A121550 -id:A121550 - cp's OEIS frontend

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).

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).

A121549 Number of ordered ways of writing n as a sum of two Fibonacci numbers (only one 1 is considered as a Fibonacci number).

Original entry on oeis.org

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

Offset: 1

Emeric Deutsch, Aug 07 2006

nonn

			a(6)=3 because we have 6=1+5=3+3=5+1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000045, A121548, A121550, A357688, A357690, A357691.

with(combinat): g:=sum(z^fibonacci(i),i=2..30)^2: gser:=series(g,z=0,130): seq(coeff(gser,z,n),n=1..126);

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

a(n) = A121548(n,2).

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).

A280210 Expansion of (Sum_{k>=1} mu(k)^2*x^k)^3, where mu(k) is the Moebius function (A008683).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 21, 30, 36, 37, 36, 48, 58, 63, 57, 70, 78, 87, 78, 96, 105, 114, 105, 123, 133, 138, 126, 148, 162, 174, 156, 195, 207, 220, 192, 234, 250, 261, 237, 280, 312, 318, 282, 330, 363, 370, 315, 375, 405, 432, 366, 421, 453, 483, 417, 468, 507, 532, 474, 537, 568, 591, 519, 601, 630, 666, 570

Offset: 0

Ilya Gutkovskiy, Dec 29 2016

nonn

Number of ordered ways of writing n as sum of three squarefree numbers (A005117).

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

Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A098235.

Cf. A098238, A121550.

nmax = 72; CoefficientList[Series[(Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}])^3, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} mu(k)^2*x^k)^3.

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).

A357732 Number of partitions of n into 3 distinct positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

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

Offset: 6

Ilya Gutkovskiy, Oct 11 2022

nonn

Cf. A000045, A000119, A121550, A319396, A357722, A357731.

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).

A359515 Number of compositions (ordered partitions) of n into at most 3 positive Fibonacci numbers (with a single type of 1).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 10, 11, 12, 12, 12, 14, 12, 12, 11, 12, 15, 12, 14, 12, 6, 12, 8, 14, 15, 9, 15, 12, 9, 14, 6, 12, 6, 0, 12, 8, 11, 17, 9, 15, 9, 6, 15, 9, 12, 9, 0, 14, 6, 6, 12, 0, 6, 0, 0, 12, 8, 11, 14, 9, 17, 9, 6, 15, 6, 9, 6, 0, 15, 9, 9, 12, 0, 9, 0, 0, 14, 6, 6, 6, 0, 12

Offset: 0

Ilya Gutkovskiy, Jan 03 2023

nonn

Cf. A000045, A076739, A121548, A121550, A359512, A359514, 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, 3):
seq(a(n), n=0..81);  # Alois P. Heinz, Jan 03 2023

g[n_] := With[{t = 5 n^2}, IntegerQ @ Sqrt[t+4] || IntegerQ @ Sqrt[t-4]];
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, 3];
Table[a[n], {n, 0, 81}] (* Jean-François Alcover, May 28 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..3} A121548(n,k). - Alois P. Heinz, Jan 03 2023

cp's OEIS Frontend

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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A121549 Number of ordered ways of writing n as a sum of two Fibonacci numbers (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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A280210 Expansion of (Sum_{k>=1} mu(k)^2*x^k)^3, where mu(k) is the Moebius function (A008683).

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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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A357732 Number of partitions of n into 3 distinct positive Fibonacci numbers (with a single type of 1).

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

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