A319394 -id:A319394 - cp's OEIS frontend

A319503 Number of partitions of Fibonacci(n) into exactly n positive Fibonacci numbers.

1, 1, 0, 0, 0, 1, 2, 6, 16, 43, 117, 305, 769, 1907, 4686, 11587, 28580, 70451, 172880, 423629, 1036332, 2533559, 6186635, 15092985, 36784586, 89590410, 218069921, 530551804, 1290218120, 3136385254, 7621522229, 18515039477, 44966884766, 109184448962

Offset: 0

Alois P. Heinz, Sep 20 2018

nonn

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(5) = 1: 11111.
a(6) = 2: 221111, 311111.
a(7) = 6: 2222221, 3222211, 3322111, 3331111, 5221111, 5311111.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000045, A098641, A259254, A319394, A319435.

(* Program not suitable for a large number of terms. *)
a[n_] := a[n] = If[n < 2, 1, IntegerPartitions[Fibonacci[n], {n}, Fibonacci[Range[2, n - 1]]] //Length];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 08 2020 *)

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

a(n) = A319394(A000045(n),n).

A359511 Number of partitions of n into at most 2 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Cf. A000045, A003107, A319394, A319395, A359512, A359513, A359514.

a(n) = Sum_{k=0..2} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A359512 Number of partitions of n into at most 3 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 3, 4, 4, 3, 4, 3, 1, 4, 2, 4, 4, 2, 4, 3, 2, 4, 1, 3, 1, 0, 4, 2, 3, 4, 2, 4, 2, 1, 4, 2, 3, 2, 0, 4, 1, 1, 3, 0, 1, 0, 0, 4, 2, 3, 3, 2, 4, 2, 1, 4, 1, 2, 1, 0, 4, 2, 2, 3, 0, 2, 0, 0, 4, 1, 1, 1, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 2, 3, 3

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Cf. A000045, A003107, A319394, A319396, A359511, A359513, A359515.

a(n) = Sum_{k=0..3} A319394(n,k). - Alois P. Heinz, Jan 03 2023

A319402 Number of partitions of n into exactly nine positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 13, 14, 18, 18, 22, 23, 27, 26, 31, 30, 36, 36, 39, 39, 45, 43, 49, 49, 55, 52, 58, 56, 63, 62, 65, 64, 71, 68, 73, 72, 79, 77, 82, 81, 87, 86, 90, 89, 96, 93, 96, 96, 101, 100, 101, 100, 107, 103, 108

Offset: 0

Alois P. Heinz, Sep 18 2018

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

Column k=9 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))(9):
seq(a(n), n=0..120);

a(n) = [x^n y^9] 1/Product_{j>=2} (1-y*x^A000045(j)).

A319403 Number of partitions of n into exactly ten positive Fibonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 14, 15, 19, 20, 24, 25, 31, 30, 35, 36, 42, 42, 48, 47, 54, 54, 59, 60, 69, 66, 73, 72, 80, 79, 86, 85, 92, 91, 97, 96, 107, 103, 110, 110, 118, 117, 123, 123, 132, 130, 135, 134, 142, 141, 146, 145

Offset: 0

Alois P. Heinz, Sep 18 2018

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

Column k=10 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))(10):
seq(a(n), n=0..120);

a(n) = [x^n y^10] 1/Product_{j>=2} (1-y*x^A000045(j)).

A136343 Number of partitions of n such that each summand is greater than or equal to the sum of the next two summands.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 7, 10, 11, 14, 16, 21, 23, 29, 32, 40, 43, 52, 57, 69, 75, 88, 96, 113, 122, 141, 153, 177, 190, 216, 233, 265, 285, 320, 345, 387, 415, 461, 495, 551, 589, 650, 695, 767, 818, 896, 957, 1048, 1116, 1214, 1293, 1407, 1495, 1620, 1721, 1864

Offset: 0

David S. Newman, May 11 2008

nonn

This sequence was suggested by Moshe Shmuel Newman. The idea came to him while reading a paper of Lev Shneerson.

Number of partitions of 2n into exactly n positive Fibonacci numbers. a(8) = 10: 82111111, 55111111, 53311111, 53221111, 52222111, 33331111, 33322111, 33222211, 32222221, 22222222. - Alois P. Heinz, Sep 18 2018

			a(5) = 4 because 4 of the 7 partitions of 5 have the required property: {5} {4,1} {3,2} {3,1,1}. The other 3 partitions of 5: {2,2,1} {2,1,1,1} and {1,1,1,1,1} each have an element which is < the sum of next two.

Alois P. Heinz, Table of n, a(n) for n = 0..3500

Cf. A000045, A319394.

b:= proc(n, i, j) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n-i, min(n-i, i,
      `if`(j=0, i, j-i)), i) +b(n, i-1, j)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 29 2017

b[n_, i_, j_]:=b[n, i, j]=If[n==0, 1, If[i<1, 0, b[n - i, Min[n - i, i, If[j==0, i, j - i]], i] + b[n, i - 1, j]]]; Table[b[n, n, 0], {n, 0, 60}] (* Indranil Ghosh, Aug 01 2017, after Maple code *)

From Alois P. Heinz, Sep 18 2018: (Start)
a(n) = [x^(2n) y^n] 1/Product_{j>=2} (1-y*x^A000045(j)).
a(n) = A319394(2n,n). (End)

Conjectured g.f. removed and a(0), a(35)-a(56) added by Alois P. Heinz, Jul 29 2017

A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 42, 62, 86, 119, 159, 211, 273, 352, 446, 562, 697, 864, 1054, 1284, 1550, 1860, 2220, 2639, 3114, 3669, 4293, 5011, 5823, 6745, 7783, 8956, 10268, 11747, 13390, 15237, 17281, 19561, 22089, 24889, 27979, 31405, 35157, 39309, 43856, 48849, 54319, 60309, 66840, 73992, 81760, 90243

Offset: 1

Ilya Gutkovskiy, Jan 27 2017

nonn

Total number of parts in all partitions of n into Fibonacci parts (with a single type of 1).

Convolution of A003107 and A005086.

			a(5) = 18 because we have [5], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1] and 1 + 2 + 3 + 3 + 4 + 5 = 18.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Index entries for related partition-counting sequences

Cf. A000045, A003107, A005086, A240225, A319394.

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, [1, n],
       b(n, h(i-1))+(p->p+[0, p[1]])(b(n-i, h(min(n-i, i)))))
    end:
a:= n-> b(n, h(n))[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Sep 18 2018

Rest[CoefficientList[Series[Sum[x^Fibonacci[k]/(1 - x^Fibonacci[k]), {k, 2, 20}]/Product[1 - x^Fibonacci[k], {k, 2, 20}], {x, 0, 52}], x]]

G.f.: Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

a(n) = Sum_{k=1..n} k * A319394(n,k). - Alois P. Heinz, Sep 18 2018

A343537 Number of partitions of the n-th Fibonacci number into a Fibonacci number of Fibonacci parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 16, 41, 135, 632, 4091, 37020, 478852, 8897512, 240133480, 9489055662, 552854898873, 47794151866058, 6165361571608551, 1192709563056788508, 347571453153709529743, 153189847887607116894958

Offset: 0

Alois P. Heinz, May 26 2021

nonn

			a(5) = 5: [5], [3,2], [3,1,1], [2,2,1], [1,1,1,1,1].  Partition [2,1,1,1] is not counted because 4 (the number of parts) is not a Fibonacci number.
a(6) = 7: [8], [5,3], [5,2,1], [3,3,2], [3,2,1,1,1], [2,2,2,1,1], [1,1,1,1,1,1,1,1].
a(7) = 16: [13], [8,5], [8,3,2], [8,2,1,1,1], [5,5,3], [5,5,1,1,1], [5,3,3,1,1], [5,3,2,2,1], [5,2,2,2,2], [5,2,1,1,1,1,1,1], [3,3,3,3,1], [3,3,3,2,2], [3,3,2,1,1,1,1,1], [3,2,2,2,1,1,1,1], [2,2,2,2,2,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1].

Cf. A000045, A098641, A316154, A319394, A344790.

f:= n-> (t-> issqr(t+4) or issqr(t-4))(5*n^2):
h:= proc(n) option remember; `if`(f(n), n, h(n-1)) end:
b:= proc(n, i, c) option remember; `if`(n=0 or i=1, `if`(
      f(c+n), 1, 0), b(n-i, h(min(n-i, i)), c+1)+b(n, h(i-1), c))
    end:
a:= n-> b((<<0|1>, <1|1>>^n)[1, 2]$2, 0):
seq(a(n), n=0..17);

$RecursionLimit = 10000;
f[n_] := With[{t = 5 n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]];
h[n_] := h[n] = If[f[n], n, h[n - 1]] ;
b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1, If[f[c+n], 1, 0], b[n-i, h[Min[n-i, i]], c+1] + b[n, h[i-1], c]];
a[n_] := a[n] = With[{m = MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]]}, b[m, m, 0]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 09 2022, after Alois P. Heinz *)

a(n) = Sum_{k in {A000045}} A319394(A000045(n),k).

A356977 a(n) is the number of solutions, j >= 0 and 2 <= m_1 <= ... <= m_n, of the equation Sum_{k=1..n} F(m_k) = 2^j where F(i) is the i-th Fibonacci number.

Original entry on oeis.org

0, 3, 6, 10, 36, 66

Offset: 0

Peter Munn and Jon E. Schoenfield, Sep 07 2022

The difficulty of this sequence comes in determining which is the largest j in a solution for a(n), equivalently the last nonzero term in each sum from the A319394-based formula in the formula section.
a(2) derives from Bravo and Luca, a(3) from Bravo and Bravo, a(4) from Pagdame Tiebekabe and Diouf. Pagdame has indicated A356928(5), from which a(5) is derived, has been determined.
a(6) >= 178, a(7) >= 478.

			For n = 2, the a(2) = 6 solutions are j = 1 with (2,2), j = 2 with (2,4) and (3,3), j = 3 with (4,5), j = 4 with (4,7) and (6,6) according to the paper of Bravo and Luca. [That is, 2 = 1+1, 4 = 1+3 = 2+2, 8 = 3+5, 16 = 3+13 = 8+8.]

J. J. Bravo, and F. Luca, On the Diophantine equation F_n+F_m=2^a, Quaest. Math. 39 (2016) 391-400.
P. Tiebekabe and I. Diouf, On solutions of Diophantine equation F_{n_1}+F_{n_2}+F_{n_3}+F_{n_4}=2^a, Journal of Algebra and Related Topics, Volume 9, Issue 2 (2021), 131-148.

E. F. Bravo and J. J. Bravo, Powers of two as sums of three Fibonacci numbers, Lithuanian Mathematical Journal, 55, pp. 301-311 (2015).

Cf. A020908, A319394, A356928.

a(n) = Sum_{i >= 0} A319394(2^i, k).

cp's OEIS Frontend

A319503 Number of partitions of Fibonacci(n) into exactly n positive Fibonacci numbers.

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A359511 Number of partitions of n into at most 2 positive Fibonacci numbers (with a single type of 1).

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A359512 Number of partitions of n into at most 3 positive Fibonacci numbers (with a single type of 1).

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A319402 Number of partitions of n into exactly nine positive Fibonacci numbers.

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A319403 Number of partitions of n into exactly ten positive Fibonacci numbers.

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A136343 Number of partitions of n such that each summand is greater than or equal to the sum of the next two summands.

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A281689 Expansion of Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)) / Product_{k>=2} (1 - x^Fibonacci(k)).

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A343537 Number of partitions of the n-th Fibonacci number into a Fibonacci number of Fibonacci parts.

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A356977 a(n) is the number of solutions, j >= 0 and 2 <= m_1 <= ... <= m_n, of the equation Sum_{k=1..n} F(m_k) = 2^j where F(i) is the i-th Fibonacci number.

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