A001299 -id:A001299 - cp's OEIS frontend

A174139 Numbers congruent to {0,1,2,3,4,10,11,12,13,14,20,21,22,23,24} mod 25.

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 85, 86, 87, 88, 89, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 110, 111, 112

Offset: 1

Rick L. Shepherd, Mar 09 2010

Numbers whose partition into parts of sizes 1, 5, 10, and 25 having a minimal number of parts does not include a part of size 5.
For each number the partition is unique.
Complement of A174138.
Amounts in cents not including a nickel when the minimal number of coins is selected from pennies, nickels, dimes, and quarters (whether usage of bills for whole-dollar amounts is permitted or not).
For each n >= 0, floor(n/25) parts of size 25 (quarters) occur in the partition with minimal number of these parts (regardless of whether partition includes part of size 5).
First differs from A032955 at n = 76. - Avi Mehra, Oct 08 2020

Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A174138, A174140, A174141, A047201 (requires at least one part of size 1 (penny)), A008587, A053344 (minimal number of parts), A001299 (number of all such partitions).

Select[Range[0, 112], Mod[Mod[#, 25], 10] < 5 &] (* Amiram Eldar, Oct 08 2020 *)

{ my(table=[0,1,2,3,4, 10,11,12,13,14, 20,21,22,23,24]);
a(n) = my(r);[n,r]=divrem(n-1,15); 25*n + table[r+1]; } \\ Kevin Ryde, Oct 08 2020

a(15+n) = a(n) + 25 for n >= 1.
From  R. J. Mathar, Oct 08 2011: (Start)
a(n) = +a(n-1) +a(n-15) -a(n-16).
G.f.: x^2*(1 +x +x^2 +x^3 +6*x^4 +x^5 +x^6 +x^7 +x^8 +6*x^9 +x^10 +x^11 +x^12 +x^13+x^14) / ( (1+x+x^2) *(x^4+x^3+x^2+x+1) *(x^8-x^7+x^5-x^4+x^3-x+1) *(x-1)^2). (End)

A208216 Number of ways of making change for n cents using coins of 1, 5, 10, and 25 cents, where each coin that appears is used a different number of times.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 3, 4, 4, 5, 5, 5, 4, 5, 8, 7, 6, 8, 7, 12, 9, 11, 11, 12, 14, 14, 15, 14, 14, 20, 16, 18, 18, 20, 25, 24, 23, 24, 24, 32, 27, 30, 31, 30, 39, 38, 36, 39, 37, 47, 43, 47, 45, 48, 55, 53, 53, 56, 57, 69, 61, 65, 65, 67, 78, 77

Offset: 0

Matthew C. Russell, Apr 24 2012

nonn

			For n=6 the a(6)=1 solution is six copies of the 1-cent coin. (Taking one 1-cent coin and one 5-cent coin is not allowed, as those two coins have the same nonzero multiplicity.)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Doron Zeilberger, Using Generatingfunctionology to enumerate distinct-multiplicity partitions.

Cf. A001299, A208217.

# for a g.f. in Maple format see the Doron Zeilberger link.

A057536 Minimal number of coins needed to pay n Euro-cents using the Euro currency.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 3, 4, 4, 5, 5, 4, 5, 5, 6, 6, 3, 4, 4, 5, 5, 4, 5, 5, 6, 6, 1, 2, 2, 3, 3, 2

Offset: 0

Thomas Brendan Murphy (murphybt(AT)tcd.ie), Sep 06 2000

Euro currency has coins and bills of size 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000 cents.

			a(57) = 3 because to pay 57 cents at least 3 coins are needed: 1 of 50 cents, 1 of 5 cents and 1 of 2 cents.

Joan Ludevid, Table of n, a(n) for n = 0..10000
Index entries for sequences related to making change.

Cf. A011542, A001299.

Table[Min[Map[Total,FrobeniusSolve[{1,2,5,10,20,50,100,200,500,1000,2000,5000,10000,20000,50000},n]]],{n, 0, 105}] (* Joan Ludevid, Jun 15 2022 *)
numCoins[n_]:=(amount = n; coins = {50000, 20000, 10000, 5000, 2000, 1000, 500, 200, 100, 50, 20, 10, 5, 2, 1}; total=0; For[i = 1, i <= Length[coins], i++, total+=Quotient[amount, coins[[i]]]; amount = Mod[amount, coins[[i]]]]; total);
Table[numCoins[n], {n, 0, 105}] (* Joan Ludevid, Jun 16 2022 *)

a(n) = floor(n/50000) + floor((n mod 50000)/20000) + floor(((n mod 50000) mod 20000)/10000) + ... + floor(((n mod 50000) mod 20000 ... mod 5)/2) + ((n mod 50000) mod 20000)... mod 2.

a(0)=0 prepended by Alois P. Heinz, May 26 2015

A351740 Total number of parts in all partitions of n into parts of size 1, 5, 10 or 25.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 19, 23, 27, 31, 35, 44, 50, 56, 62, 68, 83, 92, 101, 110, 119, 141, 154, 167, 180, 193, 226, 244, 262, 280, 298, 343, 367, 391, 415, 439, 500, 531, 562, 593, 624, 702, 741, 780, 819, 858, 959, 1008, 1057, 1106, 1155, 1280, 1340

Offset: 0

Alois P. Heinz, Feb 17 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001299, A351725.

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
     (p->`if`(p>n, 0, (t->t+[0, t[1]])(b(n-p, i))))([1, 5, 10, 25][i]))
    end:
a:= n-> b(n, 4)[2]:
seq(a(n), n=0..100);

a(n) = Sum_{k=0..n} k * A351725(n,k).

A351724 Number of compositions of n into parts of size 1, 5, 10 or 25.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 9, 13, 18, 24, 31, 42, 58, 80, 109, 146, 197, 268, 366, 499, 676, 916, 1243, 1690, 2299, 3122, 4237, 5751, 7811, 10614, 14418, 19580, 26587, 36106, 49043, 66614, 90473, 122869, 166866, 226632, 307810, 418060, 567784, 771122, 1047296, 1422396, 1931845

Offset: 0

R. J. Mathar, Feb 17 2022

Starts to differ from A114044 at n=50.

			a(8)=5 counts 5 compositions 1+1+1+1+1+1+1+1 = 1+1+1+5 = 1+1+5+1 = 1+5+1+1 = 5+1+1+1.

Index entries for sequences related to making change.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A114044 (parts 50 and 100 admitted), A001299 (partitions).

Row sums of A351726.

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

a(n) = +a(n-1) +a(n-5) +a(n-10) +a(n-25).

A129263 Skylar (age 7) counts change by stacking all coins of the same type then arranging the stacks in a row. a(n) is the number of distinct Skylar stackings of n cents using any combination of pennies, nickels, dimes or quarters.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 3, 3, 5, 7, 7, 7, 7, 10, 15, 15, 15, 15, 19, 25, 25, 25, 25, 31, 41, 41, 41, 41, 49, 63, 63, 63, 63, 74, 95, 95, 95, 95, 111, 147, 147, 147, 147, 166, 209, 209, 209, 209, 234, 293, 293, 293, 293, 322, 391, 391, 391, 391, 427, 515, 515, 515, 515

Offset: 0

Andrew V. Sutherland, Aug 20 2007

nonn

Sequence definition and Scratch program to compute the 100 terms due to Skylar Sutherland. Generating function contributed by Andrew V. Sutherland. Related to A001299, but distinguishes permutations of coin types.

			a(16) = 15 = 1+2*4+6*1 since the distinct Skylar stackings of 16 cents are:
16p, 11p1n, 1n11p, 6p2n, 2n6p, 1p3n, 3n1p, 1p1d, 1d1p, 1p1n1d, 1p1d1n, 1n1p1d, 1n1d1p, 1d1p1n, 1d1n1p

Skylar Sutherland, student presentation at "The Undiscovered Country", a course for young mathematicians. Part of MIT's Educational Studies Program.

Cf. A001299.

Let A_v(x,y) = 1-y+y/(1-x)^v and A(x,y) = A_1(x,y)A_5(x,y)A_10(x,y)A_25(x,y). Let A^(k)(x,y) denote the k-th partial derivative of A(x,y) w.r.t. y. The generating function of a(n) is A(x) = Sum A^(k)(x,0) for k from 0 to 4.

A245574 Minimal coin changing sequence for denominations 1, 2, 5, 10, 20 and 50 cents.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 3, 4, 4, 5, 5, 4, 5, 5, 6, 6, 3, 4, 4, 5, 5, 4, 5, 5, 6, 6, 2

Offset: 0

Marko Riedel, Jul 25 2014

nonn

a(100) = 2 is the first term where the sequence is different from A057536. - Colin Barker, Jul 26 2014

Coin change problem

Cf. A001299, A011542, A057536.

More terms from Colin Barker, Jul 26 2014

a(0)=0 prepended by Alois P. Heinz, May 26 2015

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

A174139 Numbers congruent to {0,1,2,3,4,10,11,12,13,14,20,21,22,23,24} mod 25.

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A208216 Number of ways of making change for n cents using coins of 1, 5, 10, and 25 cents, where each coin that appears is used a different number of times.

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A057536 Minimal number of coins needed to pay n Euro-cents using the Euro currency.

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A351740 Total number of parts in all partitions of n into parts of size 1, 5, 10 or 25.

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A351724 Number of compositions of n into parts of size 1, 5, 10 or 25.

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A129263 Skylar (age 7) counts change by stacking all coins of the same type then arranging the stacks in a row. a(n) is the number of distinct Skylar stackings of n cents using any combination of pennies, nickels, dimes or quarters.

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A245574 Minimal coin changing sequence for denominations 1, 2, 5, 10, 20 and 50 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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