A178634 a(n) = 63*((10^n - 1)/9)^2.

63, 7623, 776223, 77762223, 7777622223, 777776222223, 77777762222223, 7777777622222223, 777777776222222223, 77777777762222222223, 7777777777622222222223, 777777777776222222222223, 77777777777762222222222223, 7777777777777622222222222223, 777777777777776222222222222223

Offset: 1

Author

Reinhard Zumkeller, May 31 2010

Keywords

• nonn
• easy

Examples

			n=1: ..................... 63 = 9 * 7;
n=2: ................... 7623 = 99 * 77;
n=3: ................. 776223 = 999 * 777;
n=4: ............... 77762223 = 9999 * 7777;
n=5: ............. 7777622223 = 99999 * 77777;
n=6: ........... 777776222223 = 999999 * 777777;
n=7: ......... 77777762222223 = 9999999 * 7777777;
n=8: ....... 7777777622222223 = 99999999 * 77777777;
n=9: ..... 777777776222222223 = 999999999 * 777777777.
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 33 at p. 62.
• Walther Lietzmann, Lustiges und Merkwuerdiges von Zahlen und Formen, (F. Hirt, Breslau 1921-43), p. 149.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A059988, A075412, A075415, A178630, A178631, A178632, A178633, A178635.

Cf. A002276, A002281, A002283, A002477.

Programs

GAP

List([1..20], n -> 63*((10^n - 1)/9)^2); # G. C. Greubel, Jan 28 2019

Magma

[63*((10^n - 1)/9)^2: n in [1..20]]; // Vincenzo Librandi, Dec 28 2010

Mathematica

63((10^Range[15]-1)/9)^2 (* or *) Table[FromDigits[Join[PadRight[{},n,7],{6},PadRight[{},n,2],{3}]],{n,0,15}] (* Harvey P. Dale, Apr 23 2012 *)

PARI

a(n)=63*(10^n\9)^2 \\ Charles R Greathouse IV, Jul 02 2013

Sage

[63*((10^n - 1)/9)^2 for n in (1..20)] # G. C. Greubel, Jan 28 2019

Formula

a(n) = 63*A002477(n) = A002283(n)*A002281(n).

a(n) = ((A002281(n-1)*10 + 6)*10^(n-1) + A002276(n-1))*10 + 3.

G.f.: 63*x*(1 + 10*x)/((1 - x)*(1 - 10*x)*(1 - 100*x)). - Ilya Gutkovskiy, Feb 24 2017

E.g.f.: 7*exp(x)*(1 - 2*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 3. - Elmo R. Oliveira, Aug 01 2025