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A121343 a(n) = Fibonacci(n) mod n(n+1)/2.

%I A121343 #14 Jun 20 2017 02:11:16
%S A121343 0,0,1,2,3,5,8,13,21,34,0,23,66,51,62,10,35,67,19,1,45,89,1,229,168,
%T A121343 275,298,236,319,59,155,125,309,376,407,485,630,628,419,466,615,370,
%U A121343 517,343,663,830,988,1033,168,624,700,746,1167,158,872,1105,609,610,59,1181,0,1,125
%N A121343 a(n) = Fibonacci(n) mod n(n+1)/2.
%H A121343 Alois P. Heinz, <a href="/A121343/b121343.txt">Table of n, a(n) for n = 0..10000</a>
%F A121343 A000045(n) modulo A000217(n).
%e A121343 a(11)=23 since Fib(11)=89==23(mod (11*12/2)).
%p A121343 a:= proc(n) local r, M, p, m; r, M, p, m:=
%p A121343       <<1|0>, <0|1>>, <<0|1>, <1|1>>, n, n*(n+1)/2;
%p A121343       do if irem(p, 2, 'p')=1 then r:= r.M mod m fi;
%p A121343          if p=0 then break fi; M:= M.M mod m
%p A121343       od; r[1, 2]
%p A121343     end:
%p A121343 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 26 2016
%t A121343 f[n_] := If[n == 0, 0, Mod[Fibonacci@n, n(n + 1)/2]]; f /@ Range[0, 62] (* _Robert G. Wilson v_, Aug 31 2006 *)
%t A121343 Join[{0},Mod[First[#],Last[#]]&/@With[{nn=70},Thread[{Fibonacci[ Range[ nn]], Accumulate[Range[nn]]}]]] (* _Harvey P. Dale_, May 21 2012 *)
%o A121343 (PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
%o A121343 a(n)=lift(fibmod(n,n*(n+1)/2)) \\ _Charles R Greathouse IV_, Jun 20 2017
%Y A121343 Cf. A000045, A023173, A000217, A096535.
%K A121343 nonn,easy
%O A121343 0,4
%A A121343 _N. J. A. Sloane_, Aug 29 2006
%E A121343 Edited by _N. J. A. Sloane_, Jul 01 2008 at the suggestion of _R. J. Mathar_