A338736 -id:A338736 - cp's OEIS frontend

A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

0, 1, 1, 2, 0, 3, 2, 2, 1, 5, 1, 0, 8, 13, 10, 2, 12, 15, 5, 10, 1, 1, 1, 0, 0, 25, 1, 2, 5, 15, 27, 2, 10, 33, 20, 0, 1, 1, 34, 10, 40, 21, 18, 2, 10, 1, 1, 0, 1, 25, 1, 2, 16, 21, 5, 26, 37, 1, 7, 0, 33, 27, 1, 2, 40, 21, 5, 2, 1, 15, 1, 0, 46, 1, 25, 2, 68

Offset: 1

Alois P. Heinz, Oct 09 2015

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000045, A002708, A007570, A023172 (where a(n)=0), A263101, A274996, A338736.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
p:= (M, n, k)-> map(x-> x mod k, `if`(n=0, <<1|0>, <0|1>>,
          `if`(n::even, p(M, n/2, k)^2, p(M, n-1, k).M))):
a:= n-> p(<<0|1>, <1|1>>, F(n), n)[1, 2]:
seq(a(n), n=1..80);

F[n_] := MatrixPower[{{0, 1}, {1, 1}}, n][[1, 2]];
p[M_, n_, k_] := Mod[#, k]& /@ If[n == 0, {{1, 0}, {0, 1}}, If[EvenQ[n], MatrixPower[p[M, n/2, k], 2], p[M, n - 1, k].M]];
a[n_] := p[{{0, 1}, {1, 1}}, F[n], n][[1, 2]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Oct 29 2024, after Alois P. Heinz *)

a(n) = A007570(n) mod n.

A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 3, 1, 1, 1, 0, 1, 1, 29, 7, 1, 19679776435706023589554718882448088434898811874077010905231927243854, 1, 7

Offset: 0

Alois P. Heinz, Nov 14 2020

nonn

a(21) = 2992285359..7163788371 has 5090 decimal digits.

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A000032, A005371, A262361, A274996, A338638, A338736.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (h-> b(h$2) mod h)(b(b(n))):
seq(a(n), n=0..15);

a(n) = A262361(n) mod A005371(n).

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 1, 1, 7, 4, 1, 199, 1, 4, 843, 1, 1, 0, 1, 29, 123, 4, 1, 3, 199, 4, 39603, 29, 1, 5778, 1, 1, 7, 4, 17622890, 12752043, 1, 4, 39603, 7881196, 1, 5778, 1, 29, 7, 4, 1, 3, 1149851, 28143689044, 7, 29, 1, 0, 312119004790, 6643838879, 7, 4, 1

Offset: 0

Alois P. Heinz, Nov 04 2020

Alois P. Heinz, Table of n, a(n) for n = 0..4795

Cf. A000032, A005371, A016089, A213060, A263101, A338736, A338889.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f$2) mod f)(b(n)):
seq(a(n), n=0..60);

Table[Mod[LucasL[LucasL[n]],LucasL[n]],{n,0,60}] (* Harvey P. Dale, Jul 04 2022 *)

a(n) = A005371(n) mod A000032(n).

a(n) = 0 for n in { A016089 }.

cp's OEIS Frontend

A263112 a(n) = F(F(n)) mod n, where F = Fibonacci = A000045.

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A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

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A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

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