A048355 a(n) is the index of the smallest triangular number containing exactly n 0's.

0, 24, 200, 775, 2000, 10000, 20000, 100000, 200000, 1000000, 2000000, 10000000, 20000000, 100000000, 200000000, 1000000000, 2000000000, 10000000000, 20000000000, 100000000000, 200000000000, 1000000000000, 2000000000000, 10000000000000, 20000000000000, 100000000000000

Offset: 1

Author

Patrick De Geest, Mar 15 1999

Keywords

• nonn
• base
• easy

Examples

			From _Bernard Schott_, Mar 04 2019: (Start)
a(2) = 24: T(24) = 300 which contains exactly two 0's.
a(6) = 10000: T(10000) = 50005000 which contains exactly six 0's.
a(7) = 20000: T(20000) = 200010000 which contains exactly seven 0's.
(End)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..100
• Eric Weisstein's World of Mathematics, Triangular Number
• Index entries for linear recurrences with constant coefficients, signature (0,10).

Crossrefs

Cf. A000217, A036517.

Programs

Mathematica

nsmall = Table[Infinity, 20];
For[i = 0, i <= 10^6, i++, p = PolygonalNumber[i];
  n0 = Count[IntegerDigits[p], 0];
  If[nsmall[[n0]] > i, nsmall[[n0]] = i]];
ReplaceAll[nsmall, Infinity -> "?"] (* Robert Price, Mar 22 2020 *)
LinearRecurrence[{0,10},{0,24,200,775,2000,10000},30] (* Harvey P. Dale, Jul 26 2024 *)

PARI

Vec(x^2*(24 + 200*x + 535*x^2 + 2250*x^4) / (1 - 10*x^2) + O(x^30)) \\ Colin Barker, Mar 25 2020

Formula

From Bernard Schott, Mar 04 2019: (Start)

for n odd >= 5, a(n) = 2 * 10^((n+1)/2),

for n even >= 6, a(n) = 10^((n+2)/2).

(End)

From Colin Barker, Mar 25 2020: (Start)

G.f.: x^2*(24 + 200*x + 535*x^2 + 2250*x^4) / (1 - 10*x^2).

a(n) = 10*a(n-2) for n>4.

(End)

Extensions

a(16)-a(19) from Lars Blomberg, May 13 2011

a(20)-a(26) from Chai Wah Wu, Mar 04 2019