A048355 - OEIS

%I #39 Jul 26 2024 17:14:14

%S 0,24,200,775,2000,10000,20000,100000,200000,1000000,2000000,10000000,

%T 20000000,100000000,200000000,1000000000,2000000000,10000000000,

%U 20000000000,100000000000,200000000000,1000000000000,2000000000000,10000000000000,20000000000000,100000000000000

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

%H Chai Wah Wu, <a href="/A048355/b048355.txt">Table of n, a(n) for n = 1..100</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,10).

%F From _Bernard Schott_, Mar 04 2019: (Start)

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

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

%F (End)

%F From _Colin Barker_, Mar 25 2020: (Start)

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

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

%F (End)

%e From _Bernard Schott_, Mar 04 2019: (Start)

%e a(2) = 24: T(24) = 300 which contains exactly two 0's.

%e a(6) = 10000: T(10000) = 50005000 which contains exactly six 0's.

%e a(7) = 20000: T(20000) = 200010000 which contains exactly seven 0's.

%e (End)

%t nsmall = Table[Infinity, 20];

%t For[i = 0, i <= 10^6, i++, p = PolygonalNumber[i];

%t   n0 = Count[IntegerDigits[p], 0];

%t   If[nsmall[[n0]] > i, nsmall[[n0]] = i]];

%t ReplaceAll[nsmall, Infinity -> "?"] (* _Robert Price_, Mar 22 2020 *)

%t LinearRecurrence[{0,10},{0,24,200,775,2000,10000},30] (* _Harvey P. Dale_, Jul 26 2024 *)

%o (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 Cf. A000217, A036517.

%K nonn,base,easy

%O 1,2

%A _Patrick De Geest_, Mar 15 1999

%E a(16)-a(19) from _Lars Blomberg_, May 13 2011

%E a(20)-a(26) from _Chai Wah Wu_, Mar 04 2019