A067996 -id:A067996 - cp's OEIS frontend

A067997 Number of (unordered) ways of making change for n cents using coins of 1/2, 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage denominations up to 100 cents).

1, 2, 4, 7, 11, 17, 25, 35, 48, 64, 85, 110, 141, 178, 222, 275, 337, 409, 493, 589, 702, 830, 977, 1144, 1333, 1549, 1792, 2065, 2372, 2714, 3100, 3528, 4005, 4534, 5119, 5769, 6485, 7273, 8140, 9089, 10135, 11276, 12524, 13885, 15366, 16983, 18738

Offset: 0

Rick L. Shepherd, Feb 08 2002

The U.S.A. issued the following unusual denomination coins during the 18th and 19th centuries: 1/2-cent pieces, 1793-1857; 2-cent pieces, 1864-1873; 3-cent pieces, 1851-1889; and 20-cent pieces, 1875-1878. This sequence is also the number of ways of making change for n cents using coins of 1 (two types, say, old pre-1858 "large cents" and 1856-to-present "small cents"), 2, 3, 5, 10, 20, 25, 50, 100 cents. For present purposes, one of the two types of 1-cent piece is actually taken to be two 1/2-cent pieces.

			a(2)=4 because change can be made for 2 cents in these 4 ways: (1) 4 1/2-cent coins, (2) 2 1/2-cent, 1 1-cent, (3) 2 1-cent, (4) 1 2-cent coin.

R. S. Yeoman, A Guide Book of United States Coins, Ed. Kenneth Bressett, 53rd Edition (2000). New York: St. Martin's Press, 1999. pp. 72-77, 92-93, 104-106, 135. (also known as The Official Red Book of United States Coins)

Ron Guth, Your Online Reference For U.S. Coins
Mitch Hight, United States Coinage History
Clint Leland, U.S. Coins Quantities Minted
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, order 217.

Cf. A067996, A067995, A001314 (two kinds of nickels), A028291 (analog for 1/2, 1, 2, 3, 5 only, or 1(two types), 2, 3, 5 only).

CoefficientList[ Series[1 / ((1 - x)^2(1 - x^2)(1 - x^3)(1 - x^5)(1 - x^10)(1 - x^20)(1 - x^25)(1 - x^50)(1 - x^100)), {x, 0, 50} ], x]

a(n)=polcoeff(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100)+x*O(x^n)), n)

G.f.: 1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^25)*(1-x^50)*(1-x^100))

A067995 Minimal number of coins needed to pay exactly n cents using coins of sizes 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage from 1 to 100 cents).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 4, 4, 4, 5, 3, 4, 4, 4, 5, 3, 4, 4, 4, 5, 3, 4, 4, 4, 5, 1, 2, 2, 2, 3, 2

Offset: 1

Rick L. Shepherd, Feb 06 2002

The U.S.A. issued the following unusual denomination coins during the 19th century: 2-cent pieces, 1864-1873; 3-cent pieces, 1851-1889; and 20-cent pieces, 1875-1878.

			a(69) = 5 because to pay exactly 69 cents at least 5 coins are needed: e.g. 1 of 50 cents, 1 of 10 cents, 1 of 5 cents and 2 of 2 cents.

R. S. Yeoman, A Guide Book of United States Coins, Ed. Kenneth Bressett, 53rd Edition (2000). New York: St. Martin's Press, 1999. pp. 104-106, 135. (also known as The Official Red Book of United States Coins)

Ray Chandler, Table of n, a(n) for n = 1..1000
Jean-Paul Delahaye, Quelles pièces pour faire l'appoint ?, French edition of the Scientific American, Pour la Science, N°335, septembre 2005.
Jeffrey Shallit, What this country needs is an 18c piece, The Mathematical Intelligencer, June 2003, Volume 25, Issue 2, pp 20-23.
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A053344, A067996, A067997.

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

A020702 Expansion of (1+x^10)/((1-x)(1-x^2)(1-x^3)*(1-x^5)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 13, 16, 21, 25, 31, 37, 44, 53, 62, 72, 84, 96, 111, 126, 143, 161, 181, 203, 226, 251, 278, 306, 338, 370, 405, 442, 481, 523, 567, 613, 662, 713, 768, 824, 884, 946, 1011, 1080, 1151, 1225, 1303, 1383, 1468, 1555, 1646

Offset: 0

G. Nebe, E. Rains, N. J. A. Sloane, Apr 05 2002

nonn

Rescaled version of Molien series for self-dual Quebbemann codes over GF(4).

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,0,0,-1,0,1,1,-1).

Different from A067996.

a(n) ~ 1/90*n^3 + 1/15*n^2. - Ralf Stephan, Apr 29 2014

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

A067997 Number of (unordered) ways of making change for n cents using coins of 1/2, 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage denominations up to 100 cents).

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A067995 Minimal number of coins needed to pay exactly n cents using coins of sizes 1, 2, 3, 5, 10, 20, 25, 50, 100 cents (all historical U.S.A. coinage from 1 to 100 cents).

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

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A020702 Expansion of (1+x^10)/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)).

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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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A020702 Expansion of (1+x^10)/((1-x)(1-x^2)(1-x^3)*(1-x^5)).