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
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.
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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
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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]]
A240224
Irregular triangular array read by rows: row n gives a list of the partitions of n into distinct Fibonacci numbers. The order of the partitions is like in Abramowitz-Stegun.
Original entry on oeis.org
1, 2, 3, 2, 1, 3, 1, 5, 3, 2, 5, 1, 3, 2, 1, 5, 2, 8, 5, 3, 5, 2, 1, 8, 1, 5, 3, 1, 8, 2, 5, 3, 2, 8, 3, 8, 2, 1, 5, 3, 2, 1, 8, 3, 1, 13, 8, 5, 8, 3, 2, 13, 1, 8, 5, 1, 8, 3, 2, 1, 13, 2, 8, 5, 2, 13, 3, 13, 2, 1, 8, 5, 3, 8, 5, 2, 1, 13, 3, 1, 8, 5, 3, 1, 13, 5, 13, 3, 2, 8, 5, 3, 2, 13, 5, 1, 13, 3, 2, 1, 8, 5, 3, 2, 1, 13, 5, 2
Offset: 1
The array with separated partitions begins:
n\k 1 2 3 4 5 ...
1: 1
2: 2
3: 3 2,1
4: 3,1
5: 5 3,2
6: 5,1 3,2,1
7: 5,2
8: 8 5,3 5,2,1
9: 8,1 5,3,1
10: 8,2 5,3,2
11: 8,3 8,2,1 5,3,2,1
12: 8,3,1
13: 13 8,5 8,3,2
14: 13,1 8,5,1 8,3,2,1
15: 13,2 8,5,2
16: 13,3 13,2,1 8,5,3 8,5,2,1
17: 13,3,1 8,5,3,1
18: 13,5 13,3,2 8,5,3,2
19: 13,5,1 13,3,2,1 8,5,3,2,1
20: 13,5,2
21: 21 13,8 13,5,3 13,5,2,1
22: 21,1 13,8,1 13,5,3,1
23: 21,2 13,8,2 13,5,3,2
24: 21,3 21,2,1 13,8,3 13,8,2,1 13,5,3,2,1
25: 21,3,1 13,8,3,1
...
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A290327
Total number of parts in all partitions of n into distinct Lucas numbers (beginning with 1) (A000204).
Original entry on oeis.org
1, 0, 1, 3, 2, 0, 3, 5, 0, 2, 6, 5, 0, 5, 9, 3, 0, 6, 9, 0, 5, 12, 7, 0, 9, 12, 0, 3, 10, 9, 0, 9, 17, 7, 0, 12, 16, 0, 7, 18, 12, 0, 12, 18, 4, 0, 10, 14, 0, 9, 21, 12, 0, 17, 22, 0, 7, 21, 16, 0, 16, 27, 9, 0, 18, 23, 0, 12, 27, 15, 0, 18, 22, 0, 4, 15, 14, 0, 14, 27, 12, 0, 21, 27, 0, 12, 32, 22, 0, 22
Offset: 1
a(8) = 5 because we have [7, 1], [4, 3, 1] and 2 + 3 = 5.
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nmax = 90; Rest[CoefficientList[Series[Sum[x^LucasL[i]/(1 + x^LucasL[i]) Product[(1 + x^LucasL[j]), {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]]
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