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 A001299 Number of ways of making change for n cents using coins of 1, 5, 10, 25 cents. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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 REFERENCES 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. LINKS 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 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). FORMULA 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 EXAMPLE 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 + ... MATHEMATICA 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 *) PROG (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 -- Reinhard Zumkeller, Dec 15 2013 (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 */ CROSSREFS Cf. A001300, A169718, A000008. Sequence in context: A105674 A130496 A187243 * A001300 A169718 A001306 Adjacent sequences:  A001296 A001297 A001298 * A001300 A001301 A001302 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Mar 15 1996 STATUS approved

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)