A079971 -id:A079971 - cp's OEIS frontend

A073031 Number of ways of making change for n cents using coins of sizes 1, 2, 5, 10 cents, when order matters.

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 129, 220, 377, 644, 1101, 1883, 3219, 5505, 9412, 16093, 27517, 47049, 80448, 137553, 235195, 402148, 687611, 1175712, 2010288, 3437288, 5877241, 10049189, 17182590, 29379620, 50234693, 85893702

Offset: 0

Miklos Kristof, Aug 22 2002

			a(4)=5 because 4 = 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1 = 2 + 1 + 1 = 2 + 2: five possible exchange. a(15) = a(14) + a(13) + a(10) + a(5) = 1883 = 1101 + 644 + 129 + 9.

Peter Boros (borospet(AT)freemail.hu): Lectures on Fibonacci's World at the SOTERIA Foundation, 1999.
P. Henrici, Applied and Computational Complex Analysis. Wiley, NY, 3 vols., 1974-1986. (Vol. 1, p. 580.)

Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,1,0,0,0,0,1).

Cf. A079971.

a:= n-> (Matrix(10, (i,j)-> if i+1=j or j=1 and member (i,[1, 2, 5, 10]) then 1 else 0 fi)^n)[1, 1]: seq(a(n), n=0..35); # Alois P. Heinz, Oct 07 2008

LinearRecurrence[{1, 1, 0, 0, 1, 0, 0, 0, 0, 1}, {1, 1, 2, 3, 5, 9, 15, 26, 44, 75}, 36] (* Jean-François Alcover, Jan 25 2025 *)

a(n) = a(n-1) + a(n-2) + a(n-5) + a(n-10), a(0)=1.
G.f.: 1/(1 - x - x^2 - x^5 - x^10). - Franklin T. Adams-Watters, Oct 24 2006
With offset 1, the INVERT transform of (1 + x + x^4 + x^9). - Gary W. Adamson, Apr 04 2017

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A245367 Compositions of n into parts 3, 5 and 7.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 8, 10, 11, 17, 18, 25, 32, 37, 52, 61, 79, 102, 123, 163, 200, 254, 326, 402, 519, 649, 819, 1045, 1305, 1664, 2096, 2643, 3358, 4220, 5352, 6759, 8527, 10806, 13622, 17237, 21785, 27501, 34802, 43934, 55544, 70209, 88672, 112131, 141644, 179018, 226274, 285860, 361358

Offset: 0

David Neil McGrath, Aug 20 2014

			a(16) = 10: the compositions are the permutations of [5533] (there are 4!/2!2!=6 of them) and the permutations of [7333] (there are 4!/3!=4).

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,1,0,1).

Cf. A000073, A013979, A017818, A060945, A079956, A079957, A079971, A079973, A120400, A121833.

LinearRecurrence[{0,0,1,0,1,0,1},{1,0,0,1,0,1,1},70] (* Harvey P. Dale, Jan 27 2017 *)

Vec(1/(1-x^3-x^5-x^7) +O(x^66)) \\ Joerg Arndt, Aug 20 2014

G.f: 1/(1-x^3-x^5-x^7).

a(n) = a(n-3) + a(n-5) + a(n-7).

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A073031 Number of ways of making change for n cents using coins of sizes 1, 2, 5, 10 cents, when order matters.

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A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A245367 Compositions of n into parts 3, 5 and 7.

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