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%I #30 Nov 08 2022 10:10:24
%S 1,2,3,4,1,2,3,4,5,1,2,3,4,5,2,3,4,5,6,2,3,4,5,6,1,2,3,4,5,2,3,4,5,6,
%T 2,3,4,5,6,3,4,5,6,7,3,4,5,6,7,2,3,4,5,6,3,4,5,6,7,3,4,5,6,7,4,5,6,7,
%U 8,4,5,6,7,8,3,4,5,6,7,4,5,6,7,8,4,5,6,7,8,5,6,7,8,9,5,6,7,8,9,4
%N Minimal number of coins needed to pay n cents using coins of denominations 1, 5, 10, 25 cents.
%H Colin Barker, <a href="/A053344/b053344.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Mag#change">Index entries for sequences related to making change.</a>
%H <a href="/index/Rec#order_26">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).
%F a(n) = floor(n/25) + floor([n mod 25]/10) + floor([{n mod 25} mod 10]/5) + ([n mod 25] mod 10) mod 5.
%F G.f.: -x*(5*x^24 -x^23 -x^22 -x^21 -x^20 +4*x^19 -x^18 -x^17 -x^16 -x^15 +3*x^14 -x^13 -x^12 -x^11 -x^10 +4*x^9 -x^8 -x^7 -x^6 -x^5 +3*x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^20 +x^15 +x^10 +x^5 +1)). - _Colin Barker_, Jan 10 2015
%e a(57) = 5 because to pay 57 cents at least 5 coins are needed: 2 of 25 cents, 1 of 5 cents and 2 of 1 cent.
%t f[n_]:=Floor[n/25]+Floor[Mod[n,25]/10]+Floor[Mod[Mod[n,25],10]/5]+Mod[Mod[Mod[n,25],10],5]; lst={};Do[AppendTo[lst,f[n]],{n,6!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Sep 28 2009 *)
%t Table[Min[Total/@FrobeniusSolve[{1,5,10,25},n]],{n,100}] (* or *) LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1},{1,2,3,4,1,2,3,4,5,1,2,3,4,5,2,3,4,5,6,2,3,4,5,6,1,2},100] (* _Harvey P. Dale_, Aug 14 2014 *)
%o (PARI) Vec(-x*(5*x^24 -x^23 -x^22 -x^21 -x^20 +4*x^19 -x^18 -x^17 -x^16 -x^15 +3*x^14 -x^13 -x^12 -x^11 -x^10 +4*x^9 -x^8 -x^7 -x^6 -x^5 +3*x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^20 +x^15 +x^10 +x^5 +1)) + O(x^100)) \\ _Colin Barker_, Jan 10 2015
%o (Magma) I:=[1,2,3,4,1,2,3,4,5,1,2,3,4,5,2,3,4,5,6,2,3,4,5,6,1,2]; [n le 26 select I[n] else Self(n-1) +Self(n-25) - Self(n-26): n in [1..70]]; // _G. C. Greubel_, May 31 2018
%o (Python)
%o def A053344(n):
%o a, b = divmod(n,25)
%o c, d = divmod(b,10)
%o return a+c+sum(divmod(d,5)) # _Chai Wah Wu_, Nov 08 2022
%Y Cf. A011542, A001299.
%K easy,nonn
%O 1,2
%A Jean Fontaine (jfontain(AT)odyssee.net), Jan 06 2000
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