A076739 -id:A076739 - cp's OEIS frontend

A144172 Eigentriangle, row sums = A076739, the number of compositions into Fibonacci numbers.

1, 1, 1, 1, 1, 2, 0, 1, 2, 4, 1, 0, 2, 4, 7, 0, 1, 0, 4, 7, 14, 0, 0, 2, 0, 7, 14, 26, 1, 0, 0, 4, 0, 14, 26, 49, 0, 1, 0, 0, 7, 0, 26, 49, 94, 0, 0, 2, 0, 0, 14, 0, 49, 94, 177, 0, 0, 0, 4, 0, 0, 26, 0, 94, 177, 336, 0, 0, 0, 0, 7, 0, 0, 49, 0, 177, 336, 637

Offset: 1

Gary W. Adamson, Sep 12 2008

Row sums = A076739 starting with offset 1: (1, 2, 4, 7, 14, 26, 49,...).
Left border = A010056, the characteristic function of the Fibonacci numbers Starting with offset 1: (1, 1, 1, 0, 1,...).
Sum of n-th row terms = rightmost term of next row.
Right border = A076739.

			First few rows of the triangle =
1;
1, 1;
1, 1, 2;
0, 1, 2, 4;
1, 0, 2, 4, 7;
0, 1, 0, 4, 7, 14;
0, 0, 2, 0, 7, 14, 26;
1, 0, 0, 4, 0, 14, 26, 49;
0, 1, 0, 0, 7, 0, 26, 49, 94;
0, 0, 2, 0, 0, 14, 0, 49, 94, 177;
0, 0, 0, 4, 0, 0, 26, 0, 94, 177, 336;
0, 0, 0, 0, 7, 0, 0, 49, 0, 177, 336, 637;
1, 0, 0, 0, 0, 14, 0, 0, 94, 0, 336, 637, 1206;
...
Example: row 5 = (1, 0, 2, 4, 7) = termwise product of (1, 0, 1, 1, 1) and (1, 1, 2, 4, 7).

A076739, Cf. A010056

T(n,k) = A010056(n-k+1)*A076739(k-1). A010056, the characteristic function of the Fibonacci numbers, starts with offset 1: (1, 1, 1, 0, 1,...). A076739(k-1), the INVERTi transform of (1, 1, 1, 0, 1,...) starts with offset 0: (1, 1, 2, 4, 7, 14,...).

A121548 Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1 <= k <= n; only one 1 is considered as a Fibonacci number).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 3, 3, 1, 1, 2, 6, 4, 1, 0, 3, 7, 10, 5, 1, 0, 2, 9, 16, 15, 6, 1, 1, 2, 9, 23, 30, 21, 7, 1, 0, 2, 10, 28, 50, 50, 28, 8, 1, 0, 3, 9, 34, 71, 96, 77, 36, 9, 1, 0, 2, 12, 36, 95, 156, 168, 112, 45, 10, 1, 0, 0, 12, 43, 115, 231, 308, 274, 156, 55, 11, 1, 1, 2, 9, 48, 140, 312, 504, 560, 423, 210, 66, 12, 1

Offset: 1

Emeric Deutsch, Aug 07 2006

			T(5,3)=6 because we have [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1] and [3,1,1].
Triangle starts:
  1;
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  1,  2,  6,  4,  1;
  0,  3,  7, 10,  5,  1;
  0,  2,  9, 16, 15,  6,  1;
  ...

Alois P. Heinz, Rows n = 0..150, flattened

Cf. A076739, A010056, A121549, A121550, A121551.

T(2n,n) gives A341072.

with(combinat): G:=1/(1-t*sum(z^fibonacci(i),i=2..40))-1: Gser:=simplify(series(G,z=0,25)): for n from 1 to 23 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 1 to 15 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
# second Maple program:
g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
T:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(g(j), T(n-j, t-1), 0), j=1..n)))
    end:
seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Oct 10 2022

nmax = 14;
T = Rest@CoefficientList[#, t]& /@ Rest@(1/(1 - t*Sum[z^Fibonacci[i],
     {i, 2, nmax}]) - 1 + O[z]^(nmax+1) // CoefficientList[#, z]&);
Table[T[[n, k]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 02 2022 *)

G.f.: G(t,z) = 1 / (1 - t*Sum_{i>=2} z^Fibonacci(i)) - 1.
Sum of terms in row n = A076739(n).
T(n,1) = A010056(n) (the characteristic function of the Fibonacci numbers);
T(n,2) = A121549(n);
T(n,3) = A121550(n);
Sum_{k=1..n} k*T(n,k) = A121551(n).

A357688 Number of ways to write n as an ordered sum of four positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 4, 10, 16, 23, 28, 34, 36, 43, 48, 50, 48, 50, 56, 58, 64, 67, 60, 58, 52, 64, 64, 70, 68, 70, 76, 70, 72, 79, 60, 60, 48, 58, 68, 60, 84, 80, 64, 82, 64, 82, 88, 66, 76, 66, 64, 84, 60, 79, 60, 24, 60, 36, 60, 74, 48, 88, 76, 72, 96, 68, 88, 76, 48, 82, 60, 70

Offset: 4

Ilya Gutkovskiy, Oct 10 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..10000

Cf. A000045, A076739, A121548, A121549, A121550, A319397, A357690, A357691.

nmax = 70; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^4, {x, 0, nmax}], x] // Drop[#, 4] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^4.

a(n) = A121548(n,4).

A357690 Number of ways to write n as an ordered sum of five positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 5, 15, 30, 50, 71, 95, 115, 140, 165, 191, 205, 220, 240, 260, 285, 310, 325, 325, 320, 341, 350, 380, 385, 405, 420, 430, 450, 465, 465, 445, 410, 435, 425, 450, 481, 495, 515, 490, 510, 555, 525, 580, 540, 530, 570, 530, 580, 600, 520, 525, 440, 455, 520, 445, 555, 530

Offset: 5

Ilya Gutkovskiy, Oct 10 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..10000

Cf. A000045, A076739, A121548, A121549, A121550, A319398, A357688, A357691.

nmax = 61; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^5, {x, 0, nmax}], x] // Drop[#, 5] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^5.

a(n) = A121548(n,5).

A357691 Number of ways to write n as an ordered sum of six positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 6, 21, 50, 96, 156, 231, 312, 405, 506, 621, 726, 828, 930, 1041, 1160, 1290, 1422, 1520, 1590, 1677, 1766, 1887, 1980, 2106, 2196, 2310, 2426, 2550, 2670, 2706, 2700, 2736, 2756, 2850, 2916, 3071, 3156, 3186, 3296, 3396, 3510, 3621, 3636, 3765, 3720, 3840, 3966, 4010

Offset: 6

Ilya Gutkovskiy, Oct 10 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..17711

Cf. A000045, A076739, A121548, A121549, A121550, A319399, A357688, A357690.

nmax = 54; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^6, {x, 0, nmax}], x] // Drop[#, 6] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^6.

a(n) = A121548(n,6).

A218396 Number of compositions of n into distinct (nonzero) Fibonacci numbers.

Original entry on oeis.org

1, 1, 1, 3, 2, 3, 8, 2, 9, 8, 8, 32, 6, 9, 32, 8, 38, 30, 32, 150, 6, 33, 32, 32, 158, 30, 38, 174, 30, 176, 150, 150, 870, 24, 33, 152, 32, 182, 150, 158, 894, 30, 182, 174, 174, 1014, 144, 176, 990, 150, 1014, 864, 870, 5904, 24, 153, 152, 152, 902, 150, 182, 1014, 150, 1022, 894, 894, 6054, 144

Offset: 0

Joerg Arndt, Oct 28 2012

nonn

			There are a(37)=182 such compositions of 37. Each of the 6 partitions of 37 into distinct Fibonacci numbers corresponds to m! compositions (where m is the number of parts):
  #:  partition      ( m! compositions)
  1:  1 2 5 8 21     (120 compositions)
  2:  1 2 13 21      ( 24 compositions)
  3:  1 2 34         (  6 compositions)
  4:  3 5 8 21       ( 24 compositions)
  5:  3 13 21        (  6 compositions)
  6:  3 34           (  2 compositions)
The number of compositions is 120 + 24 + 6 + 24 + 6 + 2 = 182.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms 0..200 from Joerg Arndt)

Cf. A032021 (compositions into distinct odd numbers).
Cf. A000119 (partitions into distinct nonzero Fibonacci numbers), A000700 (partitions into distinct odd numbers).
Cf. A076739 (compositions into Fibonacci numbers).

A359514 Number of compositions (ordered partitions) of n into at most 2 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

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

Cf. A000045, A076739, A121548, A121549, A359511, A359515, A359516.

g:= proc(n) g(n):= (t-> issqr(t+4) or issqr(t-4))(5*n^2) end:
b:= proc(n, t) option remember; `if`(n=0, 1, `if`(t<1, 0,
      add(`if`(g(j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..94);  # Alois P. Heinz, Jan 03 2023

g[n_] := IntegerQ@Sqrt[# + 4] || IntegerQ@Sqrt[# - 4]&[5 n^2];
b[n_, t_] := b[n, t] = If[n == 0, 1, If[t < 1, 0, Sum[If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 94}] (* Jean-François Alcover, May 26 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..2} A121548(n,k). - Alois P. Heinz, Jan 03 2023

A288039 Number of compositions (ordered partitions) of n into Lucas numbers (beginning with 1) (A000204).

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 9, 16, 27, 43, 70, 118, 195, 318, 524, 868, 1430, 2351, 3878, 6399, 10542, 17367, 28634, 47206, 77793, 128212, 211346, 348360, 574153, 946342, 1559849, 2571016, 4237616, 6984659, 11512526, 18975464, 31276187, 51550993, 84968944, 140049801, 230836734, 380476447, 627119783, 1033648857

Offset: 0

Ilya Gutkovskiy, Jun 04 2017

nonn

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

Eric Weisstein's World of Mathematics, Lucas Number
Index entries for sequences related to compositions

Cf. A000204, A003263, A067592, A076739, A080888.

CoefficientList[Series[1/(1 - Sum[x^LucasL[k], {k, 1, 15}]), {x, 0, 43}], x]

G.f.: 1/(1 - Sum_{k>=1} x^A000204(k)).

A357694 Number of ways to write n as an ordered sum of seven positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 7, 28, 77, 168, 308, 504, 750, 1050, 1400, 1813, 2261, 2737, 3227, 3753, 4312, 4921, 5579, 6230, 6832, 7413, 8008, 8652, 9289, 9996, 10654, 11361, 12061, 12853, 13657, 14357, 14924, 15393, 15869, 16408, 16933, 17689, 18319, 18949, 19537, 20244, 21049, 21728

Offset: 7

Ilya Gutkovskiy, Oct 10 2022

nonn

Cf. A000045, A076739, A121548, A121549, A121550, A319400, A357688, A357690, A357691, A357716, A357717.

nmax = 49; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^7.

a(n) = A121548(n,7).

A357716 Number of ways to write n as an ordered sum of eight positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 8, 36, 112, 274, 560, 1008, 1640, 2479, 3536, 4844, 6392, 8170, 10136, 12308, 14680, 17291, 20160, 23248, 26440, 29674, 32992, 36456, 40040, 43834, 47712, 51752, 55840, 60250, 64856, 69560, 74088, 78331, 82440, 86500, 90616, 95074, 99568, 104188, 108528, 113304

Offset: 8

Ilya Gutkovskiy, Oct 10 2022

nonn

Cf. A000045, A076739, A121548, A121549, A121550, A319401, A357688, A357690, A357691, A357694, A357717.

nmax = 48; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^8.

a(n) = A121548(n,8).

cp's OEIS Frontend

A144172 Eigentriangle, row sums = A076739, the number of compositions into Fibonacci numbers.

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A121548 Triangle read by rows: T(n,k) is the number of compositions of n into k Fibonacci numbers (1 <= k <= n; only one 1 is considered as a Fibonacci number).

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A357688 Number of ways to write n as an ordered sum of four positive Fibonacci numbers (with a single type of 1).

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A357690 Number of ways to write n as an ordered sum of five positive Fibonacci numbers (with a single type of 1).

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A357691 Number of ways to write n as an ordered sum of six positive Fibonacci numbers (with a single type of 1).

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A218396 Number of compositions of n into distinct (nonzero) Fibonacci numbers.

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A359514 Number of compositions (ordered partitions) of n into at most 2 positive Fibonacci numbers (with a single type of 1).

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A288039 Number of compositions (ordered partitions) of n into Lucas numbers (beginning with 1) (A000204).

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A357694 Number of ways to write n as an ordered sum of seven positive Fibonacci numbers (with a single type of 1).

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