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A306809 Binomial transform of the continued fraction expansion of e. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 7 15:02 EDT 2021. Contains 343650 sequences. (Running on oeis4.)