A178635 a(n) = 72*((10^n - 1)/9)^2.

72, 8712, 887112, 88871112, 8888711112, 888887111112, 88888871111112, 8888888711111112, 888888887111111112, 88888888871111111112, 8888888888711111111112, 888888888887111111111112, 88888888888871111111111112, 8888888888888711111111111112, 888888888888887111111111111112

Offset

1,1

References

Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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

Formula

a(n) = 72*A002477(n) = A002283(n)*A002282(n).

a(n) = ((A002282(n-1)*10 + 7)*10^(n-1) + A002275(n-1))*10 + 2.

G.f.: 72*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017

From Elmo R. Oliveira, Aug 01 2025: (Start)

E.g.f.: 8*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9.

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

Example

n=1: ..................... 72 = 9 * 8;
n=2: ................... 8712 = 99 * 88;
n=3: ................. 887112 = 999 * 888;
n=4: ............... 88871112 = 9999 * 8888;
n=5: ............. 8888711112 = 99999 * 88888;
n=6: ........... 888887111112 = 999999 * 888888;
n=7: ......... 88888871111112 = 9999999 * 8888888;
n=8: ....... 8888888711111112 = 99999999 * 88888888;
n=9: ..... 888888887111111112 = 999999999 * 888888888.

Mathematica

72 (FromDigits/@Table[PadRight[{}, n, 1], {n, 40}])^2 (* Vincenzo Librandi, Mar 21 2014 *)
LinearRecurrence[{111, -1110, 1000}, {72, 8712, 887112}, 20] (* Harvey P. Dale, Feb 25 2026 *)

Prog

(Magma) [72*((10^n-1)/9)^2: n in [1..50]]; // Vincenzo Librandi, Dec 28 2010
(PARI) a(n)=72*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

Crossrefs

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178634.

Cf. A002275, A002282, A002283, A002477.

Sequence in context: A119750 A395920 A283093 * A271189 A093272 A105347

Adjacent sequences: A178632 A178633 A178634 * A178636 A178637 A178638

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 31 2010

Status

approved