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A014824 a(0) = 0, a(n) = 10*a(n-1) + n. 18
0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, 12345679011, 123456790122, 1234567901233, 12345679012344, 123456790123455, 1234567901234566, 12345679012345677 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The square roots of these numbers have some remarkable properties - see the link to Schizophrenic numbers.

Partial sums of A002275. - Jonathan Vos Post, Apr 25 2010

This sequence is the particular case of a(0) = 0, a(n) = r*a(n-1) + n, when r=10. If now the first N terms are computed for (r > N) then the resulting set of numbers is readable as the smallest k-digits permutations (1<=k<=N): Those built from the concatenation of the first k digits in base-r (see links). R. J. Cano, Jan 09 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

K. S. Brown, Schizophrenic numbers

Index entries for sequences related to linear recurrences with constant coefficients, signature (12,-21,10).

R. J. Cano, Additional information (See appendix).

FORMULA

a(n) =(10^n-1)*(10/81)-n/9. - Henry Bottomley, Jul 04 2000

a(n)/10^n converges to 10/81=0.123456790123456790...

Let b(n)=if(n=0, 1, if(n=1, 10, 10*9^(n-2))). Then a(n)=sum{k=0..n, C(n, k)b(k)} (Binomial transform). - Paul Barry, Jan 29 2004

G.f.: x/(1-12*x+21*x^2-10*x^3). - Colin Barker, Jan 08 2012

MAPLE

a:=n->sum((10^(n-j)-1^(n-j))/9, j=0..n): seq(a(n), n=0..17); - Zerinvary Lajos, Jan 15 2007

a:=n->sum(10^(n-j)*j, j=0..n): seq(a(n), n=0..16); - Zerinvary Lajos, Jun 05 2008

MATHEMATICA

Table[Sum[10^i - 1, {i, n}]/9, {n, 18}] (from Robert G. Wilson v, Nov 20 2004)

PROG

(MAGMA) [(10^n-1)*(10/81)-n/9: n in [0..20]]; // Vincenzo Librandi, Aug 23 2011

(PARI) \\ - R. J. Cano, Jan 09 2011

linrec01(p, u, base)={my(r=!p, A=1); for(j=2, u, A=A*base+r+p*j); A};

a(n)=(n!=0)*linrec01(1, n, 10); \\ With (0, n, 10) it generates repunit numbers.

(PARI) A014824(n)=(10^(n+1)\9-n)\9  \\ - M. F. Hasler, Jan 17 2013

CROSSREFS

Cf. A007908, A060011.

Cf. A002275. - Jonathan Vos Post, Apr 25 2010

Similar sequences in other bases are: (base-2) A000295, (base-3) A000340, (base-4) A014825, (base-5) A014827, (base-6) A014829. 0 R. J. Cano, Jan 11 2013

Differs from A007908, A035239, A057137, A060555, A138957 from n=10 on. - M. F. Hasler, Jan 17 2013

Sequence in context: A037610 A035239 A057137 * A060555 A138957 A007908

Adjacent sequences:  A014821 A014822 A014823 * A014825 A014826 A014827

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 22 21:10 EST 2014. Contains 252372 sequences.