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

Author

Ilya Gutkovskiy, Apr 09 2016

Keywords

• nonn
• easy

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).

Examples

			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.
		

Links

• 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).

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).

Programs

Mathematica

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

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

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.