A296518 a(1) = 3, a(2) = a(5) = 1, a(3) = a(4) = a(6) = 2; a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) for n > 6.
3, 1, 2, 2, 1, 2, 4, 8, 8, 4, 8, 12, 12, 4, 12, 16, 16, 4, 16, 20, 20, 4, 20, 24, 24, 4, 24, 28, 28, 4, 28, 32, 32, 4, 32, 36, 36, 4, 36, 40, 40, 4, 40, 44, 44, 4, 44, 48, 48, 4, 48, 52, 52, 4, 52, 56, 56, 4, 56, 60, 60, 4, 60, 64, 64, 4, 64, 68, 68, 4, 68, 72, 72, 4, 72, 76, 76, 4, 76, 80, 80, 4, 80, 84, 84, 4, 84, 88, 88, 4
Offset: 1
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics (2018) Vol. 16, Issue 1, 1490-1500.
- Nathan Fox, An exploration of nested recurrences using experimental mathematics, Ph.D. thesis, 2017 (19 MB).
- Steve Tanny, An Invitation to Nested Recurrence Relations, Slides of talk presented at CanaDAM 2013.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).
Programs
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Maple
a:= proc(n) option remember; procname(n-procname(n-1))+procname(n-procname(n-2))+procname(n-procname(n-3)) end proc: a(1):= 3: a(2):= 1: a(3):= 2: a(4):= 2: a(5):= 1: a(6):= 2: map(a, [$1..100]); # after Robert Israel at A296440
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Mathematica
Fold[Append[#1, #1[[#2 - #1[[#2 - 1]] ]] + #1[[#2 - #1[[#2 - 2]] ]] + #1[[#2 - #1[[#2 - 3]] ]] ] &, {3, 1, 2, 2, 1, 2}, Range[7, 90]] (* or *) Rest@ CoefficientList[Series[x (3 + x + 2 x^2 + 2 x^3 - 5 x^4 + 4 x^7 + 9 x^8 + x^9 + 2 x^10 - 2 x^11 - 3 x^12 - 2 x^13)/((1 - x)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 90}], x] (* Michael De Vlieger, Dec 14 2017 *) LinearRecurrence[{0,0,0,2,0,0,0,-1},{3,1,2,2,1,2,4,8,8,4,8,12,12,4},100] (* Harvey P. Dale, May 30 2018 *) -
PARI
my(q=vector(100)); q[1]=3;q[2]=1;q[3]=2;q[4]=2;q[5]=1;q[6]=2;for(n=7, #q, q[n] = q[n-q[n-1]]+q[n-q[n-2]]+q[n-q[n-3]]); q
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PARI
Vec(x*(3 + x + 2*x^2 + 2*x^3 - 5*x^4 + 4*x^7 + 9*x^8 + x^9 + 2*x^10 - 2*x^11 - 3*x^12 - 2*x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2) + O(x^100)) \\ Colin Barker, Dec 14 2017
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Scheme
;; With memoization-macro definec. (definec (A296518 n) (cond ((= 1 n) 3) ((or (= 2 n) (= 5 n)) 1) ((<= n 6) 2) (else (+ (A296518 (- n (A296518 (- n 1)))) (A296518 (- n (A296518 (- n 2)))) (A296518 (- n (A296518 (- n 3)))))))) ;; Antti Karttunen, Dec 16 2017
Formula
a(4*k) = a(4*k+1) = a(4*k+3) = 4*k, a(4*k+2) = 4 for k > 1.
From Colin Barker, Dec 14 2017: (Start)
G.f.: x*(3 + x + 2*x^2 + 2*x^3 - 5*x^4 + 4*x^7 + 9*x^8 + x^9 + 2*x^10 - 2*x^11 - 3*x^12 - 2*x^13) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8) for n > 14.
(End)
Comments