A000008 -id:A000008 - cp's OEIS frontend

A025810 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)).

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 3, 3, 3, 6, 3, 6, 3, 6, 6, 6, 6, 6, 6, 10, 6, 10, 6, 10, 10, 10, 10, 10, 10, 15, 10, 15, 10, 15, 15, 15, 15, 15, 15, 21, 15, 21, 15, 21, 21, 21, 21, 21, 21, 28, 21, 28, 21, 28, 28, 28, 28, 28, 28, 36, 28, 36, 28, 36, 36, 36, 36, 36, 36

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts of size 2, 5, and 10.

a(n) is always a triangular number.

			G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 3*x^10 + x^11 + 3*x^12 + ...

Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,1,0,-1,0,0,1,0,-1,0,0,-1,0,1).

Cf. A000008, A000217, A008616.

CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^10)), {x,0,85}], x]  (* Harvey P. Dale, Apr 06 2011 *)
a[ n_] := Module[ {m = Mod[n, 10], k}, k = n - m; If[ m == 1 || m == 3, k -= 10]; k (k + 30) / 200 + 1]; (* Michael Somos, Aug 16 2016 *)

{a(n) = if( n<-16, a(-17 - n), polcoeff( 1 / ((1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n))}; \\ Michael Somos, Mar 18 2012

{a(n) = my(m = n%10); n -= m; if( m==1 || m==3, n -= 10); n * (n + 30) / 200 + 1}; \\ Michael Somos, Aug 16 2016

a(n) = (n^2 + 17*n + (5*n+22)*(-1)^n + 200 + 4*n*[2,-1,1,-2,0][n%5+1])\200 \\ Hoang Xuan Thanh, Aug 28 2025

G.f.: 1/((1-x^2)(1-x^5)(1-x^10)).
Euler transform of length 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1]. - Michael Somos, Mar 18 2012
a(n) = a(-17 - n) = a(n - 10) + A008616(n) for all n in Z. - Michael Somos, Mar 18 2012
a(n) = A000217( A008616(n) ) = A000008(n) - A000008(n - 1). - Michael Somos, Dec 15 2002

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

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, 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

A367253 The number of ways of making change for 5n cents with Canadian coins (5, 10, 25, 100, 200).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 18, 20, 22, 24, 26, 30, 32, 36, 38, 42, 46, 50, 54, 58, 62, 68, 72, 78, 82, 88, 94, 100, 106, 112, 118, 128, 134, 144, 150, 160, 170, 180, 190, 200, 210, 224, 234, 248, 258, 272, 286, 300, 314, 328, 342, 362

Offset: 0

Johann Peters, Nov 11 2023

Since 2012 the Canadian penny has been discontinued. The coins now commonly used are the nickel (5 cents), the dime (10 cents), the quarter (25 cents), the loonie (100 cents), and the toonie (200 cents).

Number of partitions of n into parts 1, 2, 5, 20, 40. - Alois P. Heinz, Nov 11 2023

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, -1, 0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, -1, 0, 1, -1, -1, 1).
Index entries for sequences related to making change.

Cf. A000008, A001312, A182086, A307849.

a[n_]:=Length[FrobeniusSolve[{5,10,25,100,200},5*n]]; a/@Range[0,100] (* Ivan N. Ianakiev, Nov 21 2023 *)
CoefficientList[Series[1/((1-x)*(1-x^2)*(1-x^5)*(1-x^20)*(1-x^40)),{x,0,1000}],x] (* Ray Chandler, Nov 22 2023 *)

From Alois P. Heinz, Nov 11 2023: (Start)
G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^20)*(1-x^40)).
a(20*n) = A307849(n). (End)

A267419 Number of ways of making change for n cents using coins whose values are the previous terms in the sequence, starting with 1,2 cents.

Original entry on oeis.org

1, 2, 2, 3, 5, 8, 10, 14, 17, 23, 28, 35, 43, 53, 64, 78, 93, 112, 132, 158, 184, 217, 253, 295, 342, 396, 455, 526, 600, 689, 784, 893, 1014, 1150, 1299, 1468, 1651, 1860, 2084, 2339, 2613, 2921, 3257, 3628, 4034, 4482, 4967, 5508, 6087, 6731, 7426, 8188, 9017, 9920, 10898, 11969, 13120, 14382, 15737, 17215

Offset: 1

Christopher Cormier, Jan 14 2016

nonn

			For n=4, the coins available are 1,2. There are a(4)=3 ways to make 4 cents with these coins:
4 = 1+1+1+1
4 = 2+1+1
4 = 2+2
Since there are 3 ways, now the available coins are 1,2,3. For n=5, we have:
5 = 1+1+1+1+1
5 = 2+1+1+1
5 = 2+2+1
5 = 3+1+1
5 = 3+2
for 5 ways to make change, so now 1,2,3,5 are available, etc.

Cf. A000008, A001300, A089197, A208216.

a = {1, 2}; Do[AppendTo[a, Count[IntegerPartitions@ n, w_ /; AllTrue[w, MemberQ[a, #] &]]], {n, 3, 60}]; a (* Michael De Vlieger, Jan 15 2016, Version 10 *)

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

A025810 Expansion of 1/((1-x^2)*(1-x^5)*(1-x^10)).

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

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A367253 The number of ways of making change for 5n cents with Canadian coins (5, 10, 25, 100, 200).

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A267419 Number of ways of making change for n cents using coins whose values are the previous terms in the sequence, starting with 1,2 cents.

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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.

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A025810 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)).