cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A114044 Number of (ordered) sequences of coins (each of which has value 1, 5, 10, 25, 50 or 100) which add to n.

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
Offset: 0

Views

Author

David S. Johnson and N. J. A. Sloane, Feb 22 2006

Keywords

Comments

Based on US coins as of Feb 22 2006.
a(n) is the number of compositions (ordered partitions) of n into parts 1, 5, 10, 25, 50, and 100. - Joerg Arndt, Apr 19 2017

Links

Crossrefs

Cf. A114138, A073031, A114140.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - (x + x^5 + x^10 + x^25 + x^50 + x^100)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Apr 18 2017 *)

Formula

G.f.: 1/(1-(x + x^5 + x^10 + x^25 + x^50 + x^100)).

A079971 Number of compositions (ordered partitions) of n into parts 1, 2, and 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 128, 218, 372, 634, 1081, 1843, 3142, 5357, 9133, 15571, 26547, 45260, 77164, 131557, 224292, 382396, 651948, 1111508, 1895013, 3230813, 5508222, 9390983, 16010713, 27296709, 46538235, 79343166, 135272384
Offset: 0

Views

Author

Vladimir Baltic, Feb 17 2003

Keywords

Comments

Number of ways of ordered sequences of nickels, dimes and quarters that add to 5n cents.
Number of permutations satisfying -k <= p(i)-i <= r and p(i)-i not in I, i=1..n, with k=1, r=4, I={2,3}.

References

  • D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Links

Crossrefs

Cf. A002524-A002529, A072827, A072850-A072856, A079955-A080014, A073031.

Programs

  • Maple
    a:= n-> (Matrix(5, (i,j)-> if i+1=j or j=1 and member(i,[1, 2, 5]) then 1 else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 07 2008
  • Mathematica
    LinearRecurrence[{1, 1, 0, 0, 1}, {1, 1, 2, 3, 5}, 40] (* Jean-François Alcover, Nov 11 2015 *)
  • Maxima
    a(n):=sum(sum(binomial(j,n-5*k+4*j)*binomial(k,j),j,floor((5*k-n)/4),k),k,0,n); /* Vladimir Kruchinin, Dec 15 2011 */

Formula

Recurrence: a(n) = a(n-1)+a(n-2)+a(n-5).
G.f.: 1/(1-x-x^2-x^5).
a(n) = Sum_{k=0..n} Sum_{j=floor((5*k-n)/4)..k} C(j,n-5*k+4*j)*C(k,j). - Vladimir Kruchinin, Dec 15 2011
With offset 1, the INVERT transform of (1 + x + x^4). - Gary W. Adamson, Apr 01 2017

Extensions

Entry revised by N. J. A. Sloane, Feb 23 2006

A114140 Number of ordered sequences of coins (each of which has value 1, 2, 5, 10 or 20) which add to n.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 129, 220, 377, 644, 1101, 1883, 3219, 5505, 9412, 16093, 27518, 47051, 80453, 137563, 235215, 402188, 687688, 1175860, 2010567, 3437810, 5878212, 10050981, 17185883, 29385638, 50245647, 85913568, 146901103, 251181919
Offset: 0

Views

Author

David S. Johnson and N. J. A. Sloane, Feb 22 2006

Keywords

Comments

Equivalently, number of sequences of coins (each of which has value 5, 10, 25, 50 or 100) which add to 5n.
Based on "silver" US coins as of Feb 22 2006.
Number of compositions of n into parts 1, 2, 5, 10, and 20. - Joerg Arndt, Sep 19 2014

Crossrefs

Cf. A073031, A114044, A114138.

Programs

  • Maple
    A114140 := proc(n)
    coeftayl( 1/(1-(x+x^2+x^5+x^10+x^20)), x=0, n);
end proc:
seq(A114140(n), n=0..30); # Wesley Ivan Hurt, Sep 18 2014
  • Mathematica
    CoefficientList[Series[1/(1 - (x + x^2 + x^5 + x^10 + x^20)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Sep 18 2014 *)

Formula

G.f.: 1/(1-(x+x^2+x^5+x^10+x^20)).

A114138 Number of (ordered) sequences of coins (each of which has value 1, 2, 5, 10, 20, 50, 100 or 200) which add to n.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 129, 220, 377, 644, 1101, 1883, 3219, 5505, 9412, 16093, 27518, 47051, 80453, 137563, 235215, 402188, 687688, 1175860, 2010567, 3437810, 5878212, 10050981, 17185883, 29385638, 50245647, 85913568, 146901103, 251181919
Offset: 0

Views

Author

David S. Johnson and N. J. A. Sloane, Feb 22 2006

Keywords

Comments

Based on Euro coins as of Feb 22 2006.

Crossrefs

Cf. A114044, A073031, A114140.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-(x+x^2+x^5+x^10+x^20+x^50+x^100+x^200)),{x,0,50}],x] (* Harvey P. Dale, Nov 18 2013 *)

Formula

G.f.: 1/(1-(x+x^2+x^5+x^10+x^20+x^50+x^100+x^200)).

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

Views

Author

Robert G. Wilson v, Nov 25 2020

Keywords

Comments

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.

Examples

			a(5) is 4 because 1+1+1+1+1 = 2+1+1+1 = 2+2+1 = 5.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^5)*(1-x^10)*(1-x^20)*(1-x^50)*(1-x^100)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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