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%I #14 Sep 08 2022 08:45:51
%S 5,6,7,8,9,15,16,17,18,19,30,31,32,33,34,40,41,42,43,44,55,56,57,58,
%T 59,65,66,67,68,69,80,81,82,83,84,90,91,92,93,94,105,106,107,108,109,
%U 115,116,117,118,119,130,131,132,133,134,140,141,142,143,144,155,156,157
%N Numbers congruent to {5,6,7,8,9,15,16,17,18,19} mod 25.
%C Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts includes a part of size 5.
%C For each number the partition is unique and exactly one part is of size 5.
%C Complement of A174139.
%C Amounts in cents requiring a nickel when the minimal number of coins is selected from pennies, nickels, dimes, and quarters (whether usage of bills for whole-dollar amounts is permitted or not).
%C For each n >= 0, floor(n/25) parts of size 25 (quarters) occur in the partition with minimal number of these parts (regardless of whether partition includes part of size 5).
%H <a href="/index/Mag#change">Index entries for sequences related to making change.</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,1,-1).
%F a(10+n) = a(n) + 25 for n >= 1.
%F a(n) = a(n-1) + a(n-10) - a(n-11). G.f.: x*(5+x+x^2+x^3+x^4+6*x^5+x^6+x^7+x^8+x^9+6*x^10) / ( (1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011
%F a(n) = n+9+5*floor((floor((n-1)/5)-1)/2)+10*floor(floor((n-1)/5)/2). - _Wesley Ivan Hurt_, Mar 22 2015
%e As 15 = 10 + 5, 15 is a term since 5 is included and all other candidate partitions have more than two parts. Similarly, as 30 = 25 + 5, 30 is a term. However, 45 = 25 + 10 + 10 is not a term as it contains no part of size 5.
%t Table[n + 9 + 5 Floor[(Floor[(n - 1)/5] - 1)/2] + 10 Floor[Floor[(n - 1)/5]/2], {n, 100}] (* _Wesley Ivan Hurt_, Mar 22 2015 *)
%o (Magma) [n : n in [1..200] | n mod 25 in [5, 6, 7, 8, 9, 15, 16, 17, 18, 19]]; // _Vincenzo Librandi_, Mar 22 2015
%Y Cf. A174139, A174140, A174141, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).
%K easy,nonn
%O 1,1
%A _Rick L. Shepherd_, Mar 09 2010
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