A110898 -id:A110898 - cp's OEIS frontend

A110897 a(n) = 1 + a(n - a(n-1)) (n - a(n-1)) mod 2 + (n - a(n-2)) mod 3, with a(0) = 0, a(1) = 1.

0, 1, 3, 3, 2, 5, 2, 7, 1, 4, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 8, 6, 5, 7, 9, 6, 8, 9, 8, 7, 10, 8, 11, 7, 10, 9, 10, 12, 10, 9, 10, 9, 8, 12, 12, 9, 15, 11, 14, 10, 11, 10, 8, 12, 10, 12, 14, 14, 12, 13, 13, 15, 14, 12, 8, 15, 13, 12, 16, 14, 15, 15, 16, 17, 17, 14, 14, 14, 10, 15, 18, 14, 18

Offset: 0

Roger L. Bagula, Sep 20 2005

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A110898, A206012.

[n le 2 select n-1 else 1 + Self(n-Self(n-1)) - ((n+1-Self(n-1)) mod 2) + ((n+2-Self(n-2)) mod 3) : n in [1..101]]; // G. C. Greubel, Mar 30 2024

a[n_]:= a[n]= If[n<2, n, 1 +a[n-a[n-1]] +Mod[n-a[n-2],3] -Mod[n-a[n-1],2]];
Table[a[n], {n,0,100}]

@CachedFunction
def a(n): # a = A110897
    if n<2: return n
    else: return 1 +a(n-a(n-1)) -(n-a(n-1))%2 +(n-a(n-2))%3
[a(n) for n in range(101)] # G. C. Greubel, Mar 30 2024

Edited by G. C. Greubel, Mar 30 2024

A206012 Modular recursion: a(0)=a(1)=a(2)=a(3)=1, thereafter: a(n) equals a(n - 2) + a(n - 3) when n = 0 mod 5, a(n - 1) + a(n - 3) when n = 1 mod 5, a(n - 1) + a(n - 2) when n = 2 mod 5, a(n - 1) + a(n - 4) when n = 3 mod 5, and a(n - 1) + a(n - 2) + a(n - 3) otherwise.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 3, 5, 8, 16, 13, 21, 34, 50, 105, 84, 134, 218, 323, 675, 541, 864, 1405, 2080, 4349, 3485, 5565, 9050, 13399, 28014, 22449, 35848, 58297, 86311, 180456, 144608, 230919, 375527, 555983, 1162429, 931510, 1487493, 2419003, 3581432, 7487928

Offset: 0

Roger L. Bagula, Mar 19 2012

This sequence was inspired by the work of Paul Curtz on three part sequences. I did a three part version of this that gave a generating polynomial and got even more variance by adding two more modulo sequences.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,6,0,0,0,0,3,0,0,0,0,-1).

Cf. A194880, A195240, A185138, A206568, A110897, A110898.

m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1))); // Bruno Berselli, Mar 20 2012

a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1;a[n_Integer] := a[n]=If[Mod[n, 5] == 0, a[n - 2] + a[n - 3], If[Mod[n, 5] == 1, a[n - 1] + a[n - 3], If[Mod[n, 5] == 2, a[n - 1] + a[n - 2], If[Mod[n, 5] == 3, a[n - 1] + a[n - 4], a[n - 1] + a[n - 2] + a[n - 3]]]]];b = Table[a[n], {n, 0, 50}];(* FindSequenceFunction gives*);Table[c[n] = b[[n]], {n, 1, 16}];c[n_Integer] := c[n] = -c[-15 + n] + c[-10 + n] + 6 c[-5 + n];d = Table[c[n], {n, 1, Length[b]}]
CoefficientList[Series[(x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1),{x,0,1001}],x] (* Vincenzo Librandi, Apr 01 2012 *)

Vec((x^15-x^13+x^12-2*x^10-2*x^9+2*x^8-x^7-3*x^6-4*x^5+3*x^4+x^3+x^2+x+1)/(x^15-3*x^10-6*x^5+1)+O(x^99)) \\ Charles R Greathouse IV, Mar 19 2012

G.f.: (x^15 - x^13 + x^12 - 2x^10 - 2x^9 + 2x^8 - x^7 - 3x^6 - 4x^5 + 3x^4 + x^3 + x^2 + x + 1) / (x^15 - 3x^10 - 6x^5 + 1). - Alois P. Heinz, Mar 19 2012

cp's OEIS Frontend

A110897 a(n) = 1 + a(n - a(n-1)) (n - a(n-1)) mod 2 + (n - a(n-2)) mod 3, with a(0) = 0, a(1) = 1.

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