A306809 Binomial transform of the continued fraction expansion of e.

2, 3, 6, 12, 23, 46, 98, 213, 458, 972, 2051, 4322, 9098, 19113, 40054, 83748, 174767, 364086, 757298, 1572861, 3262242, 6757500, 13981019, 28894090, 59652314, 123032913, 253522382, 521957844, 1073741831, 2207135966, 4533576578, 9305762469, 19088743546, 39131924268

Offset

0,1

Comments

Taking this sequence as a continued fraction it seems to converge to 2.31601650488979...

Links

Table of n, a(n) for n=0..33.

Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev Binomial Transforms of Sequences, CU Boulder Experimental Math Lab, Spring 2019.

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*b(k), where b(k) is the k-th term of the continued fraction expansion of e.

Conjectures from Colin Barker, Mar 12 2019: (Start)

G.f.: (2 - 11*x + 27*x^2 - 41*x^3 + 40*x^4 - 22*x^5 + 6*x^6) / ((1 - x)*(1 - 2*x)^2*(1 - x + x^2)^2).

a(n) = 7*a(n-1) - 21*a(n-2) + 37*a(n-3) - 43*a(n-4) + 33*a(n-5) - 16*a(n-6) + 4*a(n-7) for n>6.

(End)

Example

For n = 3, the a(3) = binomial(3,0)*2 + binomial(3,1)*1 + binomial(3,2)*2 + binomial(3,3)*1 = 12.

Mathematica

nmax = 50; A003417 = ContinuedFraction[E, nmax+1]; Table[Sum[Binomial[n, k]*A003417[[k + 1]], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 23 2020 *)

Prog

(Sage)
def OEISbinomial_transform(N, seq):
    BT = [seq[0]]
    k = 1
    while k< N:
        next = 0
        j = 0
        while j <=k:
            next = next + ((binomial(k, j))*seq[j])
            j = j+1
        BT.append(next)
        k = k+1
    return BT
econt = oeis('A003417')
OEISbinomial_transform(50, econt)

Crossrefs

Cf. A003417 (continued fraction for e).

Sequence in context: A293363 A326021 A164363 * A103341 A023675 A029996

Adjacent sequences: A306806 A306807 A306808 * A306810 A306811 A306812

Keyword

cofr,nonn,easy

Author

Sarah Arpin, Mar 11 2019

Status

approved