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A356849
a(n) = a(n-1) - a(n-2) + 3*a(n-3) with a(0) = 1, a(1) = 2 and a(2) = 4.
1
1, 2, 4, 5, 7, 14, 22, 29, 49, 86, 124, 185, 319, 506, 742, 1193, 1969, 3002, 4612, 7517, 11911, 18230, 28870, 46373, 72193, 112430, 179356, 283505, 441439, 696002, 1105078, 1733393, 2716321, 4298162, 6782020, 10632821, 16745287, 26458526, 41611702, 65389037
OFFSET
0,2
COMMENTS
This sequence is mentioned in the Mathologer video in the links section.
FORMULA
a(n) = a(n-1) - a(n-2) + 3*a(n-3) with a(0) = 1, a(1) = 2 and a(2) = 4.
From G. C. Greubel, Aug 31 2022: (Start)
a(n) = A*(r1)^n + B*(r2)^n + C*(r3)^n, where r1 = (1 + x - y)/3, r2 = (2 - (1 + i*sqrt(3))*x +(1-i*sqrt(3))*y)/6, r3 = (2 -(1-i*sqrt(3))*x +(1-i*sqrt(3))*y)/6, x = (9*sqrt(17) + 37)^(1/3), y = (9*sqrt(17) - 37)^(1/3), A = (2-r2)*(2-r3)/((r1-r2)*(r1-r3)), B = (2-r1)*(2-r3)/((r2-r1)*(r2-r3)), C = (2-r1)*(2-r2)/((r3-r1)*(r3-r2)).
G.f.: (1 + x + 3*x^2)/(1 - x + x^2 - 3*x^3). (End)
EXAMPLE
a(4) = 7 because a(3) - a(2) + 3*a(1) = 5 - 4 + 2*3 = 7.
Number wall:
.. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ...
.. 1 2 4 5 7 14 22 29 49 86 142 185 319 506 ...
.. -1 0 6 -3 -21 42 78 -237 -93 1320 -534 -11073 8151 .......
.. 1 3 9 27 81 243 729 2187 6561 19693 ........................
.. 0 0 0 0 0 0 0 0 0 ..............................
MATHEMATICA
a[1]=1; a[2]=2; a[3]=4; a[n_]:=a[n]=a[n-1]-a[n-2]+3a[n-3]; Array[a, 50]
(* second program *)
LinearRecurrence[{1, -1, 3}, {1, 2, 4}, 40]
PROG
(Magma) [n le 3 select 2^(n-1) else Self(n-1) -Self(n-2) +3*Self(n-3): n in [1..41]]; // G. C. Greubel, Aug 31 2022
(SageMath)
@CachedFunction
def a(n): return 2^n if (n<3) else a(n-1) - a(n-2) + 3*a(n-3) # a = A356849
[a(n) for n in (0..40)] # G. C. Greubel, Aug 31 2022
CROSSREFS
Sequence in context: A332296 A115883 A063508 * A101724 A123210 A019277
KEYWORD
nonn,easy
AUTHOR
STATUS
approved