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A151945 Denomination sequence. Start with the 0th and first coins of value 1 cent: a(0)=a(1)=1. Thereafter a(n), the value of the n-th coin (n>=2), is the number of ways to make change for n cents in earlier coins. The two one-cent coins are considered distinct. 1
1, 1, 3, 5, 7, 10, 14, 19, 25, 32, 42, 53, 66, 82, 101, 124, 150, 181, 216, 257, 306, 361, 424, 495, 577, 671, 776, 895, 1029, 1180, 1350, 1540, 1752, 1988, 2252, 2547, 2872, 3231, 3630, 4071, 4558, 5093, 5683, 6330, 7040, 7822, 8674, 9606, 10625, 11738 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of nonnegative solutions to the Diophantine equation 1*x_0 + 1*x_1 + ... + a(n-1)*x_(n-1) = n. - Melvin Peralta, Jan 03 2016

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..3500

FORMULA

G.f: g(x) = Product_{n >= 0} 1/(1-x^a(n)) - x.

EXAMPLE

Call the two one-cent coins c and d.

Then we can make change for 2 cents in three ways: cc,cd,dd, so a(2) = 3.

Then we can make change for 3 cents in five ways: ccc,ccd,cdd,ddd,3, so a(3) = 5.

MAPLE

b:= proc(n, i) option remember;

      if n<0 then 0

    elif n=0 then 1

    elif i<0 then 0

    elif i=1 then n+1

    else b(n, i-1) +b(n-a(i), i)

      fi

    end:

a:= n-> b(n, n-1):

seq(a(n), n=0..100);  # Alois P. Heinz, Aug 14 2009

MATHEMATICA

b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 0, 0, If[i == 1, n + 1, b[n, i - 1] + b[n - a[i], i]] ]]]; a[0] = a[1] = 1; a[n_] := a[n] = b[n, n - 1]; Table[ a@n, {n, 0, 50}] (* Robert G. Wilson v, Aug 17 2009 *)

PROG

(Haskell)

a151945 n = a151945_list !! n

a151945_list = 1 : 1 : f [2..] where

   f (x:xs) = p (take x a151945_list) x : f xs

   p _ 0 = 1; p [] _ = 0

   p ds'@(d:ds) m = if m < d then 0 else p ds' (m - d) + p ds m

-- Reinhard Zumkeller, Jan 21 2014

CROSSREFS

Sequence in context: A054040 A011848 A131673 * A140261 A015827 A088635

Adjacent sequences:  A151942 A151943 A151944 * A151946 A151947 A151948

KEYWORD

nonn,nice

AUTHOR

David W. Wilson, Aug 14 2009

EXTENSIONS

More terms from Alois P. Heinz, Aug 14 2009

STATUS

approved

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Last modified August 20 11:44 EDT 2017. Contains 290835 sequences.