937, 43692, 87578, 174764, 349884, 699098, 1398786, 2796204, 5593886, 11184823, 22372354, 44739418, 89483996, 178956972, 357925242, 715827884, 1431677702, 2863312218, 5726666754, 11453246178, 22906581193
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

Comments

Inverse Moebius transform of Jacobsthal numbers (A001045).

Crossrefs

Cf. A001045, A007435, A055895, A100107, A104723, A256281.

Programs

  • Maple
    seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # Robert Israel, Aug 14 2019
  • Mathematica
    nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]
  • PARI
    a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ Andrew Howroyd, Aug 14 2019
  • Python
    n = 1
while n <= 36:
    s, d = 0, 1
    while d <= n:
        if n%d == 0:
            s = s+2**d-(-1)**d
        d = d+1
    print(n,s//3)
n = n+1 # A.H.M. Smeets, Aug 14 2019

Formula

G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).
a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).
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