A319394 -id:A319394 - cp's OEIS frontend

A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).

1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 41, 49, 59, 71, 83, 99, 115, 134, 157, 180, 208, 239, 272, 312, 353, 400, 453, 509, 573, 642, 717, 803, 892, 993, 1102, 1219, 1350, 1489, 1640, 1808, 1983, 2178, 2386, 2609, 2854, 3113, 3393, 3697, 4017, 4367, 4737

Offset: 0

N. J. A. Sloane, Herman P. Robinson

The partitions allow repeated items but the order of items is immaterial (1+2=2+1). - Ron Knott, Oct 22 2003

A098641(n) = a(A000045(n)). - Reinhard Zumkeller, Apr 24 2005

			a(4) = 4 since the 4 partitions of 4 using only Fibonacci numbers, repetitions allowed, are 1+1+1+1, 2+2, 2+1+1, 3+1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
G. Almkvist, Partitions with Parts in a Finite Set and with Parts Outside a Finite Set, Exper. Math. vol 11 no 4 (2002) p 449-456.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
Herman P. Robinson, Letter to N. J. A. Sloane, Jan 1974.

Cf. A007000, A005092, A028290 (where the only Fibonacci numbers allowed are 1, 2, 3, 5 and 8).
Cf. A000045, A000119, A102848, A238998.
Row sums of A319394.

import Data.MemoCombinators (memo2, integral)
a003107 n = a003107_list !! n
a003107_list = map (p' 2) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m | m < fib   = 0
         | otherwise = p' k (m - fib) + p' (k + 1) m where fib = a000045 k
-- Reinhard Zumkeller, Dec 09 2015

F:= combinat[fibonacci]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0,
       b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 11 2013

CoefficientList[ Series[1/ Product[1 - x^Fibonacci[i], {i, 2, 21}], {x, 0, 53}], x] (* Robert G. Wilson v, Mar 28 2006 *)
nmax = 53;
s = Table[Fibonacci[n], {n, nmax}];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Jul 31 2020 *)
F = Fibonacci;
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 2, 0,
     b[n, i - 1] + If[F[i] > n, 0, b[n - F[i], i]]]];
a[n_] := Module[{j}, For[j = Floor@Log[(1+Sqrt[5])/2, n+1],
     F[j + 1] <= n, j++]; b[n, j]];
a /@ Range[0, 100] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

a(n) = (1/n)*Sum_{k=1..n} A005092(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Jan 21 2002
G.f.: Product_{i>=2} 1/(1-x^fibonacci(i)). - Ron Knott, Oct 22 2003
a(n) = f(n,1,1) with f(x,y,z) = if xReinhard Zumkeller, Nov 11 2009
G.f.: 1 + Sum_{i>=2} x^Fibonacci(i) / Product_{j=2..i} (1 - x^Fibonacci(j)). - Ilya Gutkovskiy, May 07 2017

More terms from Vladeta Jovovic, Jan 21 2002

A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1

Offset: 0

Alois P. Heinz, Sep 28 2018

Equals A181506 when the first column is removed. - Georg Fischer, Jul 26 2023

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 1, 0, 1;
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1;
  0, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A010054 (for n>0), A052344, A063993, A319814, A319815, A319816, A319817, A319818, A319819, A319820.
Row sums give A007294.
T(2n,n) gives A319799.
Cf. A000217, A117278, A181506, A243148, A319394.

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(8*n+1), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..20);

h[n_] := h[n] = If[n < 1, 0, If[IntegerQ @ Sqrt[8*n + 1], n, h[n - 1]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[ x*b[n - i, h[Min[n - i, i]]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, n}]& @ b[n, h[n]];
Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

T(n,k) = [x^n y^k] 1/Product_{j>=1} (1-y*x^A000217(j)).

A319395 Number of partitions of n into exactly two positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 0, 1, 0, 0, 1, 1, 1, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Alois P. Heinz, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=2 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(2):
seq(a(n), n=0..120);

h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
a[n_] := With[{k = 2}, b[n, h[n], k] - b[n, h[n], k - 1]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = [x^n y^2] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319397 Number of partitions of n into exactly four positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 4, 3, 4, 4, 5, 4, 5, 4, 5, 5, 5, 5, 7, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 5, 7, 4, 4, 3, 5, 5, 4, 6, 6, 5, 6, 4, 6, 6, 5, 5, 5, 5, 5, 4, 7, 4, 1, 4, 2, 4, 6, 3, 6, 5, 5, 6, 5, 6, 5, 3, 6, 3, 5, 6, 5, 6, 5, 2, 5, 3, 6, 5, 2, 5, 4, 3, 7, 1, 4

Offset: 0

Alois P. Heinz, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=4 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(4):
seq(a(n), n=0..120);

h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
a[n_] := With[{k = 4}, b[n, h[n], k] - b[n, h[n], k - 1]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) = [x^n y^4] 1/Product_{j>=2} (1-y*x^A000045(j)).

A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 6, 8, 7, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 8, 8, 7, 8, 9, 7, 10, 8, 8, 9, 7, 8, 8, 4, 8, 5, 8, 9, 6, 10, 8, 6, 10, 6, 9, 8, 5, 9, 6, 6, 8, 4, 8, 4, 1, 8, 4, 7, 9, 5, 10, 7, 6, 10, 6, 8, 6, 3, 10, 5, 7, 9, 5, 8, 5, 2, 9, 4, 7, 6, 2, 8, 4, 3, 8, 1, 4, 1

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000045, A003107, A319394, A319397, A359511, A359512, A359516.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
a:= n-> (p-> add(coeff(p, x, i), i=0..4))(b(n, h(n))):
seq(a(n), n=0..87);  # Alois P. Heinz, Jan 03 2023

h[n_] := h[n] = If[n < 1, 0, With[{t = 5 n^2}, If[IntegerQ @ Sqrt[t + 4] || IntegerQ @ Sqrt[t - 4], n, h[n - 1]]]];
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, x^n, b[n, h[i - 1]] + Expand[x*b[n - i, h[Min[n - i, i]]]]];
a[n_] := Sum[Coefficient[#, x, i], {i, 0, 4}]&[b[n, h[n]]];
Table[a[n], {n, 0, 87}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..4} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A319396 Number of partitions of n into exactly three positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3, 2, 3, 3, 3, 1, 2, 1, 3, 3, 2, 2, 3, 2, 3, 1, 3, 1, 0, 2, 1, 2, 3, 2, 3, 2, 1, 2, 2, 3, 2, 0, 3, 1, 1, 3, 0, 1, 0, 0, 2, 1, 2, 2, 2, 3, 2, 1, 3, 1, 2, 1, 0, 2, 2, 2, 3, 0, 2, 0, 0, 3, 1, 1, 1, 0, 3, 0, 0, 1, 0, 0

Offset: 0

Alois P. Heinz, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=3 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(3):
seq(a(n), n=0..120);

h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
a[n_] := With[{k = 3}, b[n, h[n], k] - b[n, h[n], k - 1]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

a(n) = [x^n y^3] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319398 Number of partitions of n into exactly five positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 5, 5, 7, 6, 7, 7, 8, 7, 9, 8, 10, 9, 9, 8, 11, 8, 10, 10, 11, 10, 11, 10, 13, 10, 11, 8, 10, 10, 10, 11, 12, 11, 11, 11, 13, 11, 12, 11, 12, 12, 11, 11, 13, 12, 10, 8, 10, 9, 9, 12, 11, 10, 13, 10, 14, 14, 11, 11, 11, 11, 13

Offset: 0

Alois P. Heinz, Sep 18 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=5 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(5):
seq(a(n), n=0..120);

h[n_] := h[n] = If[n < 1, 0, If[Function[t, IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4]][5 n^2], n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
a[n_] := With[{k = 5}, b[n, h[n], k] - b[n, h[n], k - 1]];
a /@ Range[0, 120] (* Jean-François Alcover, Dec 08 2020 *)

a(n) = [x^n y^5] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319399 Number of partitions of n into exactly six positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 6, 8, 8, 9, 9, 12, 10, 12, 12, 14, 13, 15, 13, 16, 15, 16, 15, 19, 16, 18, 18, 20, 18, 20, 17, 20, 17, 19, 19, 21, 21, 20, 20, 24, 21, 23, 21, 23, 22, 22, 23, 24, 23, 23, 20, 22, 21, 20, 21, 24, 22, 22, 23, 25, 25, 27, 23

Offset: 0

Alois P. Heinz, Sep 18 2018

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=6 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(6):
seq(a(n), n=0..120);

a(n) = [x^n y^6] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319400 Number of partitions of n into exactly seven positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 9, 9, 11, 11, 14, 14, 16, 15, 19, 17, 20, 20, 22, 21, 24, 22, 27, 25, 27, 26, 31, 28, 30, 29, 32, 29, 32, 30, 34, 33, 34, 34, 37, 36, 38, 36, 41, 37, 38, 39, 41, 41, 40, 39, 41, 38, 41, 38, 41, 42, 40, 41, 46, 43

Offset: 0

Alois P. Heinz, Sep 18 2018

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=7 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(7):
seq(a(n), n=0..120);

a(n) = [x^n y^7] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319401 Number of partitions of n into exactly eight positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 10, 12, 13, 16, 16, 20, 19, 23, 22, 25, 25, 30, 28, 31, 31, 35, 34, 39, 36, 42, 40, 43, 42, 47, 44, 47, 46, 51, 48, 52, 51, 56, 55, 57, 56, 62, 59, 62, 60, 65, 64, 64, 65, 67, 64, 67, 65, 70, 67, 69, 68, 72

Offset: 0

Alois P. Heinz, Sep 18 2018

Alois P. Heinz, Table of n, a(n) for n = 0..17711

Column k=8 of A319394.

Cf. A000045.

h:= proc(n) option remember; `if`(n<1, 0, `if`((t->
      issqr(t+4) or issqr(t-4))(5*n^2), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(8):
seq(a(n), n=0..120);

a(n) = [x^n y^8] 1/Product_{j>=2} (1-y*x^A000045(j)).

cp's OEIS Frontend

A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).

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A319797 Number T(n,k) of partitions of n into exactly k positive triangular numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A319395 Number of partitions of n into exactly two positive Fibonacci numbers.

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A319397 Number of partitions of n into exactly four positive Fibonacci numbers.

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A359513 Number of partitions of n into at most 4 positive Fibonacci numbers (with a single type of 1).

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A319396 Number of partitions of n into exactly three positive Fibonacci numbers.

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A319398 Number of partitions of n into exactly five positive Fibonacci numbers.

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A319399 Number of partitions of n into exactly six positive Fibonacci numbers.