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).

1, 1, 2, 3, 5, 9, 14, 23, 37, 60, 98, 158, 256, 414, 670, 1085, 1755, 2840, 4595, 7435, 12031, 19466, 31497, 50963, 82460, 133424, 215884, 349308, 565192, 914500, 1479693, 2394193, 3873886, 6268079, 10141965, 16410045, 26552010, 42962055, 69514065, 112476120

Offset: 1

Author

N. J. A. Sloane, May 25 2008

Keywords

• nonn

Comments

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.

Examples

			a(6) = a(5) + a(4) + 1 = 5 + 3 + 1 = 9 because n=5 is a multiple of 5.
a(7) = a(6) + a(5) = 9 + 5 = 14 because n=6 is not a multiple of 5.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,1,-1,-1).

Crossrefs

Cf. A052952, A004695, A080239, A131132.

Programs

Maple

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

Mathematica

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 *)
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 *)

Formula

O.g.f.: x/((1-x)*(x^4 + x^3 + x^2 + x + 1)*(1 - x - x^2)). - R. J. Mathar, May 30 2008

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

a(n) = 12*a(n-5) - 10*a(n-10) - a(n-15). - Gary Detlefs, Dec 10 2010

Extensions

Typo in Maple code corrected by R. J. Mathar, May 30 2008

More specific name from R. J. Mathar, Dec 09 2009

Indices in definition corrected by N. J. A. Sloane, Nov 25 2010