A309636 -id:A309636 - cp's OEIS frontend

A309704 a(1) = 3, a(2) = 4, a(3) = 5, a(4) = 4, a(5) = 5; a(6) = 6; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

3, 4, 5, 4, 5, 6, 7, 7, 8, 8, 9, 10, 10, 10, 11, 11, 13, 12, 14, 14, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 24, 23, 23, 25, 24, 26, 27, 27, 26, 28, 28, 28, 29, 29, 30, 31, 31, 32, 32, 33, 33, 34, 35, 35, 35, 36, 36, 37, 37, 38, 39, 39, 39, 40, 40, 40, 41, 41, 42, 43, 43, 45, 45, 45, 45, 48, 44, 48, 49, 47, 52, 47, 51, 50, 47, 52, 50, 54, 52, 54, 55, 54, 54, 56

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 13 2019

This sequence is finite but has an exceptionally long life: a(3080193026) = 3101399868 is its last term since a(3080193027) refers to a nonpositive index and thus fails to exist. See plots in Links section to fractal-like structure of a(n)-n/2.

Altug Alkan, Table of n, a(n) for n = 1..20000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.
Rémy Sigrist, Density plot of a(n)-n/2 for n <= 10^8
Rémy Sigrist, Scatterplot of a(n)-n/2 for n <= 10^8
Rémy Sigrist, Density plot of the whole sequence (n = 1..3080193026)

Cf. A005185, A063882, A309567, A309636.

Nest[Append[#, #[[-#[[-1]] ]] + #[[-#[[-4]] ]]] &, {3, 4, 5, 4, 5, 6}, 94] (* Michael De Vlieger, May 08 2020 *)

q=vector(100); q[1]=3; q[2]=4; q[3]=5; q[4]=4; q[5]=5; q[6]=6; for(n=7, #q, q[n] = q[n-q[n-1]] + q[n-q[n-4]]); q

a(3080193026) from Giovanni Resta, Aug 13 2019

A309967 a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 8, a(6) = 6, a(7) = a(8) = 4; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 8.

Original entry on oeis.org

1, 1, 2, 3, 8, 6, 4, 4, 9, 4, 8, 7, 9, 12, 6, 13, 7, 14, 17, 6, 18, 7, 19, 22, 6, 23, 7, 24, 27, 6, 28, 7, 29, 32, 6, 33, 7, 34, 37, 6, 38, 7, 39, 42, 6, 43, 7, 44, 47, 6, 48, 7, 49, 52, 6, 53, 7, 54, 57, 6, 58, 7, 59, 62, 6, 63, 7, 64, 67, 6, 68, 7, 69, 72, 6, 73, 7, 74, 77, 6, 78, 7

Offset: 1

Altug Alkan, Aug 25 2019

A quasilinear solution sequence for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).

Cf. A005185, A063882, A244477, A309567, A309636.

q=vector(100); q[1]=q[2]=1; q[3]=2; q[4]=3; q[5]=8; q[6]=6; q[7]=q[8]=4; for(n=9, #q, q[n]=q[n-q[n-1]]+q[n-q[n-4]]); q

Vec(x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2) + O(x^40)) \\ Colin Barker, Aug 25 2019

For k > 2:
a(5*k-4) = 5*k-7,
a(5*k-3) = 7,
a(5*k-2) = 5*k-6,
a(5*k-1) = 5*k-3,
a(5*k) = 6.
From Colin Barker, Aug 25 2019: (Start)
G.f.: x*(1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 4*x^5 + 2*x^6 + 3*x^8 - 12*x^9 - 3*x^10 +  3*x^12 - 3*x^13 + 6*x^14 + 3*x^15 - 3*x^16 + 2*x^18 - 2*x^19) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)^2).
a(n) = 2*a(n-5) - a(n-10) for n > 20.
(End)

cp's OEIS Frontend

A309704 a(1) = 3, a(2) = 4, a(3) = 5, a(4) = 4, a(5) = 5; a(6) = 6; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

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A309967 a(1) = a(2) = 1, a(3) = 2, a(4) = 3, a(5) = 8, a(6) = 6, a(7) = a(8) = 4; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 8.

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