A261544 a(n) = Sum_{k=0..n} 1000^k.

1, 1001, 1001001, 1001001001, 1001001001001, 1001001001001001, 1001001001001001001, 1001001001001001001001, 1001001001001001001001001, 1001001001001001001001001001, 1001001001001001001001001001001, 1001001001001001001001001001001001

Offset: 0

Author

Ilya Gutkovskiy, Aug 24 2015

Keywords

• nonn
• base
• easy

Comments

A sequence of palindromic numbers.

Examples

			From _Bruno Berselli_, Aug 25 2015: (Start)
a(n)   is the binary representation of    A023001
-------------------------------------------------
1  ...........................................  1
1001  ........................................  9
1001001 .....................................  73
1001001001  ................................  585
1001001001001  ............................  4681
1001001001001001  ........................  37449
1001001001001001001  ....................  299593
1001001001001001001001  ................  2396745
1001001001001001001001001  ............  19173961, etc.
(End)
		

Links

• Colin Barker, Table of n, a(n) for n = 0..333 (corrected by Michel Marcus, Jan 19 2019)
• John Rafael M. Antalan, A Recreational Application of Two Integer Sequences and the Generalized Repetitious Number Puzzle, arXiv:1908.06014 [math.HO], 2019.
• Eric Weisstein's World of Mathematics, Palindromic Number.
• Index entries for linear recurrences with constant coefficients, signature (1001,-1000).

Crossrefs

Cf. A000533, A002113, A023001.

Subsequence of A033146.

Sums of 100^k: A094028; sums of 10^k: A000042.

Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

Programs

Magma

[(1000^(n+1)-1)/999: n in [0..30]]; // Vincenzo Librandi, Aug 24 2015

Mathematica

Table[(1000^(n + 1) - 1)/999, {n, 0, 15}]
LinearRecurrence[{1001, -1000}, {1, 1001}, 20] (* Vincenzo Librandi, Aug 24 2015 *)

PARI

Vec(1 / ((x-1)*(1000*x-1)) + O(x^20)) \\ Colin Barker, Aug 24 2015

Formula

a(n) = (1000^(n + 1) - 1)/999.

a(n) = 1001*a(n-1) - 1000*a(n-2). - Colin Barker, Aug 24 2015

G.f.: 1 / ((x-1)*(1000*x-1)). - Colin Barker, Aug 24 2015

E.g.f.: (1/999)*(1000000*exp(1000*x) - exp(x)). - G. C. Greubel, Aug 29 2015