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A124502 a(1)=a(2)=1; thereafter, a(n+1) = a(n) + a(n-1) + 1 if n is a multiple of 5, otherwise a(n+1) = a(n) + a(n-1). 4

%I

%S 1,1,2,3,5,9,14,23,37,60,98,158,256,414,670,1085,1755,2840,4595,7435,

%T 12031,19466,31497,50963,82460,133424,215884,349308,565192,914500,

%U 1479693,2394193,3873886,6268079,10141965,16410045,26552010,42962055,69514065,112476120

%N a(1)=a(2)=1; thereafter, a(n+1) = a(n) + a(n-1) + 1 if n is a multiple of 5, otherwise a(n+1) = a(n) + a(n-1).

%C If we split this sequence into 5 separate sequences of n mod 5, each individual sequence is of the form a(n) = 12*a(n-1) - 10*a(n-2) - a(n-3). For example, 12*98 - 10*9 - 1 = 1085. This is the same recurrence exhibited in A138134 and the n mod 5 =0 sequence...5, 60, 670, 7435 is A138134.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,1,-1,-1).

%F O.g.f.: x/((1-x)*(x^4 + x^3 + x^2 + x + 1)*(1 - x - x^2)). - _R. J. Mathar_, May 30 2008

%F a(n+5) = a(n) + Fibonacci(n+5), n>5.

%F a(n) = 12*a(n-5) - 10*a(n-10) - a(n-15). - _Gary Detlefs_, Dec 10 2010

%e a(6) = a(5) + a(4) + 1 = 5 + 3 + 1 = 9 because n=5 is a multiple of 5.

%e a(7) = a(6) + a(5) = 9 + 5 = 14 because n=6 is not a multiple of 5.

%p A124502:=proc(n) option remember; local t1; if n <= 2 then return 1; fi: if n mod 5 = 1 then t1:=1 else t1:=0; fi: procname(n-1)+procname(n-2)+t1; end proc; [seq(A124502(n), n=1..100)]; # _N. J. A. Sloane_, May 25 2008

%t a=0; b=0; lst={a,b}; Do[z=a+b+1; AppendTo[lst,z]; a=b; b=z; z=a+b; AppendTo[lst,z]; a=b; b=z; z=a+b; AppendTo[lst,z]; a=b; b=z; z=a+b; AppendTo[lst,z]; a=b; b=z; z=a+b; AppendTo[lst,z]; a=b; b=z,{n,4!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2010 *)

%t nxt[{n_,a_,b_}]:={n+1,b,If[Divisible[n,5],a+b+1,a+b]}; NestList[nxt,{2,1,1},40][[All,2]] (* or *) LinearRecurrence[{1,1,0,0,1,-1,-1},{1,1,2,3,5,9,14},40] (* _Harvey P. Dale_, Jun 15 2017 *)

%Y Cf. A052952, A004695, A080239, A131132.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, May 25 2008

%E Typo in Maple code corrected by _R. J. Mathar_, May 30 2008

%E More specific name from _R. J. Mathar_, Dec 09 2009

%E Indices in definition corrected by _N. J. A. Sloane_, Nov 25 2010

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Last modified July 22 16:43 EDT 2019. Contains 325225 sequences. (Running on oeis4.)