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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.
7
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
1,1
COMMENTS
A quasi-periodic solution to the three-term Hofstadter recurrence a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)). For a quasi-quadratic solution, see also A268368 that uses the convention that evaluating at a nonpositive index gives zero (this sequence and A244477 do not use this convention). See page 119 at the Fox reference for the detailed analysis of solutions with initial conditions with 1 through N. The three-term Hofstadter recurrence also has a slow solution (A278055) and chaotic solutions such as A292351, A296413 and A296440. So the recurrence a(n) = a(n-a(n-1)) + a(n-a(n-2)) + a(n-a(n-3)) has all known types of behaviors that are mentioned on page 7 of the Tanny reference in the Links section.
LINKS
Altug Alkan, On a conjecture about generalized Q-recurrence, Open Mathematics (2018) Vol. 16, Issue 1, 1490-1500.
Steve Tanny, An Invitation to Nested Recurrence Relations, Slides of talk presented at CanaDAM 2013.
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)
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
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 *)
PROG
(PARI) q=vector(10^5); 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
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Dec 14 2017
STATUS
approved