A000064 Partial sums of (unordered) ways of making change for n cents using coins of 1, 2, 5, 10 cents.
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
References
- 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).
Links
- 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).
Crossrefs
Cf. A000008.
Programs
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Maple
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
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Mathematica
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 *) -
PARI
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))
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PARI
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
Formula
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
Extensions
Corrected and extended by Simon Plouffe
Comments