A341072
Number of compositions of 2n into n Fibonacci parts.
Original entry on oeis.org
1, 1, 3, 7, 23, 71, 231, 750, 2479, 8251, 27673, 93248, 315515, 1071097, 3646618, 12445982, 42571327, 145895599, 500855361, 1722062265, 5929045173, 20439121983, 70539320558, 243695962031, 842704577995, 2916613479471, 10102511916071, 35018749192885
Offset: 0
-
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, `if`(t=0, 1, 0), `if`(t<1, 0, add(
`if`(g(j), b(n-j, t-1), 0), j=1..n)))
end:
a:= n-> b(2*n, n):
seq(a(n), n=0..35);
-
g[n_] := g[n] = With[{t = 5*n^2}, IntegerQ@Sqrt[t+4] || IntegerQ@Sqrt[t-4]];
b[n_, t_] := b[n, t] =
If[n == 0, If[t == 0, 1, 0], If[t < 1, 0, Sum[
If[g[j], b[n - j, t - 1], 0], {j, 1, n}]]];
a[n_] := b[2n, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 02 2022, after Alois P. Heinz *)
A357717
Number of ways to write n as an ordered sum of nine positive Fibonacci numbers (with a single type of 1).
Original entry on oeis.org
1, 9, 45, 156, 423, 954, 1878, 3321, 5409, 8251, 11979, 16686, 22446, 29250, 37134, 46107, 56259, 67671, 80407, 94338, 109269, 125118, 141930, 159723, 178608, 198522, 219510, 241338, 264438, 288810, 314550, 341010, 367785, 394596, 421443, 448650, 476614, 505404, 534978
Offset: 9
Cf.
A000045,
A076739,
A121548,
A121549,
A121550,
A319402,
A357688,
A357690,
A357691,
A357694,
A357716.
-
nmax = 47; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^9, {x, 0, nmax}], x] // Drop[#, 9] &
A121551
Number of parts in all the compositions of n into Fibonacci numbers (i.e., in all ordered sequences of Fibonacci numbers having sum n; only one 1 is considered as a Fibonacci number).
Original entry on oeis.org
1, 3, 8, 19, 44, 98, 213, 457, 965, 2018, 4183, 8604, 17594, 35780, 72428, 146024, 293335, 587386, 1172836, 2335761, 4640947, 9201531, 18208325, 35967145, 70929855, 139667107, 274630886, 539309530, 1057789244, 2072370716, 4055782140, 7929563974, 15488792843
Offset: 1
a(4)=19 because the compositions of 8 into Fibonacci numbers are [1,3],[2,2],[3,1],[1,1,2],[1,2,1],[2,1,1] and [1,1,1,1], having a total of 2+2+2+3+3+3+4 = 19 parts.
-
with(combinat): g:=sum(z^fibonacci(i),i=2..20)/(1-sum(z^fibonacci(i),i=2..20))^2: gser:=series(g,z=0,48): seq(coeff(gser,z,n),n=1..35);
A357730
Number of ways to write n as an ordered sum of ten positive Fibonacci numbers (with a single type of 1).
Original entry on oeis.org
1, 10, 55, 210, 625, 1542, 3300, 6310, 11040, 17980, 27673, 40660, 57475, 78520, 104175, 134742, 170620, 212220, 260035, 314290, 374933, 441790, 514855, 594210, 680070, 772582, 871920, 977790, 1090680, 1210960, 1339417, 1475340, 1618020, 1766080, 1918785, 2076012
Offset: 10
Cf.
A000045,
A076739,
A121548,
A121549,
A121550,
A319403,
A357688,
A357690,
A357691,
A357694,
A357716,
A357717.
-
nmax = 45; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^10, {x, 0, nmax}], x] // Drop[#, 10] &
Comments