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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.

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A096654 Denominators of self-convergents to 1/(e-2).

Original entry on oeis.org

1, 2, 8, 38, 222, 1522, 11986, 106542, 1054766, 11506538, 137119578, 1772006854, 24681524038, 368577425634, 5874202721042, 99515904921182, 1785757627196766, 33835407673201882, 675016383080377546, 14143200407398386678, 310507536216973671158, 7128173005328786885714
Offset: 0

Views

Author

Clark Kimberling, Jul 01 2004

Keywords

Comments

The self-continued fraction of r>0 is here introduced as the sequence (b(0), b(1), b(2), ...) defined as follows: put r(0)=r, b(0)=[r(0)] and for n>=1, put r(n)=b(n-1)/(r(n-1)-b(n-1)) and b(n)=[r(n)]. This differs from simple continued fraction, for which r(n)=1/(r(n-1)-b(n-1)). Now r=lim(p(n)/q(n)), where p(0)=b(1), q(0)=1, p(1)=b(0)(b(1)+1), q(1)=b(1) and for n>=2, p(n)=b(n)*p(n-1)+b(n-1)*p(n-2), q(n)=b(n)*q(n-1)+b(n-1)*q(n-2); p(0),p(1),... are the numerators of the self-convergents to r; q(0),q(1),... are the denominators of the self-convergents to r. Thus A096654 is given by a(n)=(n+1)*a(n-1)+n*a(n-2), a(0)=1, a(1)=2.
Number of increasing runs of odd length in all permutations of [n+1]. Example: a(2) = 8 because we have (123), 13(2), (3)12, (2)13, 23(1), (3)(2)(1) (the runs of odd length are shown between parentheses). - Emeric Deutsch, Aug 29 2004

Examples

			a(2)=q(2)=3*2+2*1=8, a(3)=q(3)=4*8+3*2=38. The convergents p(0)/q(0) to p(4)/q(4) are 1/1, 3/2, 11/8, 53/38, 309/222.

Crossrefs

Cf. A000255, A096655, A096656, A096657, A096658, A097591, A233590.

Programs

  • Maple
    G:=(3-x-2*(1+x)*exp(-x))/(1-x)^3: Gser:=series(G,x=0,22): 1,seq(n!*coeff(Gser,x^n),n=1..21);
  • Mathematica
    With[{g = (3 - x - 2*(1 + x)*Exp[-x])/(1 - x)^3},CoefficientList[Series[g, {x, 0, 21}], x]*Table[k!, {k, 0, 21}]] (* Shenghui Yang, Oct 15 2024 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace((3-x-2*(1+x)*exp(-x))/(1-x)^3)) /* Joerg Arndt, Aug 06 2012 */
  • Python
    prpr = 1
prev = 2
for n in range(2, 77):
    print(prpr, end=', ')
    curr = (n+1)*prev + n*prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 05 2012

Formula

a(n) = (n+1)*a(n-1) + n*a(n-2), with a(0)=1, a(1)=2. - Alex Ratushnyak, Aug 05 2012
E.g.f.: (3-x-2*(1+x)*exp(-x))/(1-x)^3. - Emeric Deutsch, Aug 29 2004
From Gary Detlefs, Apr 12 2010: (Start)
a(n) = A055596(n+1) + A055596(n+2).
a(n) = (n+1)!+(n+2)! -2*( A000166(n+1) + A000166(n+2)).
a(n) = (n+1)! - 2*floor(((n+1)!+1)/e) + (n+2)!-2*floor(((n+2)!+1)/e). (End)
a(n) = ((n+3)!-2*floor(((n+3)!+1)/e))/(n+2). - Gary Detlefs, Jul 11 2010 [corrected by Gary Detlefs, Oct 26 2020]
a(n) = Sum_{k=1..n+1} A097591(n+1,k). - Alois P. Heinz, Jul 03 2019

Extensions

More terms from Emeric Deutsch, Aug 29 2004

A096656 a(n) = F(n+2)*a(n-1) + F(n+1)*a(n-2), where F = A000045 (Fibonacci numbers), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 46, 408, 5672, 124416, 4349256, 243439224, 21905300016, 3176029293240, 743169188527224, 280914798900088368, 171638202113128667928, 169578263512987049149416, 270985893735725975486862288
Offset: 0

Views

Author

Clark Kimberling, Jul 01 2004

Keywords

Comments

This is the sequence of denominators of self-convergents to the number 1.389805... whose self-continued fraction is (1,2,3,5,8,...) (Fibonacci numbers). See A096655 for numerators and A096654 for definitions.

Examples

			a(2)=F(4)*2+F(3)*1=8, a(3)=F(5)*8+F(4)*2=46.

Crossrefs

Cf. A000045, A096654, A096655.

Programs

  • Mathematica
    a[0] = 1; a[1] = 2; a[n_] := Fibonacci[n + 2]*a[n - 1] + Fibonacci[n + 1]*a[n - 2]; Table[ a[n], {n, 0, 16}] (* Robert G. Wilson v, Jul 09 2004 *)

Formula

a(n) ~ c * ((1+sqrt(5))/2)^((n+2)*(n+3)/2) / 5^(n/2) where c = 0.5018252861856573838264566231631563920610293670131098212588... . - Vaclav Kotesovec, Nov 27 2015

Extensions

More terms from Robert G. Wilson v, Jul 09 2004

A105804 a(n)=F(n-1)a(n-1)+F(n)a(n-2), where F = A000045 (Fibonacci numbers), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 3, 9, 42, 282, 2802, 42348, 984576, 35804724, 2056887084, 188218830732, 27582766315980, 6497743050809304, 2466474617607855408, 1510962789131940581928, 1495259232837545099449512, 2391833322688676458284572616, 6186748984680033744268134049416
Offset: 0

Views

Author

Clark Kimberling, Apr 20 2005

Keywords

Crossrefs

Cf. A096655.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==0,a[1]==1,a[n]==Fibonacci[n-1]a[n-1]+ Fibonacci[ n] a[n-2]},a,{n,20}] (* Harvey P. Dale, Apr 25 2014 *)

Formula

a(n) ~ c * ((1+sqrt(5))/2)^(n*(n-1)/2) / 5^(n/2), where c = 49.6023707313141860163673593923904868464364654572509980004... . - Vaclav Kotesovec, Nov 27 2015

Extensions

Corrected and extended by Harvey P. Dale, Apr 25 2014
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