A309492 -id:A309492 - cp's OEIS frontend

A309494 a(1) = a(2) = a(3) = a(5) = 1, a(4) = 2; a(n) = a(n-a(n-3)) + a(n-a(n-4)) for n > 5.

1, 1, 1, 2, 1, 2, 3, 5, 8, 8, 7, 5, 2, 5, 13, 12, 18, 3, 5, 6, 4, 23, 21, 9, 5, 2, 5, 26, 14, 31, 3, 5, 6, 4, 36, 34, 9, 5, 2, 5, 39, 14, 44, 3, 5, 6, 4, 49, 47, 9, 5, 2, 5, 52, 14, 57, 3, 5, 6, 4, 62, 60, 9, 5, 2, 5, 65, 14, 70, 3, 5, 6, 4, 75, 73, 9, 5, 2, 5, 78, 14, 83, 3, 5, 6, 4, 88, 86, 9, 5, 2, 5, 91

Offset: 1

Altug Alkan, Aug 04 2019

A well-defined solution sequence for recurrence a(n) = a(n-a(n-3)) + a(n-a(n-4)).

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,-1).

Cf. A064657, A244477, A309492.

for n from 1 to 5 do a[n]:= `if`(n=4,2,1) od:
for n from 6 to 100 do a[n]:= a[n-a[n-3]] + a[n-a[n-4]] od:
seq(a[n],n=1..100); # Robert Israel, Aug 07 2019

a[1]=a[2]=a[3]=a[5]=1; a[4]=2; a[n_] := a[n] = a[n - a[n-3]] + a[n - a[n-4]]; Array[a, 93] (* Giovanni Resta, Aug 07 2019 *)

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

Vec(x*(1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + 7*x^10 + 5*x^11 + 2*x^12 + 3*x^13 + 11*x^14 + 10*x^15 + 14*x^16 + x^17 + x^18 - 6*x^20 + 7*x^21 + 5*x^22 - 5*x^23 - 5*x^24 - 2*x^25 - 4*x^26 + x^27 - 9*x^28 - 3*x^29 - 2*x^30 - 3*x^31 - 3*x^32 + x^33 - 2*x^34 - 2*x^36 - 2*x^41) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)^2) + O(x^80)) \\ Colin Barker, Aug 08 2019

For k > 1,
  a(13*k-9) = 13*k-8,
  a(13*k-8) = 3,
  a(13*k-7) = 5,
  a(13*k-6) = 6,
  a(13*k-5) = 4,
  a(13*k-4) = 13*k-3,
  a(13*k-3) = 13*k-5,
  a(13*k-2) = 9,
  a(13*k-1) = 5,
  a(13*k)   = 2,
  a(13*k+1) = 5,
  a(13*k+2) = 13*k,
  a(13*k+3) = 14.
From Colin Barker, Aug 05 2019: (Start)
G.f.: x*(1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 5*x^7 + 8*x^8 + 8*x^9 + 7*x^10 + 5*x^11 + 2*x^12 + 3*x^13 + 11*x^14 + 10*x^15 + 14*x^16 + x^17 + x^18 - 6*x^20 + 7*x^21 + 5*x^22 - 5*x^23 - 5*x^24 - 2*x^25 - 4*x^26 + x^27 - 9*x^28 - 3*x^29 - 2*x^30 - 3*x^31 - 3*x^32 + x^33 - 2*x^34 - 2*x^36 - 2*x^41) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12)^2).
a(n) = 2*a(n-13) - a(n-26) for n > 42.
(End)

A309567 a(1) = 4, a(2) = 2, a(3) = 5, a(4) = 3, a(5) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 5.

Original entry on oeis.org

4, 2, 5, 3, 1, 4, 7, 5, 8, 6, 4, 12, 5, 13, 6, 9, 17, 5, 18, 6, 9, 22, 5, 23, 11, 9, 27, 5, 28, 11, 9, 32, 5, 33, 11, 14, 37, 5, 38, 11, 14, 42, 5, 43, 11, 14, 47, 5, 48, 16, 14, 52, 5, 53, 16, 14, 57, 5, 58, 16, 14, 62, 5, 63, 16, 19, 67, 5, 68, 16, 19, 72, 5, 73, 16, 19, 77, 5, 78, 16, 19, 82, 5, 83, 21, 19, 87, 5

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 08 2019

A well-defined quasi-periodic solution for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Robert Israel, Table of n, a(n) for n = 1..10000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063882, A244477, A296518, A309492, A309494, A309496, A309554.

f:= proc(n) local k,j;
  j:= n mod 5;
  k:= (n-j)/5;
  if j=0 then 5*floor(sqrt(k-1))+1
  elif j=1 then 5*round(sqrt(k))-1
  elif j=2 then 5*k+2
  elif j=3 then 5
  else 5*k+3
  fi
end proc:
f(1):= 4:
map(f, [$1..100]); # Robert Israel, Aug 08 2019

a[n_] := a[n] = If[n < 6, {4, 2, 5, 3, 1}[[n]], a[n - a[n-1]] + a[n - a[n-4]]]; Array[a, 88] (* Giovanni Resta, Aug 08 2019 *)

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

For k >= 1:
a(5*k)   = 5*floor(sqrt(k-1))+1,
a(5*k+1) = 5*round(sqrt(k))-1,
a(5*k+2) = 5*k+2,
a(5*k+3) = 5,
a(5*k+4) = 5*k+3.

A309496 a(1) = 1, a(2) = 3, a(3) = a(4) = a(6) = 6, a(5) = 2, a(7) = 4; a(n) = a(n-a(n-4)) + a(n-a(n-5)) for n > 7.

Original entry on oeis.org

1, 3, 6, 6, 2, 6, 4, 6, 10, 12, 6, 12, 10, 9, 16, 18, 6, 16, 18, 9, 22, 24, 6, 22, 24, 9, 28, 30, 6, 28, 30, 9, 34, 36, 6, 34, 36, 9, 40, 42, 6, 40, 42, 9, 46, 48, 6, 46, 48, 9, 52, 54, 6, 52, 54, 9, 58, 60, 6, 58, 60, 9, 64, 66, 6, 64, 66, 9, 70, 72, 6, 70, 72, 9, 76, 78, 6, 76, 78, 9, 82, 84, 6, 82, 84, 9, 88

Offset: 1

Altug Alkan, Aug 05 2019

A well-defined solution sequence for recurrence a(n) = a(n-a(n-4)) + a(n-a(n-5)).

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

Cf. A244477, A309492, A309494.

I:=[1,3,6,6,2,6,4];[n le 7 select I[n] else Self(n-Self(n-4))+Self(n-Self(n-5)): n in [1..90]]; // Marius A. Burtea, Aug 07 2019

a[n_] := a[n] = If[n < 8, {1, 3, 6, 6, 2, 6, 4}[[n]], a[n - a[n-4]] + a[n - a[n-5]]]; Array[a, 87] (* Giovanni Resta, Aug 07 2019 *)

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

Vec(x*(1 + 3*x + 6*x^2 + 5*x^3 - x^4 - 3*x^6 + x^7 - 2*x^8 + 3*x^9 + x^10 + 2*x^11 - x^13 - 3*x^16 - 2*x^17 + 2*x^18 + 2*x^20 - 2*x^21) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)^2) + O(x^80)) \\ Colin Barker, Aug 15 2019

For k > 2:
  a(6*k-4) = 9,
  a(6*k-3) = 6*k-2,
  a(6*k-2) = 6*k,
  a(6*k-1) = 6,
  a(6*k)   = 6*k-2,
  a(6*k+1) = 6*k.
From Colin Barker, Aug 05 2019: (Start)
G.f.: x*(1 + 3*x + 6*x^2 + 5*x^3 - x^4 - 3*x^6 + x^7 - 2*x^8 + 3*x^9 + x^10 + 2*x^11 - x^13 - 3*x^16 - 2*x^17 + 2*x^18 + 2*x^20 - 2*x^21) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)^2).
a(n) = a(n-3) + a(n-6) - a(n-9) for n > 22.
(End)

A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

Original entry on oeis.org

1, 2, 2, 7, 5, 1, 7, 2, 9, 3, 11, 3, 6, 4, 14, 5, 9, 16, 6, 15, 6, 10, 8, 21, 21, 21, 2, 28, 3, 30, 3, 6, 4, 33, 5, 9, 35, 6, 34, 6, 10, 8, 40, 40, 40, 2, 47, 3, 49, 3, 6, 4, 52, 5, 9, 54, 6, 53, 6, 10, 8, 59, 59, 59, 2, 66, 3, 68, 3, 6, 4, 71, 5, 9, 73, 6, 72, 6, 10, 8, 78, 78, 78, 2, 85, 3, 87, 3, 6, 4

Offset: 1

Altug Alkan and Rémy Sigrist, Aug 07 2019

A well-defined solution sequence for recurrence a(n) = a(n-a(n-1)) + a(n-a(n-3)).

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

Cf. A046700, A244477, A296518, A309492, A309494, A309496.

I:=[1,2,2,7,5,1,7,2];[n le 8 select I[n] else Self(n-Self(n-1))+Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 08 2019

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

Vec(x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2) + O(x^90)) \\ Colin Barker, Aug 11 2019

For k >= 1:
a(19*k-11) = 2,
a(19*k-10) = 19*k-10,
a(19*k-9) = 3,
a(19*k-8) = 19*k-8,
a(19*k-7) = 3,
a(19*k-6) = 6,
a(19*k-5) = 4,
a(19*k-4) = 19*k-5,
a(19*k-3) = 5,
a(19*k-2) = 9,
a(19*k-1) = 19*k-3,
a(19*k) = 6,
a(19*k+1) = 19*k-4,
a(19*k+2) = 6,
a(19*k+3) = 10,
a(19*k+4) = 8,
a(19*k+5) = a(19*k+6) = a(19*k+7) = 19*k+2.
From Colin Barker, Aug 08 2019: (Start)
G.f.: x*(1 + x)*(1 + x + x^2 + 6*x^3 - x^4 + 2*x^5 + 5*x^6 - 3*x^7 + 12*x^8 - 9*x^9 + 20*x^10 - 17*x^11 + 23*x^12 - 19*x^13 + 33*x^14 - 28*x^15 + 37*x^16 - 21*x^17 + 27*x^18 - 14*x^19 + 16*x^20 - 10*x^21 + 4*x^22 + 7*x^23 + 12*x^24 - 5*x^25 + 3*x^26 + 7*x^27 - 10*x^28 + 18*x^29 - 21*x^30 + 15*x^31 - 19*x^32 + 24*x^33 - 29*x^34 + 20*x^35 - 17*x^36 + 11*x^37 - 6*x^38 + 2*x^39 - 10*x^40 + 9*x^41 - 6*x^42 + 5*x^43) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18)^2).
a(n) = 2*a(n-19) - a(n-38) for n > 45.
(End)

A309636 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2, a(5) = 5; a(6) = 3; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

Original entry on oeis.org

3, 1, 4, 2, 5, 3, 6, 4, 7, 10, 8, 6, 9, 7, 10, 13, 6, 14, 12, 10, 18, 6, 14, 17, 10, 23, 11, 14, 22, 10, 28, 16, 14, 27, 10, 33, 16, 14, 32, 10, 38, 16, 19, 37, 10, 43, 16, 24, 42, 10, 48, 16, 24, 47, 10, 53, 16, 24, 52, 10, 58, 16, 24, 57, 10, 63, 21, 24, 62, 10, 68, 26, 24, 67, 10

Offset: 1

Altug Alkan and Nathan Fox, Aug 10 2019

A well-defined quasi-periodic solution for Hofstadter V recurrence (a(n) = a(n-a(n-1)) + a(n-a(n-4))).

Michael De Vlieger, Table of n, a(n) for n = 1..10006
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063882, A244477, A296518, A309492, A309494, A309496, A309554, A309567.

I:=[3,1,4,2,5,3]; [n le 6 select I[n] else  Self(n-Self(n-1)) + Self(n-Self(n-4)): n in [1..80]]; // Marius A. Burtea, Aug 11 2019

Nest[Append[#, #[[-#[[-1]] ]] + #[[-#[[-4]] ]]] &, {3, 1, 4, 2, 5, 3}, 69] (* Michael De Vlieger, May 08 2020 *)

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

For k > 1:
a(5*k)   = 10,
a(5*k+1) = 5*k-2,
a(5*k+2) = 5*(floor((sqrt(2*k-1)-1)/2) + floor((sqrt(2*k-3)-1)/2)) + 6,
a(5*k+3) = 5*(floor(sqrt(k/2)) + floor(sqrt((k-1)/2))) + 4,
a(5*k+4) = 5*k-3.
Also, a(5*k+2) = 5*f(k)+1 and a(5*k+3) = 5*g(k)-1 where f(k) = g(k-g(k-1)) and g(k) = f(k-f(k))+2 with f(1) = g(1) = 1, g(2) = 2.

A309650 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2; a(n) = a(n-a(n-2)) + a(n-a(n-3)) for n > 4.

Original entry on oeis.org

3, 1, 4, 2, 5, 3, 6, 9, 7, 5, 3, 11, 14, 7, 5, 8, 16, 19, 7, 5, 8, 21, 24, 12, 5, 8, 26, 29, 12, 5, 8, 31, 34, 12, 5, 13, 36, 39, 12, 5, 13, 41, 44, 12, 5, 13, 46, 49, 17, 5, 13, 51, 54, 17, 5, 13, 56, 59, 17, 5, 13, 61, 64, 17, 5, 18, 66, 69, 17, 5, 18, 71, 74, 17, 5, 18, 76, 79, 17, 5, 18, 81, 84, 22, 5

Offset: 1

Altug Alkan and Nathan Fox, Aug 11 2019

A well-defined quasi-periodic solution for recurrence (a(n) = a(n-a(n-2)) + a(n-a(n-3))).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Altug Alkan, Nathan Fox, Orhan Ozgur Aybar, Zehra Akdeniz, On Some Solutions to Hofstadter's V-Recurrence, arXiv:2002.03396 [math.DS], 2020.

Cf. A063892, A244477, A309492, A309567.

I:=[3,1,4,2]; [n le 4 select I[n] else  Self(n-Self(n-2)) + Self(n-Self(n-3)): n in [1..90]]; // Marius A. Burtea, Aug 11 2019

Nest[Append[#, #[[-#[[-2]] ]] + #[[-#[[-3]] ]]] &, {3, 1, 4, 2}, 81] (* Michael De Vlieger, May 08 2020 *)

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

For k >= 1:
a(5*k)   = 5,
a(5*k+1) = 5*floor(sqrt(k)+1/2)-2,
a(5*k+2) = 5*k+1,
a(5*k+3) = 5*k+4,
a(5*k+4) = 5*floor(sqrt(k))+2.

cp's OEIS Frontend

A309494 a(1) = a(2) = a(3) = a(5) = 1, a(4) = 2; a(n) = a(n-a(n-3)) + a(n-a(n-4)) for n > 5.

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A309567 a(1) = 4, a(2) = 2, a(3) = 5, a(4) = 3, a(5) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 5.

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A309496 a(1) = 1, a(2) = 3, a(3) = a(4) = a(6) = 6, a(5) = 2, a(7) = 4; a(n) = a(n-a(n-4)) + a(n-a(n-5)) for n > 7.

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A309554 a(1) = a(6) = 1, a(2) = a(3) = a(8) = 2, a(4) = a(7) = 7, a(5) = 5; a(n) = a(n-a(n-1)) + a(n-a(n-3)) for n > 8.

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A309636 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2, a(5) = 5; a(6) = 3; a(n) = a(n-a(n-1)) + a(n-a(n-4)) for n > 6.

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A309650 a(1) = 3, a(2) = 1, a(3) = 4, a(4) = 2; a(n) = a(n-a(n-2)) + a(n-a(n-3)) for n > 4.

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