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 1-9 of 9 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

References

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

Links

Crossrefs

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.

Programs

  • Haskell
    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
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    f(x,y,z)=if(xCharles R Greathouse IV, Dec 14 2015

Formula

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

Extensions

More terms from Vladeta Jovovic, Jan 21 2002

A065033 1 appears three times, other numbers twice.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35
Offset: 0

Views

Author

N. J. A. Sloane, Nov 04 2001

Keywords

Comments

Gives the number of terms in n-th row of many common tables.
Number of partitions of the (n+1)-th Fibonacci number into distinct Fibonacci numbers: a(n) = A000119(A000045(n)), see also A098641. - Reinhard Zumkeller, Apr 24 2005
a(n) = length of run n+1 of consecutive 4s in A254338. - Reinhard Zumkeller, Feb 27 2015
This is the Engel expansion of A070910 + A096789. - Benedict W. J. Irwin, Dec 16 2016

Links

Crossrefs

Cf. A004526, A008619.
Cf. A254338.

Programs

Formula

From Philippe Deléham, Sep 28 2006: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3) for n>3.
G.f.: (1-x^2+x^3)/(1-x-x^2+x^3). (End)
a(n) = floor((n+1)/2) + 0^n. - Reinhard Zumkeller, Feb 27 2015
E.g.f.: (2 + exp(x)*x + sinh(x))/2. - Stefano Spezia, Aug 05 2025

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

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Sep 20 2018

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    (* 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 *)

Formula

a(n) = [x^A000045(n) y^n] 1/Product_{j>=2} (1-y*x^A000045(j)).
a(n) = A319394(A000045(n),n).

A098644 Number of partitions of Fibonacci(n) into Fibonacci numbers, some of which are odd and some of which are even.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 30, 129, 710, 5233, 51819, 709352, 13710800, 379963761, 15289965453, 902291244954, 78717614351420, 10220774050567926, 1986419713778008167, 580763090016320217550, 256553735019174430718651
Offset: 1

Views

Author

Marcel Dubois de Cadouin (dubois.ml(AT)club-internet.fr), Oct 27 2004

Keywords

Crossrefs

Cf. A098641-A098643.

Extensions

Corrected and extended by Franklin T. Adams-Watters, Robert G. Wilson v and Don Reble, Mar 28 2006

A098643 Number of partitions of Fibonacci(n) into even Fibonacci numbers.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 6, 0, 0, 52, 0, 0, 1471, 0, 0, 135029, 0, 0, 41906834, 0, 0, 45729175098, 0, 0
Offset: 1

Views

Author

Marcel Dubois de Cadouin (dubois.ml(AT)club-internet.fr), Oct 27 2004

Keywords

Comments

a(3i-1)=a(3i+1)=0.

Crossrefs

Cf. A098641-A098644.

Programs

  • Mathematica
    cl = CoefficientList[ Series[1/Product[(1 - x^Fibonacci[3i]), {i, 8}], {x, 0, 46368}], x]; cl[[Table[Fibonacci[i] + 1, {i, 21}]]] (* Robert G. Wilson v *)

Extensions

More terms from Franklin T. Adams-Watters and Robert G. Wilson v, Mar 28 2006

A098642 Number of partitions of Fibonacci(n) into odd Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 11, 28, 87, 331, 1565, 9440, 72908, 712687, 8940809, 146775560, 3135694178, 86745468962, 3153522201617, 151857397825012, 9589042822673342, 797159462260375958, 88429042471401581924, 13020775433175768654191, 2532864078831923248348591
Offset: 0

Views

Author

Marcel Dubois de Cadouin (dubois.ml(AT)club-internet.fr), Oct 27 2004

Keywords

Crossrefs

Cf. A000045, A098641-A098644.

Programs

  • Mathematica
    cl = CoefficientList[ Series[1/Product[(1 - x^Fibonacci[3i + 1])(1 - x^Fibonacci[3i - 1]), {i, 8}], {x, 0, 46370}], x]; cl[[Table[Fibonacci[i] + 1, {i, 24}]]] (* Robert G. Wilson v *)

Extensions

Corrected and extended by Franklin T. Adams-Watters and Robert G. Wilson v, Mar 28 2006
a(0)=1 prepended and a(25) from Alois P. Heinz, Jun 02 2023

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

Views

Author

Alois P. Heinz, May 26 2021

Keywords

Examples

			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].

Crossrefs

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

Programs

  • Maple
    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);
  • Mathematica
    $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 *)

Formula

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

A137273 Number of partitions of n-th Fibonacci number into Fibonacci parts obtained by iteratively dividing F(k) into F(n-1) and F(n-2); number of sub-Fibonacci sequences of length n starting with 1,0.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 37, 134, 659, 4416, 41343, 546577, 10345970, 283128770, 11306821624, 664047579721, 57753201767477, 7483309752358051
Offset: 1

Views

Author

Franklin T. Adams-Watters, Apr 05 2008

Keywords

Comments

By a sub-Fibonacci sequence we mean a sequence of nonnegative integers b(i) with b(i) <= b(i-1) + b(i-2). Here we are taking b(1) = 1 and b(2) = 0.
In the above, b(i) (for i >= 2) is the number of times F(n-i+2) is divided into the next two smaller Fibonacci numbers in forming the partition.

Examples

			For the sub-Fibonacci sequence 1,0,1,1,1,2, we split F(6)=8 into 5,3; split the 5 into 3,2; split one 3 into 2,1; and split both 2's into 1,1. This gives the partition [3,1^5].
[2^4] is the smallest partition of a Fibonacci number into Fibonacci parts that cannot be obtained in this way.

Links

Crossrefs

Cf. A098641, A008934, A002449.

Programs

  • PARI
    nextfibpart(m) = local(s); s=matsize(m);matrix(s[2],s[1]+s[2]-1,i,j,sum(k=max(j-i+1,1),s[1],m[k,i]))
alist(n) = {local(v,m); v=vector(n,j,1); m=[0;1]; for(i=3,n, m=nextfibpart(m);v[i]=sum(j=1,matsize(m)[1],sum(k=1,matsize(m)[2],m[j,k]))); v}

A290248 Number of partitions of the n-th Lucas number into Lucas parts (beginning with 1) (A000204).

Original entry on oeis.org

1, 2, 3, 6, 13, 39, 147, 755, 5230, 50282, 677730, 13010007, 359551127, 14457741910, 853120090801, 74437567936635, 9666377127590346, 1878877762201043122, 549363336929733878734, 242695457366120511255070, 16263199149257162654631846
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 24 2017

Keywords

Examples

			a(4) = 6 because Lucas(4) = 7 and we have [7], [4, 3], [4, 1, 1, 1], [3, 3, 1], [3, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000204, A003263, A067592, A098641.

Programs

  • Mathematica
    Rest[Table[SeriesCoefficient[Product[1/(1 - x^LucasL[k]), {k, 1, n}], {x, 0, LucasL[n]}], {n, 0, 21}]]

Formula

a(n) = [x^A000204(n)] Product_{k>=1} 1/(1 - x^A000204(k)).
a(n) = A067592(A000204(n)).
Showing 1-9 of 9 results.

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

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