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Monetary amounts in U.S. cents such that the smallest number of coins summing to the amount uses only denominations of 10 cents and above.
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%I #15 Mar 17 2022 08:42:39

%S 10,20,25,35,45,50,60,70,75,85,95,100,110,120,125,135,145,150,160,170,

%T 175,185,195,200,210,220,225,235,245,250,260,270,275,285,295,300,310,

%U 320,325,335,345,350,360,370,375,385,395,400,410,420,425,435,445,450,460,470,475,485,495,500,510,520,525,535,545,550,560

%N Monetary amounts in U.S. cents such that the smallest number of coins summing to the amount uses only denominations of 10 cents and above.

%C U.S. coins are the cent ("penny") (1 cent), nickel (5 cents), dime (10 cents), quarter dollar ("quarter") (25 cents), half dollar (50 cents), and dollar (100 cents).

%C For values above 100, one can simply add 100 to the previous terms: a(n) = a(n-12) + 100. - _R. J. Mathar_, Aug 05 2010

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F From _R. J. Mathar_, Aug 05 2010: (Start)

%F a(n) = +a(n-1) +a(n-3) -a(n-4).

%F G.f.: 5*x*(2+2*x+x^2) / ( (1+x+x^2)*(x-1)^2 ). (End)

%e A total amount of 30 cents can be obtained using 3 coins as 10+10+10, but the only way to obtain 30 cents using 2 coins (which is the minimum number of coins) is as 25+5 (which includes a coin smaller than a 10-cent piece), so 30 is not in the sequence.

%e From _R. J. Mathar_, Aug 05 2010: (Start)

%e Shortest partitions of multiples of 5 into parts of 5, 10, 25, 50, 100:

%e 5 = 5

%e 10 = 10 a(1)

%e 15 = 5 + 10

%e 20 = 10 + 10 a(2)

%e 25 = 25 a(3)

%e 30 = 5 + 25

%e 35 = 10 + 25 a(4)

%e 40 = 5 + 10 + 25

%e 45 = 10 + 10 + 25 a(5)

%e 50 = 50 a(6)

%e 55 = 5 + 50

%e 60 = 10 + 50 a(7)

%e 65 = 5 + 10 + 50

%e 70 = 10 + 10 + 50 a(8)

%e 75 = 25 + 50 a(9)

%e 80 = 5 + 25 + 50

%e 85 = 10 + 25 + 50 a(10)

%e 90 = 5 + 10 + 25 + 50

%e 95 = 10 + 10 + 25 + 50 a(11)

%e 100 = 100. a(12)

%e (End)

%K nonn,easy

%O 1,1

%A _J. Lowell_, Jul 28 2010

%E More terms from _R. J. Mathar_, Aug 05 2010

%E More terms from _Max Alekseyev_, Oct 15 2012