A036517 Smallest triangular number containing exactly n 0's.

0, 300, 20100, 300700, 2001000, 50005000, 200010000, 5000050000, 20000100000, 500000500000, 2000001000000, 50000005000000, 200000010000000, 5000000050000000, 20000000100000000, 500000000500000000, 2000000001000000000, 50000000005000000000

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• base
• easy

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..100
• Index entries for linear recurrences with constant coefficients, signature (0,110,0,-1000).

Crossrefs

Cf. A000217, A048355.

Programs

Mathematica

nsmall = Table[Infinity, 20];
For[i = 0, i <= 10^6, i++, p = PolygonalNumber[i];
  n0 = Count[IntegerDigits[p], 0];
  If[nsmall[[n0]] > p, nsmall[[n0]] = p]];
ReplaceAll[nsmall, Infinity -> "?"] (* Robert Price, Mar 22 2020 *)
LinearRecurrence[{0,110,0,-1000},{0,300,20100,300700,2001000,50005000,200010000,5000050000},30] (* Harvey P. Dale, Jul 31 2021 *)

PARI

concat(0, Vec(100*x^2*(3 + 201*x + 2677*x^2 - 2100*x^3 + 172280*x^4 - 1998000*x^6) / ((1 - 10*x)*(1 + 10*x)*(1 - 10*x^2)) + O(x^20))) \\ Colin Barker, Mar 25 2020

Formula

a(n) = A000217(A048355(n)).

From Colin Barker, Mar 25 2020: (Start)

G.f.: 100*x^2*(3 + 201*x + 2677*x^2 - 2100*x^3 + 172280*x^4 - 1998000*x^6) / ((1 - 10*x)*(1 + 10*x)*(1 - 10*x^2)).

a(n) = 110*a(n-2) - 1000*a(n-4) for n>6.

(End)