A173284
Triangle by columns, Fibonacci numbers in every column shifted down twice, for k > 0.
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 5, 2, 1, 8, 3, 1, 13, 5, 2, 21, 8, 3, 1, 34, 13, 5, 2, 1, 55, 21, 8, 3, 1, 89, 34, 13, 5, 2, 1, 144, 55, 21, 8, 3, 1, 233, 89, 34, 13, 5, 2, 1, 377, 144, 55, 21, 8, 3, 1, 610, 233, 89, 34, 13, 5, 2, 1
Offset: 0
First few rows of the triangle:
1;
1;
2, 1;
3, 1;
5, 2, 1;
8, 3, 1;
13, 5, 2, 1;
21, 8, 3, 1;
34, 13, 5, 2, 1;
55, 21, 8, 3, 1;
89, 34, 13, 5, 2, 1;
144, 55, 21, 8, 3, 1;
233, 89, 34, 13, 5, 2, 1;
377, 144, 55, 21, 8, 3, 1;
610, 233, 89, 34, 13, 5, 2, 1;
...
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T := proc(n, k): if n<0 then return(0) elif k < 0 or k > floor(n/2) then return(0) else combinat[fibonacci](n-2*k+1) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..14); # Johannes W. Meijer, Sep 05 2013
A309702
G.f. A(x) satisfies: A(x) = A(x^2) / (1 - x - x^2 - x^3).
Original entry on oeis.org
1, 1, 3, 5, 12, 20, 42, 74, 148, 264, 506, 918, 1730, 3154, 5876, 10760, 19938, 36574, 67536, 124048, 228664, 420248, 773878, 1422790, 2618646, 4815314, 8859904, 16293864, 29974958, 55128726, 101408308, 186511992, 343068964, 630989264, 1160606794, 2134665022
Offset: 0
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nmax = 35; A[] = 1; Do[A[x] = A[x^2]/(1 - x - x^2 - x^3) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 35; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)) - x^(3 2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
A309703
G.f. A(x) satisfies: A(x) = A(x^2) / (1 - x - x^2 - x^3 - x^4).
Original entry on oeis.org
1, 1, 3, 5, 13, 22, 48, 88, 184, 342, 684, 1298, 2556, 4880, 9506, 18240, 35366, 67992, 131446, 253044, 488532, 941014, 1815334, 3497924, 6745360, 12999632, 25063130, 48306046, 93123674, 179492482, 346003572, 666925774, 1285580868, 2478002696, 4776580902, 9207090240
Offset: 0
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nmax = 35; A[] = 1; Do[A[x] = A[x^2]/(1 - x - x^2 - x^3 - x^4) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 35; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)) - x^(3 2^k) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
A357366
Expansion of Product_{k>=0} 1 / (1 - x^(2^k) - x^(2^(k+1)))^(2^k).
Original entry on oeis.org
1, 1, 4, 5, 18, 23, 59, 82, 203, 285, 610, 895, 1838, 2733, 5217, 7950, 14763, 22713, 40526, 63239, 110652, 173891, 297529, 471420, 796706, 1268126, 2116508, 3384634, 5606444, 8991078, 14791302, 23782380, 38955441, 62737821, 102388280, 165126101, 268844542, 433970643
Offset: 0
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nmax = 37; CoefficientList[Series[Product[1/(1 - x^(2^k) - x^(2^(k + 1)))^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 37; A[] = 1; Do[A[x] = A[x^2]^2/(1 - x - x^2) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Showing 1-4 of 4 results.
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