A001300 -id:A001300 - cp's OEIS frontend

A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 13, 13, 13, 13, 13, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 31, 31, 31, 31, 31, 39, 39, 39, 39, 39, 49, 49, 49, 49, 49, 60, 60, 60, 60, 60, 73, 73, 73, 73, 73, 87, 87, 87, 87, 87, 103, 103, 103, 103, 103

Offset: 0

N. J. A. Sloane, Mar 15 1996

a(n) = A001300(n) = A169718(n) for n < 50. - Reinhard Zumkeller, Dec 15 2013

Number of partitions of n into parts 1, 5, 10, and 25. - Joerg Arndt, Sep 05 2014

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

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

T. D. Noe, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 175
Gerhard Kirchner, Derivation of formulas
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003.
Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 1, -1).

Cf. A001300, A169718, A000008.

a001299 = p [1,5,10,25] where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 15 2013

CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 65} ], x ]
Table[Length[FrobeniusSolve[{1,5,10,25},n]],{n,0,80}] (* Harvey P. Dale, Dec 01 2015 *)
a[ n_] := With[ {m = Quotient[n, 5] / 10}, Round[ (4 m + 3) (5 m + 1) (5 m + 2) / 6]]; (* Michael Somos, Feb 23 2017 *)

a(n)=floor((n\5+1)*((n\5+2)*(2-n%5)/100+[54,27,-2,-33,-66][n%5+1]/500)+(2-5*(n%5%2))*(-1)^n/40+(2*n^3+123*n^2+2146*n+16290)/15000) \\ Tani Akinari, May 09 2014

{a(n) = my(m=n\5 / 10); round((4*m + 3) * (5*m + 1) * (5*m + 2) / 6)}; /* Michael Somos, Feb 23 2017 */

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

a(n) = round((100*x^3 + 135*x^2 +53*x)/6) + 1 with x= floor(n/5)/10. See link "Derivation of formulas". - Gerhard Kirchner, Feb 23 2017

A000008 Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

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, 341, 356, 377, 392, 413, 434, 455, 476, 497, 518, 546

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, 5, and 10.

There is a unique solution to this puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a semiprime number of ways that I can make change for n-1 cents and for n+1 cents." There is a unique solution to this related puzzle: "There are a prime number of ways that I can make change for n cents using coins of 1, 2, 5, 10 cents; but a 3-almost prime number of ways that I can make change for n-1 cents and for n+1 cents." - Jonathan Vos Post, Aug 26 2005

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

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

William Boyles, Table of n, a(n) for n = 0..10000 (terms 0...1000 from T. D. Noe)
Henry Bottomley, Initial terms of A000008, A001301, A001302, A001312, A001313
X. Gourdon and B. Salvy, Effective asymptotics of linear recurrences with rational coefficients, Discrete Mathematics, vol. 153, no. 1-3, 1996, pages 145-163.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 174
Gerhard Kirchner, Derivation of formulas
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,-1,-1,1,0,1,-1,-1,1,0,-1,1,1,-1).

Cf. A001299, A025810.

Cf. A001300, A169718.

a000008 = p [1,2,5,10] where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 15 2013

[#RestrictedPartitions(n,{1,2,5,10}):n in [0..60]]; // Marius A. Burtea, May 07 2019

M:= Matrix(18, (i,j)-> if(i=j-1 and i<17) or (j=1 and member(i, [2,5,10,17,18])) or (i=18 and j=18) then 1 elif j=1 and member(i, [7,12,15]) then -1 else 0 fi); a:= n-> (M^(n+1))[18,1]; seq(a(n), n=0..51); # Alois P. Heinz, Jul 25 2008
# second Maple program:
a:= proc(n) local m, r; m := iquo(n, 10, 'r'); r:= r+1; ([23, 26, 35, 38, 47, 56, 65, 74, 83, 92][r]+ (3*r+ 24+ 10*m) *m) *m/6+ [1, 1, 2, 2, 3, 4, 5, 6, 7, 8][r] end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 05 2008

a[ n_] := SeriesCoefficient[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)), {x, 0, n}]
a[n_, d_] := SeriesCoefficient[1/(Times@@Map[(1-x^#)&, d]), {x, 0, n}] (* general case for any set of denominations represented as a list d of coin values in cents *)
Table[Length[FrobeniusSolve[{1,2,5,10},n]],{n,0,70}] (* Harvey P. Dale, Apr 02 2012 *)
LinearRecurrence[{1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1}, {1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 11, 12, 15, 16, 19, 22, 25, 28}, 100] (* Vincenzo Librandi, Feb 10 2016 *)
a[ n_] := Quotient[ With[{r = Mod[n, 10, 1]}, n^3 + 27 n^2 + (191 + 3 {4, 13, 0, 5, 8, 9, 8, 5, 0, 13}[[r]]) n + 25], 600] + 1; (* Michael Somos, Mar 06 2018 *)
Table[Length@IntegerPartitions[n,All,{1,2,5,10}],{n,0,70}] (* Giorgos Kalogeropoulos, May 07 2019 *)

a(n):=floor(((n+17)*(2*n^2+20*n+81)+15*(n+1)*(-1)^n+120*((floor(n/5)+1)*((1+(-1)^mod(n,5))/2-floor(((mod(n,5))^2)/8))))/1200); /* Tani Akinari, Jun 21 2013 */

{a(n) = if( n<-17, -a(-18-n), if( n<0, 0, polcoeff( 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n)))}; /* Michael Somos, Apr 01 2003 */

Vec( 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)) + O(x^66) ) \\ Joerg Arndt, Oct 02 2013

{a(n) = my(r = (n-1)%10 + 1); (n^3 + 27*n^2 + (191 + 3*[4, 13, 0, 5, 8, 9, 8, 5, 0, 13][r])*n + 25)\600 + 1}; /* Michael Somos, Mar 06 2018 */

G.f.: 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)). - Michael Somos, Nov 17 1999
a(n) - a(n-1) = A025810(n). - Michael Somos, Dec 15 2002
a(n) = a(n-2) + a(n-5) - a(n-7) + a(n-10) - a(n-12) - a(n-15) + a(n-17) + 1. - Michael Somos, Apr 01 2003
a(n) = -a(-18-n). - Michael Somos, Apr 01 2003
a(n) = (q+1)*(h(n) - q*(3n-10q+7)/6) with q = floor(n/10) and h(n) = A000115(n) = round((n+4)^2/20). See link "Derivation of formulas". - Gerhard Kirchner, Feb 10 2017
a(n) = floor((2*n^3 + 54*n^2 + 421*n + 15*n*(-1)^n + 24*n * ((-1)^[(n mod 5)>2] - [(n mod 5)=1]) + 1248)/1200). - Hoang Xuan Thanh, Jun 27 2025

A169718 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 and 100 cents.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 13, 13, 13, 13, 13, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 31, 31, 31, 31, 31, 39, 39, 39, 39, 39, 50, 50, 50, 50, 50, 62, 62, 62, 62, 62, 77, 77, 77, 77, 77, 93, 93, 93, 93, 93, 112, 112, 112, 112, 112, 134, 134

Offset: 0

N. J. A. Sloane, Apr 20 2010

nonn

a(n) = A001300(n) for n < 100; a(n) = A001299(n) for n < 50. - Reinhard Zumkeller, Dec 15 2013

Number of partitions of n into parts 1, 5, 10, 25, 50, and 100. - Joerg Arndt, Sep 05 2014

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Erickson, Change for a dollar, change for a million, Math. Horizons, Feb 2010, pp. 22-25.
B. Polster, Explaining the bizarre pattern in making change for a googol dollars (infinite generating functions), Mathologer video (2021).
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, order 191.

Cf. A001299, A001300, A000008.

a169718 = p [1,5,10,25,50,100] where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Dec 15 2013

Table[Length[FrobeniusSolve[{1,5,10,25,50,100},n]],{n,0,80}] (* or *) CoefficientList[Series[1/((1-x)(1-x^5)(1-x^10)(1-x^25)(1-x^50)(1-x^100)),{x,0,80}],x] (* Harvey P. Dale, Dec 25 2011 *)

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

A001302 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents.

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, 65, 71, 78, 84, 91, 102, 109, 120, 127, 138, 151, 162, 175, 186, 199, 217, 230, 248, 261, 279, 300, 318, 339, 357, 378, 407, 428, 457, 478, 507, 540, 569, 602, 631, 664

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n into parts 1, 2, 5, 10, 25, and 50. - Joerg Arndt, Sep 05 2014

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 316.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 1.

T. D. Noe, Table of n, a(n) for n = 0..1000
H. Bottomley, Initial terms of A000008, A001301, A001302, A001312, A001313
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 178
Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, 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, 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, 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).

CoefficientList[ Series[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)(1 - x^25)(1 - x^50)), {x, 0, 55} ], x ]
Array[Length@IntegerPartitions[#, All, {1, 2, 5, 10, 25, 50}]&, 100, 0] (* Giorgos Kalogeropoulos, Apr 24 2021 *)

Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50))+ O(x^100)) \\ Michel Marcus, Sep 05 2014

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

a(n) = Sum_{k=0..floor(n/2)} A001300(n-2*k). - Christian Krause, Apr 24 2021

A187243 Number of ways of making change for n cents using coins of 1, 5, and 10 cents.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 9, 9, 9, 9, 9, 12, 12, 12, 12, 12, 16, 16, 16, 16, 16, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 30, 30, 30, 30, 30, 36, 36, 36, 36, 36, 42, 42, 42, 42, 42, 49, 49, 49, 49, 49, 56, 56, 56, 56, 56, 64, 64, 64, 64, 64, 72, 72, 72, 72, 72

Offset: 0

T. D. Noe, Mar 07 2011

a(n) is the number of partitions of n into parts 1, 5, and 10. - Joerg Arndt, Feb 02 2017
From Gerhard Kirchner, Jan 25 2017: (Start)
There is a simple recurrence for solving such problems given coin values 1 = c(1) < c(2) < ... < c(k).
Let f(n, j), 1 < j <= k, be the number of ways of making change for n cents with coin values c(i), 1 <= i <= j. Then any number m of c(j)-coins with 0 <= m <= floor(n/c(j)) can be used, and the remaining amount of change to be made using coins of values smaller than c(j) will be n - m*c(j) cents. This leads directly to the recurrence formula with a(n) = f(n, k).
For k = 3 with c(1) = 1, c(2) = 5, c(3) = 10, the recurrence can be reduced to an explicit formula; see link "Derivation of formulas".
By the way, a(n) is also the number of ways of making change for n cents using coins of 2, 5, 10 cents and at most one 1-cent coin. That is because any coin combination is, as in the original problem, fixed by the numbers of 5-cent and 10-cent coins.
(End)

			From _Gerhard Kirchner_, Jan 25 2017: (Start)
Recurrence:
a(11)  = f(11, 3) = f(11 - 0, 2) + f(11 - 10, 2)
       = f(11 - 0, 1) + f(11 - 5, 1) + f(11 - 10, 1) + f(1, 2)
       = 1 + 1 + 1 + 1 = 4.
Explicitly: a(79) = (7 + 1)*(7 + 1 + 1) = 72.
(End)

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

Cf. A001299, A001300, A001306, A169718, A002266, A002620, A245573.

CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)), {x, 0, 75} ], x ]

Vec( 1/((1-x)*(1-x^5)*(1-x^10))+O(x^99)) \\ Charles R Greathouse IV, Aug 22 2011

a(n)=(n^2+16*n+97+10*(n\5+1)*(5*(n\5)+2-n))\100 \\ Tani Akinari, Sep 10 2015

a(n) = {my(q=n\10, s=(n%10)\5); (q+1)*(q+1+s); } \\ (Kirchner's explicit formula) Joerg Arndt, Feb 02 2017

G.f.: 1/((1-x)*(1-x^5)*(1-x^10)).
From Gerhard Kirchner, Jan 25 2017: (Start)
General recurrence: f(n, 1) = 1; j > 1: f(0, j) = 1 or f(n, j) = Sum_{m=0..floor(n/c(j))} f(n-m*c(j), j-1);
a(n) = f(n, k).
Note: f(n, j) = f(n, j-1) for n < c(j) => f(1, j) = 1.
Explicit formula:
a(n) = (q+1)*(q+1+s) with q = floor(n/10) and s = floor((n mod 10)/5). (End)
a(n) = A002620(A002266(n)+2) = floor((floor(n/5) + 2)^2 / 4). - Hoang Xuan Thanh, Jun 26 2025

A212774 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which, when using the minimal number of coins, have equal numbers of all denominations used.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 10, 11, 15, 16, 20, 22, 25, 26, 30, 31, 35, 36, 40, 41, 50, 51, 55, 56, 60, 61, 65, 66, 75, 76, 80, 81, 85, 86, 90, 91, 100, 102, 120, 122, 150, 153, 200, 204, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900, 950

Offset: 1

Rick L. Shepherd, May 29 2012

nonn

Nonnegative integers representable as a linear combination of 1, 5, 10, 25, and 50 with nonnegative coefficients, minimal sum of coefficients, and all nonzero coefficients equal.
Includes all nonnegative multiples of 50 and every term > 204 is a multiple of 50.
Unlike A212773, here it is permitted--and necessary--to use a single denomination for some amounts; otherwise, this sequence would be finite.

			a(37) = 91 is a term because the minimal number of coins to equal the amount 91 is five, 91 = 1*1 + 1*5 + 1*10 + 1*25 + 1*50, and there is one of each of the five denominations used.
a(45) = 204 is a term because the minimal number of coins for 204 is eight, 204 = 4*1 + 4*50, and there are four of each of the two denominations used.
Although 12 can be represented as 12*1 or 2*1 + 2*5, requiring 12 or 4 coins and each otherwise meeting the criteria, three (2*1 + 1*10) is the minimal number of coins required and 2 does not equal 1, so 12 is not a term.

Index entries for sequences related to making change.

Cf. A212773, A011542, A001300.

a(n) = (n-41)*50 for n >= 46.

A085502 Number of (unordered) ways of making change for n dollars using coins of denominations 1, 5, 10, 25, 50 and 100.

Original entry on oeis.org

1, 293, 2728, 12318, 38835, 98411, 215138, 422668, 765813, 1302145, 2103596, 3258058, 4870983, 7066983, 9991430, 13812056, 18720553, 24934173, 32697328, 42283190, 53995291, 68169123, 85173738, 105413348, 129328925, 157399801, 190145268, 228126178, 271946543

Offset: 0

Jason Earls, Aug 15 2003

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A001300.

{a(n)=if(n<0,0,polcoeff(1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50)*(1-x^100))+ x*O(x^n),n))}
for(n=0,30,print1(a(n*100)","))

Vec((1 + 287*x + 985*x^2 + 325*x^3 + 2*x^4) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Feb 21 2017

a(n) = (n + 1) (80 n^4 + 310 n^3 + 362 n^2 + 121 n + 6) / 6. - Dean Hickerson
From Colin Barker, Feb 21 2017: (Start)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>5.
G.f.: (1 + 287*x + 985*x^2 + 325*x^3 + 2*x^4) / (1 - x)^6.
(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

A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.

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A000008 Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

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A169718 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 and 100 cents.

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A001302 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents.

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A187243 Number of ways of making change for n cents using coins of 1, 5, and 10 cents.

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A212774 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which, when using the minimal number of coins, have equal numbers of all denominations used.

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