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.

A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

Original entry on oeis.org

1, 1, 5, 85, 2928, 314925, 84974760, 63327890015, 123670531939440, 644385861467631972, 8853970669063185618000, 321538767413685546538468385, 30768712746239178236068160093280, 7755868453482819803691622493685140880, 5144106193113274410507722020733942141881664
Offset: 1

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Author

Paul D. Hanna, Jan 06 2012

Keywords

Examples

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^85 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^314925 * (1-29*x^7-x^14)^84974760 * (1-47*x^8+x^16)^63327890015 * (1-76*x^9-x^18)^123670531939440 *...).
where F(x) = exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = g.f. of A156216:
F(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
so that the logarithm of F(x) begins:
log(F(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 + 29^7*x^7/7 + 47^8*x^8/8 + 76^9*x^9/9 + 123^10*x^10/10 +...+ Lucas(n)^n*x^n +...

Links

Crossrefs

Cf. A156216, A000032 (Lucas), A203318, A203803, A203804, A203805, A203806, A203807, A203808.

Programs

  • Mathematica
    a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^(#-1)&]; Array[a, 15] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^(d-1))/n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^m*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

Formula

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n), which is the g.f. of A156216.
G.f.: Product_{n>=1} G_n(x^n)^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A203850 G.f.: Product_{n>=1} (1 - Lucas(n)*x^n + (-x^2)^n) / (1 + Lucas(n)*x^n + (-x^2)^n) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, -2, -4, 0, 14, 16, 0, 0, 4, -152, -188, 0, 0, -44, 0, 0, 4414, 5456, -4, 0, 1288, 0, 0, 0, 0, -335406, -414728, 0, 0, -97904, 0, 0, 4, 0, -8828, 0, 66770564, 82532956, 0, 0, 19483388, -304, 0, 0, 0, 1756816, 0, 0, 0, -34787592002, -42999828492, 0, 60508, -10150882544, 0, 0, 0, 0, -915304508, 0, 0, 796
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2012

Keywords

Comments

Compare to: Product_{n>=1} (1-q^n)/(1+q^n) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2), the Jacobi theta_4 function, which has the g.f: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * q^n/n ).

Examples

			G.f.: A(x) = 1 - 2*x - 4*x^2 + 14*x^4 + 16*x^5 + 4*x^8 - 152*x^9 - 188*x^10 +...
-log(A(x)) = 2*x + 4*3*x^2/2 + 8*4*x^3/3 + 8*7*x^4/4 + 12*11*x^5/5 + 16*18*x^6/6 +...+ (sigma(2*n)-sigma(n))*Lucas(n)*x^n/n +...
Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*x^(n^2):
-log(H(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 8*x^4/4 + 12*x^5/5 + 16*x^6/6 + 16*x^7/7 +...+ (sigma(2*n)-sigma(n))*x^n/n +...
The g.f. equals the product:
A(x) = (1-x-x^2)/(1+x-x^2) * (1-3*x^2+x^4)/(1+3*x^2+x^4) * (1-4*x^3-x^6)/(1+4*x^3-x^6) * (1-7*x^4+x^8)/(1+7*x^4+x^8) * (1-11*x^5-x^10)/(1+11*x^5-x^10) *...* (1 - Lucas(n)*x^n + (-x^2)^n)/(1 + Lucas(n)*x^n + (-x^2)^n) *...
Positions of zeros form A022544:
[3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]
which are numbers that are not the sum of 2 squares.

Links

Crossrefs

Cf. A203860, A203861, A022544, A000204 (Lucas), A203318, A225524.

Programs

  • PARI
    /* Subroutine used in PARI programs below: */
{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n, 1 - Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 + Lucas(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}
  • PARI
    {a(n)=polcoeff(prod(m=1, n\2+1, (1 - Lucas(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - Lucas(2*m)*x^(2*m) + x^(4*m) +x*O(x^n))), n)}
  • PARI
    {a(n)=polcoeff(exp(sum(k=1, n,-(sigma(2*k)-sigma(k))*Lucas(k)*x^k/k)+x*O(x^n)), n)}

Formula

a(n) = 0 iff n is not the sum of 2 squares (A022544).
G.f.: Product_{n>=1} (1 - Lucas(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - Lucas(2*n)*x^(2*n) + x^(4*n)).
G.f.: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * Lucas(n) * x^n/n ) where Lucas(n) = A000204(n).

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

A203319 a(n) = n*Fibonacci(n) * Sum_{d|n} 1/(d*Fibonacci(d)).

Original entry on oeis.org

1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230
Offset: 1

Views

Author

Paul D. Hanna, Jan 01 2012

Keywords

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = Fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...

Links

Crossrefs

Cf. A203318, A203321; A203414 (Pell variant).
Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.

Programs

  • Mathematica
    a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)
  • PARI
    {a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
  • PARI
    {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L,n)}
  • PARI
    {a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Formula

Equals the logarithmic derivative of A203318.
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = Fibonacci(n).
(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) where Lucas(n) = A000032(n).
(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000032(n).
(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.
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