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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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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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