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

%I A048355 #40 Feb 16 2025 08:32:39
%S A048355 0,24,200,775,2000,10000,20000,100000,200000,1000000,2000000,10000000,
%T A048355 20000000,100000000,200000000,1000000000,2000000000,10000000000,
%U A048355 20000000000,100000000000,200000000000,1000000000000,2000000000000,10000000000000,20000000000000,100000000000000
%N A048355 a(n) is the index of the smallest triangular number containing exactly n 0's.
%H A048355 Chai Wah Wu, <a href="/A048355/b048355.txt">Table of n, a(n) for n = 1..100</a>
%H A048355 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>
%H A048355 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,10).
%F A048355 From _Bernard Schott_, Mar 04 2019: (Start)
%F A048355 for n odd >= 5, a(n) = 2 * 10^((n+1)/2),
%F A048355 for n even >= 6, a(n) = 10^((n+2)/2).
%F A048355 (End)
%F A048355 From _Colin Barker_, Mar 25 2020: (Start)
%F A048355 G.f.: x^2*(24 + 200*x + 535*x^2 + 2250*x^4) / (1 - 10*x^2).
%F A048355 a(n) = 10*a(n-2) for n>4.
%F A048355 (End)
%e A048355 From _Bernard Schott_, Mar 04 2019: (Start)
%e A048355 a(2) = 24: T(24) = 300 which contains exactly two 0's.
%e A048355 a(6) = 10000: T(10000) = 50005000 which contains exactly six 0's.
%e A048355 a(7) = 20000: T(20000) = 200010000 which contains exactly seven 0's.
%e A048355 (End)
%t A048355 nsmall = Table[Infinity, 20];
%t A048355 For[i = 0, i <= 10^6, i++, p = PolygonalNumber[i];
%t A048355   n0 = Count[IntegerDigits[p], 0];
%t A048355   If[nsmall[[n0]] > i, nsmall[[n0]] = i]];
%t A048355 ReplaceAll[nsmall, Infinity -> "?"] (* _Robert Price_, Mar 22 2020 *)
%t A048355 LinearRecurrence[{0,10},{0,24,200,775,2000,10000},30] (* _Harvey P. Dale_, Jul 26 2024 *)
%o A048355 (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
%Y A048355 Cf. A000217, A036517.
%K A048355 nonn,base,easy
%O A048355 1,2
%A A048355 _Patrick De Geest_, Mar 15 1999
%E A048355 a(16)-a(19) from _Lars Blomberg_, May 13 2011
%E A048355 a(20)-a(26) from _Chai Wah Wu_, Mar 04 2019