A187893
a(0)=1, a(1)=4, a(n) = a(n-1) + a(n-2) - 1.
Original entry on oeis.org
1, 4, 4, 7, 10, 16, 25, 40, 64, 103, 166, 268, 433, 700, 1132, 1831, 2962, 4792, 7753, 12544, 20296, 32839, 53134, 85972, 139105, 225076, 364180, 589255, 953434, 1542688, 2496121, 4038808, 6534928, 10573735, 17108662, 27682396, 44791057, 72473452, 117264508
Offset: 0
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Join[{a=1,b=4},Table[c=a+b-1;a=b;b=c,{n,100}]]
LinearRecurrence[{2,0,-1},{1,4,4},40] (* Harvey P. Dale, Jun 06 2020 *)
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a(n)=3*fibonacci(n)+1 \\ Charles R Greathouse IV, Oct 29 2016
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)
A187891
a(0)=0, a(1)=5, a(n)=a(n-1)+a(n-2)-1.
Original entry on oeis.org
0, 5, 4, 8, 11, 18, 28, 45, 72, 116, 187, 302, 488, 789, 1276, 2064, 3339, 5402, 8740, 14141, 22880, 37020, 59899, 96918, 156816, 253733, 410548, 664280, 1074827, 1739106, 2813932, 4553037, 7366968, 11920004, 19286971, 31206974, 50493944, 81700917
Offset: 0
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Join[{a=0,b=5},Table[c=a+b-1;a=b;b=c,{n,100}]]
nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{0,5},40][[All,1]] (* Harvey P. Dale, Nov 03 2022 *)
A187892
a(0)=0, a(1)=6, a(n)=a(n-1)+a(n-2)-1.
Original entry on oeis.org
0, 6, 5, 10, 14, 23, 36, 58, 93, 150, 242, 391, 632, 1022, 1653, 2674, 4326, 6999, 11324, 18322, 29645, 47966, 77610, 125575, 203184, 328758, 531941, 860698, 1392638, 2253335, 3645972, 5899306, 9545277, 15444582, 24989858, 40434439, 65424296, 105858734
Offset: 0
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Join[{a=0,b=6},Table[c=a+b-1;a=b;b=c,{n,100}]]
LinearRecurrence[{2,0,-1},{0,6,5},40] (* Harvey P. Dale, Aug 17 2019 *)
Showing 1-4 of 4 results.
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