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A036517 Smallest triangular number containing exactly n 0's.

%I A036517 #24 Oct 21 2022 22:05:21
%S A036517 0,300,20100,300700,2001000,50005000,200010000,5000050000,20000100000,
%T A036517 500000500000,2000001000000,50000005000000,200000010000000,
%U A036517 5000000050000000,20000000100000000,500000000500000000,2000000001000000000,50000000005000000000
%N A036517 Smallest triangular number containing exactly n 0's.
%H A036517 Chai Wah Wu, <a href="/A036517/b036517.txt">Table of n, a(n) for n = 1..100</a>
%H A036517 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,110,0,-1000).
%F A036517 a(n) = A000217(A048355(n)).
%F A036517 From _Colin Barker_, Mar 25 2020: (Start)
%F A036517 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)).
%F A036517 a(n) = 110*a(n-2) - 1000*a(n-4) for n>6.
%F A036517 (End)
%t A036517 nsmall = Table[Infinity, 20];
%t A036517 For[i = 0, i <= 10^6, i++, p = PolygonalNumber[i];
%t A036517   n0 = Count[IntegerDigits[p], 0];
%t A036517   If[nsmall[[n0]] > p, nsmall[[n0]] = p]];
%t A036517 ReplaceAll[nsmall, Infinity -> "?"] (* _Robert Price_, Mar 22 2020 *)
%t A036517 LinearRecurrence[{0,110,0,-1000},{0,300,20100,300700,2001000,50005000,200010000,5000050000},30] (* _Harvey P. Dale_, Jul 31 2021 *)
%o A036517 (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
%Y A036517 Cf. A000217, A048355.
%K A036517 nonn,base,easy
%O A036517 1,2
%A A036517 _David W. Wilson_