A374438
Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with initial values T(n, k) = k + 1 for k < 3.
Original entry on oeis.org
1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 3, 4, 3, 1, 2, 3, 6, 6, 2, 1, 2, 3, 8, 9, 6, 3, 1, 2, 3, 10, 12, 12, 9, 2, 1, 2, 3, 12, 15, 20, 18, 8, 3, 1, 2, 3, 14, 18, 30, 30, 20, 12, 2, 1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3, 1, 2, 3, 18, 24, 56, 63, 70, 60, 30, 15, 2
Offset: 0
Triangle starts:
[ 0] [1]
[ 1] [1, 2]
[ 2] [1, 2, 3]
[ 3] [1, 2, 3, 2]
[ 4] [1, 2, 3, 4, 3]
[ 5] [1, 2, 3, 6, 6, 2]
[ 6] [1, 2, 3, 8, 9, 6, 3]
[ 7] [1, 2, 3, 10, 12, 12, 9, 2]
[ 8] [1, 2, 3, 12, 15, 20, 18, 8, 3]
[ 9] [1, 2, 3, 14, 18, 30, 30, 20, 12, 2]
[10] [1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3]
Family of triangles:
A162515 (m=1, Fibonacci),
A374439 (m=2, Lucas), this triangle (m=3).
Row sums:
A187890 (apart from initial terms), also
A001060 + 1 (with 1 prepended).
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M := 3; # family index
T := proc(n, k) option remember; if k > n then 0 elif k < M then k + 1 else
T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..11);
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from functools import cache
@cache
def T(n: int, k: int) -> int:
if k > n: return 0
if k < 3: return k + 1
return T(n - 1, k) + T(n - 2, k - 2)
A224824
Smallest m such that Fibonacci(m) >= m^n.
Original entry on oeis.org
5, 12, 21, 30, 41, 51, 62, 73, 85, 97, 109, 122, 134, 147, 160, 174, 187, 200, 214, 228, 242, 256, 270, 284, 298, 312, 327, 342, 356, 371, 386, 401, 416, 431, 446, 461, 476, 491, 507, 522, 538, 553, 569, 585, 600, 616, 632, 648, 664, 680
Offset: 1
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Flatten[{5,Table[Ceiling[(n*LambertW[-1,-Log[GoldenRatio]/(n*5^(1/(2*n)))])/-Log[GoldenRatio]],{n,2,50}]}] (* Vaclav Kotesovec, Jul 24 2013 *)
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a(n) = {my(ok = 0, m = 2); until (ok, if (fibonacci(m) >= m^n, ok = 1, m++); ); return (m); } \\ Michel Marcus, Jul 21 2013; corrected Jun 13 2022
A340326
a(n) = a(n-2) + (-1)^n*a(n-1) + n*(1-(-1)^n) with a(0) = a(1) = 1.
Original entry on oeis.org
1, 1, 2, 5, 7, 8, 15, 7, 22, 3, 25, 0, 25, 1, 26, 5, 31, 8, 39, 7, 46, 3, 49, 0, 49, 1, 50, 5, 55, 8, 63, 7, 70, 3, 73, 0, 73, 1, 74, 5, 79, 8, 87, 7, 94, 3, 97, 0, 97, 1, 98, 5, 103, 8, 111, 7, 118, 3, 121, 0, 121, 1, 122, 5, 127, 8, 135, 7, 142, 3, 145, 0, 145, 1, 146, 5, 151
Offset: 0
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