A001364 -id:A001364 - cp's OEIS frontend

A182086 Number of ways of making change for n Pfennig using Deutschmark coins.

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, 64, 70, 76, 82, 88, 98, 104, 114, 120, 130, 140, 150, 160, 170, 180, 195, 205, 220, 230, 245, 260, 275, 290, 305, 320, 342, 357, 379, 394, 416, 438, 460, 482, 504, 526

Offset: 0

Reinhard Zumkeller, Apr 11 2012

nonn

The Pfennig was the subunit of the Deutsche Mark, the currency of Germany until the adoption of the Euro in 2002; the coins were (business strike): 1 Pfg, 2 Pfg, 5 Pfg, 10 Pfg, 50 Pfg, 1 DM = 100 Pfg, 2 DM and 5 DM;
a(n) = A000008(n) for n < 50; a(50) = A000008(50) + 1 = 342;
a(n) = A001312(n) for n < 200; a(200) = A001312(200) + 1 = 26905.
Number of partitions of n into parts 1, 2, 5, 10, 50, 100, 200, and 500. - Joerg Arndt, Jul 08 2013

			Number of partitions of coin values into coin values:
a(1) = #{1} = 1;
a(2) = #{2, 1+1} = 2;
a(5) = #{5, 2+2+1, 2+1+1+1, 1+1+1+1+1} = 4;
a(10) = #{10, 5+5, 5+2+2+1, 5+2+1+1+1, 5+5x1, 2+2+2+2+2, 2+2+2+2+1+1, 2+2+2+1+1+1+1, 2+2+6x1, 2+8x1, 10x1} = 11;
a(50) = #{50,10+10+10+10+10, 10+10+10+10+5+5, 10+10+10+10+5+2+2+1, 10+10+10+10+5+2+1+1+1, 10+10+10+10+5+10x1, ...} = 342;
a(100) = 2499;
a(200) = 26905;
a(500) = 1229587.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Deutsche Bundesbank, Umlaufmuenzen
Wikipedia, Deutsche Mark, Coins
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, order 868.

Cf. A001364, A067996, A082593.

a182086 = p [1,2,5,10,50,100,200,500] where
   p  0 = 1; p []  = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

CoefficientList[Series[1/((1 - x)*(1 - x^2)*(1 - x^5)*(1 - x^10)*(1 - x^50)*(1 - x^100)*(1 - x^200)*(1 - x^500)), {x, 0, 50}], x] (* G. C. Greubel, Aug 20 2017 *)

Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^50)*(1-x^100)*(1-x^200)*(1-x^500))+O(x^566)) \\ Joerg Arndt, Jul 08 2013

G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^50)*(1-x^100)*(1-x^200)*(1-x^500)). - Joerg Arndt, Jul 08 2013

A339094 Number of (unordered) ways of making change for n US Dollars using the current US denominations of $1, $2, $5, $10, $20, $50 and $100 bills.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28, 31, 34, 41, 44, 51, 54, 61, 68, 75, 82, 89, 96, 109, 116, 129, 136, 149, 162, 175, 188, 201, 214, 236, 249, 271, 284, 306, 328, 350, 372, 394, 416, 451, 473, 508, 530, 565, 600, 635, 670, 705, 740, 793, 828, 881, 916

Offset: 0

Robert G. Wilson v, Nov 25 2020

Not the same as A001313. First difference appears at A001313(100) being 4562, whereas a(100) is 4563; obviously one more than A001313(100).
Not the same as A057537.
Number of partitions of n into parts 1, 2, 5, 10, 20, 50 and 100.

			a(5) is 4 because 1+1+1+1+1 = 2+1+1+1 = 2+2+1 = 5.

Cf. A000008, A001299, A001300, A001301, A001306, A001302, A001306, A001310, A001312, A001313, A001314, A001319, A001343, A001362, A001364, A057537, A067996, A067997, A073031, A085502, A112024, A124146, A160551, A169718, A181934, A187243.

f[n_] := Length@ IntegerPartitions[n, All, {1, 2, 5, 10, 20, 50, 100}]; Array[f, 75, 0] (* or *)
CoefficientList[ Series[1/((1 - x) (1 - x^2) (1 - x^5) (1 - x^10) (1 - x^20) (1 - x^50) (1 - x^100)), {x, 0, 75}], x] (* or *)
Table[ Length@ FrobeniusSolve[{1, 2, 5, 10, 20, 50, 100}, n], {n, 0, 75}] (* much slower *)

coins(v[..])=my(x='x); prod(i=1, #v, 1/(1-x^v[i]))
Vec(coins(1, 2, 5, 10, 20, 50, 100)+O(x^99)) \\ Charles R Greathouse IV, Jan 24 2022

G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^50)*(1-x^100)).

cp's OEIS Frontend

A182086 Number of ways of making change for n Pfennig using Deutschmark coins.

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