A003107 -id:A003107 - cp's OEIS frontend

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

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

A028290 Expansion of 1/((1-x)(1-x^2)(1-x^3)(1-x^5)(1-x^8)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 40, 48, 57, 68, 79, 93, 107, 124, 142, 162, 184, 209, 235, 265, 296, 331, 368, 409, 452, 500, 550, 605, 663, 726, 792, 864, 939, 1021, 1106, 1198, 1294, 1397, 1505

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts equal to 1, 2, 3, 5 and 8. E.g. a(5)=6 because we have 5, 3+2, 3+1+1, 2+2+1, 2+1+1+1 and 1+1+1+1+1. - Emeric Deutsch, Mar 25 2005

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,0,0,-1,1,0,0,-1,1,0,0,1,0,-1,-1,1).

Cf. A003107, A029145.

import Data.MemoCombinators (memo2, integral)
a028290 n = a028290_list !! n
a028290_list = map (p' 0) [0..] where
   p' = memo2 integral integral p
   p _ 0 = 1
   p 5 _ = 0
   p k m | m < parts !! k = 0
         | otherwise = p' k (m - parts !! k) + p' (k + 1) m
   parts = [1, 2, 3, 5, 8]
-- Reinhard Zumkeller, Dec 09 2015

G:=1/(1-x)/(1-x^2)/(1-x^3)/(1-x^5)/(1-x^8): Gser:=series(G,x=0,47): 1, seq(coeff(Gser,x^n),n=1..45); # Emeric Deutsch, Mar 25 2005

CoefficientList[ Series[ 1/Product[1 - x^Fibonacci[i], {i, 2, 6}], {x, 0, 45}], x] (* Robert G. Wilson v, Oct 15 2016 *)
CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^5)(1-x^8)),{x,0,100}],x] (* Harvey P. Dale, Jan 26 2019 *)

Vec(1/((1-x)*(1-x^2)*(1-x^3)*(1-x^5)*(1-x^8))+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A239001 Irregular triangular array read by rows: row n gives a list of the partitions of n into Fibonacci numbers.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 3, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 5, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 1, 1, 3, 1, 1

Offset: 1

Clark Kimberling, Mar 08 2014

The number of partitions represented in row n is A003107(n).

The parts of a partition are nonincreasing and the order of the partitions is anti-lexicographic. As parts one uses A000045(n), n >= 2. - Wolfdieter Lang, Mar 17 2014

			1
2 1 1
3 2 1 1 1 1
3 1 2 2 2 1 1 1 1 1 1
5 3 2 3 1 1 2 2 1 2 1 1 1 1 1 1 1 1
Row 5 represents these six partitions: 5, 32, 311, 221, 2111, 11111.
From _Wolfdieter Lang_, Mar 17 2014: (Start)
The array with separated partitions begins:
n\k   1      2      3        4        5          6          7            8             9             10 ...
1:    1
2:    2    1,1
3:    3    2,1  1,1,1
4:  3,1    2,2  2,1,1  1,1,1,1
5:    5    3,2  3,1,1    2,2,1  2,1,1,1  1,1,1,1,1
6:  5,1    3,3  3,2,1  3,1,1,1    2,2,2    2,2,1,1  2,1,1,1,1  1,1,1,1,1,1
7:  5,2  5,1,1  3,3,1    3,2,2  3,2,1,1  3,1,1,1,1    2,2,2,1    2,2,1,1,1   2,1,1,1,1,1  1,1,1,1,1,1,1
...
Row n=8: 8  5,3  5,2,1  5,1,1,1  3,3,2  3,3,1,1  3,2,2,1  3,2,1,1,1  3,1,1,1,1,1   2,2,2,2   2,2,2,1,1
  2,2,1,1,1,1  2,1,1,1,1,1,1  1,1,1,1,1,1,1,1;
Row n=9  8,1  5,3,1  5,2,2   5,2,1,1   5,1,1,1,1  3,3,3   3,3,2,1   3,3,1,1,1  3,2,2,2  3,2,2,1,1
3,2,1,1,1,1   3,1,1,1,1,1,1  2,2,2,2,1  2,2,2,1,1,1  2,2,1,1,1,1,1   2,1,1,1,1,1,1,1   1,1,1,1,1,1,1,1,1;
Row n=10: 8,2  8,1,1   5,5   5,3,2  5,3,1,1  5,2,2,1  5,2,1,1,1  5,1,1,1,1,1   3,3,3,1  3,3,2,2  3,3,2,1,1
  3,3,1,1,1,1   3,2,2,2,1  3,2,2,1,1,1   3,2,1,1,1,1,1   3,1,1,1,1,1,1,1   2,2,2,2,2   2,2,2,2,1,1
  2,2,2,1,1,1,1  2,2,1,1,1,1,1,1  2,1,1,1,1,1,1,1,1  1,1,1,1,1,1,1,1,1,1.
-----------------------------------------------------------------------------------------------------------
(End)

Cf. A003107, A000045, A239512.

f = Table[Fibonacci[n], {n, 2, 60}]; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; s[n_, k_] := If[Union[f, DeleteDuplicates[p[n, k]]] == f, p[n, k], 0]; t[n_] := Table[s[n, k], {k, 1, PartitionsP[n]}]; TableForm[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* shows partitions *)
y = Flatten[Table[DeleteCases[t[n], 0], {n, 1, 12}]] (* A239001 *)
(* also *)
FibonacciQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]]; Attributes[FibonacciQ] = {Listable}; TableForm[t = Map[Select[IntegerPartitions[#], And @@ FibonacciQ[#] &] &, Range[0, 12]]]
Flatten[t] (* Peter J. C. Moses, Mar 24 2014 *)

A238998 Number of partitions of n that such that no part is a Fibonacci number.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 2, 4, 4, 6, 5, 9, 8, 11, 11, 16, 16, 22, 22, 29, 31, 40, 42, 54, 57, 71, 77, 95, 103, 127, 137, 165, 182, 218, 238, 285, 313, 369, 408, 479, 530, 619, 684, 794, 883, 1019, 1130, 1304, 1446, 1658, 1843, 2107, 2340, 2670

Offset: 0

Clark Kimberling, Mar 08 2014

			a(15) counts these partitions:  [15], [11,4], [9,6], [7,4,4]; a(16) counts these:  [16], [12,4], [10,6], [9,7], [6,6,4], [4,4,4,4].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A003107, A000045.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`((f-> issqr(f+4) or issqr(f-4))(5*d^2), 0, d),
        d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 31 2017

p[n_] := IntegerPartitions[n, All, Complement[Range@n, Fibonacci@Range@15]]; Table[p[n], {n, 0, 20}] (* shows partitions *)
a[n_] := Length@p@n; a /@ Range[0, 80] (* counts partitions *)
(* Second program: *)
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[
    If[Function[f, IntegerQ@Sqrt[f+4] || IntegerQ@Sqrt[f-4]][5*d^2], 0, d],
    {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
a /@ Range[0, 100] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

N=66; q='q+O('q^N); Vec( prod(n=1,11,1-q^fibonacci(n+1))/eta(q) ) \\ Joerg Arndt, Mar 11 2014

G.f.: A(x) = sum(1/product(1 - x^c(i))), i >=1, where c(i) are the non-Fibonacci numbers.

A239000 Number of partitions of n using Fibonacci numbers > 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 2, 1, 3, 2, 2, 4, 2, 4, 4, 3, 7, 4, 5, 8, 5, 9, 8, 7, 12, 9, 11, 13, 11, 17, 14, 15, 20, 16, 22, 22, 20, 29, 24, 27, 33, 28, 37, 36, 35, 45, 40, 46, 50, 47, 60, 55, 58, 69, 62, 75, 76, 73, 91, 84, 91, 102, 95, 114, 112, 113

Offset: 0

Clark Kimberling, Mar 08 2014

			a(21) counts these partitions: [21], [13,8], [13,5,3], [8,8,5], [8,5,5,3], [5,5,5,3,3], [3,3,3,3,3,3,3].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000045, A003107, A238999.

p[n_] := IntegerPartitions[n, All, Fibonacci@Range[4, 60]]; Table[p[n], {n, 0, 12}] (*shows partitions*)
a[n_] := Length@p@n; a /@ Range[0, 80] (*counts partitions, A239000*)

N=66; q='q+O('q^N); Vec( 1/prod(n=1,11,1-q^fibonacci(n+3)) ) \\ Joerg Arndt, Mar 11 2014

G.f.: 1/Product_{i>=4} (1 - x^Fibonacci(i)).

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

A290807 Number of partitions of n into Pell parts (A000129).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 14, 15, 18, 20, 23, 26, 29, 32, 36, 39, 44, 47, 53, 57, 63, 68, 74, 81, 88, 95, 103, 110, 120, 128, 139, 148, 159, 170, 182, 195, 208, 221, 236, 250, 267, 282, 300, 317, 336, 355, 375, 396, 418, 440, 464, 487, 514, 539, 568, 595, 625, 655, 687, 720, 754, 788

Offset: 0

Ilya Gutkovskiy, Aug 11 2017

nonn

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

Eric Weisstein's World of Mathematics, Pell Number
Index entries for sequences related to partitions

Cf. A000129, A007000, A003107, A067592, A067593, A290808.

CoefficientList[Series[Product[1/(1 - x^Fibonacci[k, 2]), {k, 1, 15}], {x, 0, 67}], x]

G.f.: Product_{k>=1} 1/(1 - x^A000129(k)).

A357452 Number of partitions of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 27, 36, 37, 46, 48, 60, 62, 74, 78, 94, 98, 114, 120, 140, 147, 168, 178, 204, 215, 242, 256, 288, 304, 338, 358, 398, 420, 462, 488, 537, 567, 619, 654, 714, 753, 816, 860, 932, 982, 1058, 1114

Offset: 0

Ilya Gutkovskiy, Sep 29 2022

nonn

Cf. A000078, A003107, A240844, A287656, A357453, A357454.

A000078[0] = A000078[1] = A000078[2] = 0; A000078[3] = 1; A000078[n_] := A000078[n] = A000078[n - 1] + A000078[n - 2] + A000078[n - 3] + A000078[n - 4]; nmax = 55; CoefficientList[Series[Product[1/(1 - x^A000078[k]), {k, 4, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=4} 1 / (1 - x^A000078(k)).

A357454 Number of partitions of n into pentanacci numbers 1,2,4,8,16,31, ... (A001591).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 26, 36, 36, 46, 46, 60, 60, 74, 74, 94, 94, 114, 114, 140, 140, 166, 167, 202, 203, 238, 240, 284, 286, 330, 334, 390, 394, 450, 456, 524, 530, 598, 608, 692, 702, 786, 800, 900, 914, 1014, 1034

Offset: 0

Ilya Gutkovskiy, Sep 29 2022

nonn

Cf. A001591, A003107, A240844, A288120, A357452, A357455.

A001591[0] = A001591[1] = A001591[2] = A001591[3] = 0; A001591[4] = 1; A001591[n_] := A001591[n] = A001591[n - 1] + A001591[n - 2] + A001591[n - 3] + A001591[n - 4] + A001591[n - 5]; nmax = 55; CoefficientList[Series[Product[1/(1 - x^A001591[k]), {k, 5, 20}], {x, 0, nmax}], x]

G.f.: Product_{k>=5} 1 / (1 - x^A001591(k)).

cp's OEIS Frontend

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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A028290 Expansion of 1/((1-x)(1-x^2)(1-x^3)(1-x^5)(1-x^8)).

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A239001 Irregular triangular array read by rows: row n gives a list of the partitions of n into Fibonacci numbers.

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A238998 Number of partitions of n that such that no part is a Fibonacci number.

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Maple

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A239000 Number of partitions of n using Fibonacci numbers > 2.

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Mathematica

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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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A290807 Number of partitions of n into Pell parts (A000129).

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A357452 Number of partitions of n into tetranacci numbers 1,2,4,8,15,29, ... (A000078).