A102807 a(n) is the square of one plus the number consisting of n 3's.

1, 16, 1156, 111556, 11115556, 1111155556, 111111555556, 11111115555556, 1111111155555556, 111111111555555556, 11111111115555555556, 1111111111155555555556, 111111111111555555555556, 11111111111115555555555556, 1111111111111155555555555556, 111111111111111555555555555556

Offset: 0

Author

Ron Knott, Feb 27 2005

Keywords

• nonn
• base
• easy

Comments

Old name was: The number (333...334)^2.

An infinite sequence of squares with no zeros in base 10.

a(n) = A104265(2n) for n > 0. - Chai Wah Wu, Mar 24 2020

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 31 at p. 61.
• Italo Ghersi, Matematica dilettevole e curiosa, pp. 111-112, Hoepli, Milano, 1967. [Vincenzo Librandi, Dec 31 2008]

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Emile Fourrey, Récréations arithmétiques, Vuibert, 1899 and after, Paris, pages 72-73.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A052041, A093137, A075415, A109344, A104265.

Programs

Maple

a:= n-> (1+parse(cat(0, 3$n)))^2:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 03 2018

Mathematica

Table[(10^n + 2)^2/9, {n, 0, 20}] (* Paolo Xausa, Jun 26 2024 *)

Formula

From R. J. Mathar, Jan 06 2009: (Start)

a(n) = (100^n + 4*10^n + 4)/9.

G.f.: (1 - 95*x + 490*x^2)/((1-x)*(100*x-1)*(10*x-1)). (End)

E.g.f.: exp(x)*(4 + 4*exp(9*x) + exp(99*x))/9. - Stefano Spezia, Jul 31 2024

Extensions

New name from Alois P. Heinz, Sep 03 2018