A271528 a(n) = 2*(10^n - 1)^2/27.

0, 6, 726, 73926, 7405926, 740725926, 74073925926, 7407405925926, 740740725925926, 74074073925925926, 7407407405925925926, 740740740725925925926, 74074074073925925925926, 7407407407405925925925926, 740740740740725925925925926, 74074074074073925925925925926

Offset

0,2

Comments

All terms are multiple of 6.

Converges in a 10-adic sense to ...925925925926.

A transformation of the Wonderful Demlo numbers (A002477).

More generally, the ordinary generating function for the transformation of the Wonderful Demlo numbers, is k*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

Links

Table of n, a(n) for n=0..15.

Ilya Gutkovskiy, Transformation of the Wonderful Demlo numbers

Eric Weisstein's World of Mathematics, Demlo Number

Eric Weisstein's World of Mathematics, Repunit

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Formula

O.g.f.: 6*x*(1 + 10*x)/(1 - 111*x + 1110*x^2 - 1000*x^3).

E.g.f.: 2 (exp(x) - 2*exp(10*x) + exp(100*x))/27.

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3).

a(n) = 6*A002477(n) = 6*A002275(n)^2 = A002276(n)*A002277(n) = sqrt(A075411(n)*A075412(n)).

Sum_{n>=1} 1/a(n) = 0.1680577405662077350849154881928636039793563...

Lim_{n -> infinity} a(n + 1)/a(n) = 100.

Example

n=1:                  6 = 2 * 3;
n=2:                726 = 22 * 33;
n=3:              73926 = 222 * 333;
n=4:            7405926 = 2222 * 3333;
n=5:          740725926 = 22222 * 33333;
n=6:        74073925926 = 222222 * 333333;
n=7:      7407405925926 = 2222222 * 3333333;
n=8:    740740725925926 = 22222222 * 33333333;
n=9:  74074073925925926 = 222222222 * 333333333, etc.

Mathematica

Table[2 ((10^n - 1)^2/27), {n, 0, 15}]
LinearRecurrence[{111, -1110, 1000}, {0, 6, 726}, 16]

Prog

(PARI) x='x+O('x^99); concat(0, Vec(6*x*(1+10*x)/(1-111*x+1110*x^2-1000*x^3))) \\ Altug Alkan, Apr 09 2016
(Python)
for n in range(0, 10**1):print((int)((2*(10**n-1)**2)/27))
# Soumil Mandal, Apr 10 2016

Crossrefs

Cf. A002275, A002276, A002277, A002477.

Cf. similar sequences of the form k*((10^n - 1)/9)^2: A075411 (k=4), this sequence (k=6), A075412 (k=9), A075413 (k=16), A178630 (k=18), A075414 (k=25), A178631 (k=27), A075415 (k=36), A178632 (k=45), A075416 (k=49), A178633 (k=54), A178634 (k=63), A075417 (k=64), A178635 (k=72), A059988 (k=81).

Sequence in context: A036981 A343434 A202080 * A316748 A354251 A232130

Adjacent sequences: A271525 A271526 A271527 * A271529 A271530 A271531

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Apr 09 2016

Status

approved