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A001301 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25 cents. 5

%I

%S 1,1,2,2,3,4,5,6,7,8,11,12,15,16,19,22,25,28,31,34,40,43,49,52,58,65,

%T 71,78,84,91,102,109,120,127,138,151,162,175,186,199,217,230,248,261,

%U 279,300,318,339,357,378,406,427,455,476,504,536,564,596,624,656

%N Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25 cents.

%C Number of partitions of n into parts 1, 2, 5, 10, and 25. - _Joerg Arndt_, Sep 05 2014

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.

%D G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

%H T. D. Noe, <a href="/A001301/b001301.txt">Table of n, a(n) for n = 0..1000</a>

%H H. Bottomley, <a href="/A000008/a000008.gif">Initial terms of A000008, A001301, A001302, A001312, A001313</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=177">Encyclopedia of Combinatorial Structures 177</a>

%H <a href="/index/Rec#order_43">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1).

%H <a href="/index/Mag#change">Index entries for sequences related to making change.</a>

%F G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)).

%p M := Matrix(43, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 2, 5, 8, 10, 13, 16, 17, 25, 28, 31, 32, 36, 37, 40, 43])) then 1 elif j=1 and member(i, [3, 6, 7, 11, 12, 15, 18, 26, 27, 30, 33, 35, 38, 41, 42]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..51); # _Alois P. Heinz_, Jul 25 2008

%t CoefficientList[ Series[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 55} ], x ]

%t Table[Length[FrobeniusSolve[{1,2,5,10,25},n]],{n,0,60}] (* _Harvey P. Dale_, Jan 19 2020 *)

%o (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)) + O(x^100)) \\ _Michel Marcus_, Sep 05 2014

%K nonn

%O 0,3

%A _N. J. A. Sloane_.

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Last modified July 11 05:49 EDT 2020. Contains 335609 sequences. (Running on oeis4.)