Search: a001300 -id:a001300
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A001299
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Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents.
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+10
19
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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
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OFFSET
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0,6
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COMMENTS
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Number of partitions of n into parts 1, 5, 10, and 25. - Joerg Arndt, Sep 05 2014
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REFERENCES
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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.
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LINKS
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Ed Pegg, Jr., Sequence Pictures, Math Games column, Dec 08 2003 [Cached copy, with permission (pdf only)]
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).
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
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
(PARI) 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
(PARI) {a(n) = my(m=n\5 / 10); round((4*m + 3) * (5*m + 1) * (5*m + 2) / 6)}; /* Michael Somos, Feb 23 2017 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A000008
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Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.
(Formerly M0280 N0099)
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+10
16
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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).
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FORMULA
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G.f.: 1 / ((1 - x) * (1 - x^2) * (1 - x^5) * (1 - x^10)). - Michael Somos, Nov 17 1999
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) = (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
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EXAMPLE
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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 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {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 */
(PARI) Vec( 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)) + O(x^66) ) \\ Joerg Arndt, Oct 02 2013
(PARI) {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 */
(Maxima) 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 */
(Haskell)
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
(Magma) [#RestrictedPartitions(n, {1, 2, 5, 10}):n in [0..60]]; // Marius A. Burtea, May 07 2019
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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A169718
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Number of ways of making change for n cents using coins of 1, 5, 10, 25, 50 and 100 cents.
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+10
7
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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
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OFFSET
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0,6
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COMMENTS
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Number of partitions of n into parts 1, 5, 10, 25, 50, and 100. - Joerg Arndt, Sep 05 2014
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REFERENCES
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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.
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LINKS
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FORMULA
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G.f.: 1/((1-x)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50)*(1-x^100)).
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A001302
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Number of ways of making change for n cents using coins of 1, 2, 5, 10, 25, 50 cents.
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+10
6
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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
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OFFSET
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0,3
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COMMENTS
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Number of partitions of n into parts 1, 2, 5, 10, 25, and 50. - Joerg Arndt, Sep 05 2014
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REFERENCES
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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.
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LINKS
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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).
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FORMULA
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G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^25)*(1-x^50)).
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A187243
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Number of ways of making change for n cents using coins of 1, 5, and 10 cents.
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+10
5
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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
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OFFSET
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0,6
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COMMENTS
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a(n) is the number of partitions of n into parts 1, 5, and 10. - Joerg Arndt, Feb 02 2017
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)
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LINKS
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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).
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FORMULA
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G.f.: 1/((1-x)*(1-x^5)*(1-x^10)).
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)
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EXAMPLE
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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)
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MATHEMATICA
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CoefficientList[ Series[ 1 / ((1 - x)(1 - x^5)(1 - x^10)), {x, 0, 75} ], x ]
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PROG
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(PARI) a(n)=(n^2+16*n+97+10*(n\5+1)*(5*(n\5)+2-n))\100 \\ Tani Akinari, Sep 10 2015
(PARI) a(n) = {my(q=n\10, s=(n%10)\5); (q+1)*(q+1+s); } \\ (Kirchner's explicit formula) Joerg Arndt, Feb 02 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A212774
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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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+10
4
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = (n-41)*50 for n >= 46.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A085502
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Number of (unordered) ways of making change for n dollars using coins of denominations 1, 5, 10, 25, 50 and 100.
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+10
3
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = (n + 1) (80 n^4 + 310 n^3 + 362 n^2 + 121 n + 6) / 6. - Dean Hickerson
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)
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PROG
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(PARI) {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)", "))
(PARI) Vec((1 + 287*x + 985*x^2 + 325*x^3 + 2*x^4) / (1 - x)^6 + O(x^30)) \\ Colin Barker, Feb 21 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A267419
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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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+10
0
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A339094
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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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+10
0
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Not the same as A001313. First difference appears at A001313(100) being 4562, whereas a(100) is 4563; obviously one more than A001313(100).
Number of partitions of n into parts 1, 2, 5, 10, 20, 50 and 100.
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LINKS
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FORMULA
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G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^50)*(1-x^100)).
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EXAMPLE
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a(5) is 4 because 1+1+1+1+1 = 2+1+1+1 = 2+2+1 = 5.
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MATHEMATICA
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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 *)
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PROG
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(PARI) coins(v[..])=my(x='x); prod(i=1, #v, 1/(1-x^v[i]))
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CROSSREFS
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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.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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