A131132 -id:A131132 - cp's OEIS frontend

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

N. J. A. Sloane, May 25 2008

nonn

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.

			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.

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

Cf. A052952, A004695, A080239, A131132.

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

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 *)

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

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

A293553 a(n) is the integer k that minimizes |k/Fibonacci(n) - 1/4|.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 14, 22, 36, 58, 94, 152, 247, 399, 646, 1045, 1691, 2736, 4428, 7164, 11592, 18756, 30348, 49104, 79453, 128557, 208010, 336567, 544577, 881144, 1425722, 2306866, 3732588, 6039454, 9772042, 15811496, 25583539, 41395035, 66978574

Offset: 0

Clark Kimberling, Oct 14 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, 0, 1, -1, -1)

Cf. A000045, A004697, A131132, A293552.

z = 120; r = 1/4; f[n_] := Fibonacci[n];
Table[Floor[r*f[n]], {n, 0, z}];   (* A004697 *)
Table[Ceiling[r*f[n]], {n, 0, z}]; (* A293552 *)
Table[Round[r*f[n]], {n, 0, z}];   (* A293553 *)

G.f.: x^4/((-1 + x) (1 + x) (1 - x + x^2) (-1 + x + x^2) (1 + x + x^2)).
a(n) = a(n-1) + a(n-2) + a(n-6) - a(n-7) - a(n-8) for n >= 9.
a(n) = floor(1/2 + Fibonacci(n)/4).
a(n) = A004697(n) if (fractional part of Fibonacci(n)/4) < 1/2, otherwise a(n) = A293552(n).
a(n) = A131132(n-4) for n > 3. - Georg Fischer, Oct 22 2018

cp's OEIS Frontend

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).

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