A000008 -id:A000008 - 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

A001300 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 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

Offset: 0

N. J. A. Sloane, Mar 15 1996

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

a(n) = A001299(n) for n < 50; a(n) = A169718(n) for n < 100. - Reinhard Zumkeller, Dec 15 2013

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, Problems 1 and 2.

T. D. Noe, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 176
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, 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, 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. A001299, A169718, A000008.

a001300 = p [1,5,10,25,50] 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

1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50));

CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)(1 - x^25)(1 - x^50)), {x, 0, 65} ], x ]

a(n)=floor(((n\5)^4+38*(n\5)^3+476*(n\5)^2+2185*(n\5)+3735)/2400+(n\5+1)*(-1)^(n\5)/160+(n\5\5+1)*[0,0,1,0,-1][n\5%5+1]/10) \\ Tani Akinari, May 10 2014

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

A001312 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 50, 100 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, 342, 357, 379, 394, 416, 438, 460, 482, 504, 526

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, 5, 10, 50, and 100. - Joerg Arndt, Sep 05 2014

			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.

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 181
Index entries for linear recurrences with constant coefficients, order 168.
Index entries for sequences related to making change.

a[ n_] := SeriesCoefficient[1/((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)(1 - x^50)(1 - x^100)), {x, 0, n}]
Table[Length[FrobeniusSolve[{1,2,5,10,50,100},n]],{n,0,60}] (* Harvey P. Dale, Dec 29 2017 *)

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

A001313 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 20, 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, 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

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, 5, 10, 20, 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 182
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, 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, 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, -1, 1, 1, -1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, -1, 1, 1, -1).

Cf. A001319.

CoefficientList[Series[1/((1 - x) (1 - x^2) (1 - x^5) (1 - x^10) (1 - x^20) (1 - x^50)), {x, 0, 50}], x]
Table[Length[FrobeniusSolve[{1,2,5,10,20,50},n]],{n,0,60}] (* (very slow) Harvey P. Dale, Dec 25 2011 *)

Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^50))+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)).

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)).

A001301 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25 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, 406, 427, 455, 476, 504, 536, 564, 596, 624, 656

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n into parts 1, 2, 5, 10, and 25. - 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 177
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).
Index entries for sequences related to making change.

M := Matrix(43, (i,j)-> if (i=j-1) or (j=1 and member(i, [1, 2, 5, 8, 10, 13, 16, 17, 25, 28, 31, 32, 36, 37, 40, 43])) then 1 elif j=1 and member(i, [3, 6, 7, 11, 12, 15, 18, 26, 27, 30, 33, 35, 38, 41, 42]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..51); # Alois P. Heinz, Jul 25 2008

CoefficientList[ Series[ 1 / ((1 - x)(1 - x^2)(1 - x^5)(1 - x^10)(1 - x^25)), {x, 0, 55} ], x ]
Table[Length[FrobeniusSolve[{1,2,5,10,25},n]],{n,0,60}] (* Harvey P. Dale, Jan 19 2020 *)

Vec(1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)) + 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)).

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

A091967 a(n) is the n-th term of sequence A_n, ignoring the offset, or -1 if A_n has fewer than n terms.

Original entry on oeis.org

0, 2, 1, 0, 2, 3, 0, 6, 6, 4, 44, 1, 180, 42, 16, 1096, 7652, 13781, 8, 24000, 119779, 458561, 152116956851941670912, 1054535, -53, 26, 27, 59, 4806078, 2, 35792568, 3010349, 2387010102192469724605148123694256128, 2, 0, -53, 43, 0, -4097, 173, 37338, 111111111111111111111111111111111111111111, 30402457, 413927966

Offset: 1

Proposed by several people, including Jeff Burch and Michael Joseph Halm

sign

This version ignores the offset of A_n and just counts from the beginning of the terms shown in the OEIS entry.
Thus a(8) = 6 because A_8 begins 1,1,2,2,3,4,5,6,... [even though A_8(8) is really 7].
The value a(n) = -1 could arise in two different ways, but it will be easy to decide which. - N. J. A. Sloane, Nov 27 2016
From M. F. Hasler, Sep 22 2013: (Start)
The value of a(91967) can be chosen at will.
Note that this sequence may change if the initial terms in A_n are altered, which does happen from time to time, usually because of the addition of an initial term.
After a(47), currently unknown, the sequence continues with a(48) = A48(47) = 1497207322929, a(49) = A49(48) = unknown, a(50) = A50(49) = unknown, a(51) = A51(50) = 1125899906842625, a(52)=97, a(53) = -1 (since A000053 has only 29 terms). (End)
a(58) = A000058(57) = 138752...985443 (29334988649136302 digits) is too large to include in the b-file. - Pontus von Brömssen, May 21 2022

			a(1) = 0 since A000001 has offset 0, and begins with A000001(0) = 0.
a(26) = 26 because the 26th term of A000026 = 26.

Pontus von Brömssen, Table of n, a(n) for n = 1..57 (terms n = 1..46 from M. F. Hasler)
OEIS, List of finite sequences, sorted by A-number.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences whose definition involves A_n (or An).

See A051070, A107357, A102288 for other versions.

Cf. A000001, A000002, A000003, A000004, A000005, A000006, A000007, A000008, A000009, A000010, A000011, A000012, A000013, A000014, A000015, etc.

Corrected and extended by Jud McCranie; further extended by N. J. A. Sloane and E. M. Rains, Dec 08 1998
Corrected and extended by N. J. A. Sloane, May 25 2005
a(26), a(36) and a(42) corrected by M. F. Hasler, Jan 30 2009
a(43) and a(44) added by Daniel Sterman, Nov 27 2016
a(1) corrected by N. J. A. Sloane, Nov 27 2016 at the suggestion of Daniel Sterman
Definition and comments changed by N. J. A. Sloane, Nov 27 2016

A067996 Number of ways of making change for n cents using coins of 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, 2, 3, 4, 6, 8, 10, 13, 16, 21, 25, 31, 37, 44, 53, 62, 72, 84, 96, 113, 128, 147, 167, 189, 216, 243, 273, 307, 342, 386, 428, 477, 529, 585, 650, 716, 788, 867, 949, 1046, 1141, 1248, 1361, 1481, 1617, 1755, 1904, 2065, 2232, 2424, 2614, 2824, 3045, 3278

Offset: 0

Rick L. Shepherd, Feb 07 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(5)=6 because change can be made for 5 cents in these 6 ways: (1) 5 1-cent coins, (2) 3 1-cent, 1 2-cent, (3) 2 1-cent, 1 3-cent, (4) 1 1-cent, 2 2-cent, (5) 1 2-cent, 1 3-cent, (6) 1 5-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. 104-106, 135. (also known as The Official Red Book of United States Coins)

Cf. A067995, A067997, A000008.

CoefficientList[ Series[1/((1 - x)(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, 55} ], x ]

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

Offset corrected to 0 by Ray Chandler, Dec 04 2023

A000064 Partial sums of (unordered) ways of making change for n cents using coins of 1, 2, 5, 10 cents.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 18, 24, 31, 39, 50, 62, 77, 93, 112, 134, 159, 187, 218, 252, 292, 335, 384, 436, 494, 558, 628, 704, 786, 874, 972, 1076, 1190, 1310, 1440, 1580, 1730, 1890, 2060, 2240, 2435, 2640, 2860, 3090, 3335, 3595, 3870, 4160, 4465, 4785, 5126

Offset: 0

N. J. A. Sloane

Number of partitions of n into two kinds of part 1 and one kind of parts 2, 5, and 10. - Joerg Arndt, May 10 2014

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).

Christian G. Bower, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1,1,-2,0,2,-1,1,-2,0,2,-1,-1,2,0,-2,1).

Cf. A000008.

1/(1-x)^2/(1-x^2)/(1-x^5)/(1-x^10)
a:= proc(n) local m, r; m := iquo(n, 10, 'r'); r:= r+1; (55+(119+(95+ 25*m) *m) *m) *m/6+ [1, 2, 4, 6, 9, 13, 18, 24, 31, 39][r]+ [0, 26, 61, 99, 146, 202, 267, 341, 424, 516][r]*m/6+ [0, 10, 21, 33, 46, 60, 75, 91, 108, 126][r]*m^2/2+ (5*r-5) *m^3/3 end: seq(a(n), n=0..100); # Alois P. Heinz, Oct 05 2008

CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^5)(1-x^10)),{x,0,100}],x] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)

a(n)=if(n<0,0,polcoeff(1/((1-x)^2*(1-x^2)*(1-x^5)*(1-x^10))+x*O(x^n),n))

a(n)=floor((n^4+38*n^3+476*n^2+2185*n+3735)/2400+(n+1)*(-1)^n/160+(n\5+1)*[0,0,1,0,-1][n%5+1]/10) \\ Tani Akinari, May 10 2014

G.f.: 1 / ( ( 1 - x )^2 * ( 1 - x^2 ) * ( 1 - x^5 ) * ( 1 - x^10 ) ).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9) + a(n-10) - 2*a(n-11) + 2*a(n-13) - a(n-14) - a(n-15) + 2*a(n-16) - 2*a(n-18) + a(n-19). - Fung Lam, May 07 2014
a(n) ~ n^4 / 2400 as n->oo. - Daniel Checa, Jul 11 2023

Corrected and extended by Simon Plouffe

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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A001300 Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 cents.

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A001312 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 50, 100 cents.

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A001313 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 20, 50 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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A001301 Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25 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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