A076843 -id:A076843 - cp's OEIS frontend

A076844 a(1) = a(2) = a(3) = 1; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1

Offset: 1

N. J. A. Sloane, Nov 21 2002

nonn

Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3) has period 8. - James Propp, Nov 20 2002. This is the 8-cycle discovered by H. Todd - see Lyness, Note 1847. - N. J. A. Sloane, Jul 19 2020

R. C. Lyness, Note 1581. Cycles, Math. Gazette, 26 (1942), 62.
R. C. Lyness, Note 1847. Cycles, Math. Gaz., 29 (1945), 231-233.
R. C. Lyness, Note 2952. Cycles, Math. Gaz., 45 (1961), 207-209.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A076840, A076841, A076839, A076842, A076843.

a := 1; b := 1; c := 1; f := proc(n) option remember; global a,b,c; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n=3 then RETURN(c); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-3)); end;

nxt[{a_,b_,c_}]:={b,c,(b+c+1)/a}; Transpose[NestList[nxt,{1,1,1},110]][[1]] (* or *) PadRight[{},110,{1,1,1,3,5,9,5,3}] (* Harvey P. Dale, Jan 13 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 1, 3, 5, 9, 5, 3},105] (* Ray Chandler, Aug 25 2015 *)

A076842 Numerators of sequence of fractions defined by a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1)+a(n-2)+1)/a(n-2).

Original entry on oeis.org

1, 1, 3, 5, 3, 9, 29, 71, 751, 5095, 117707, 8786157, 1495204807, 869159527657, 10563805576326283, 3058448407404141530011, 21945047981309619478239602921, 78849616271246494204031567399858651677, 163732240861295155285004461265839986271194799721161

Offset: 1

N. J. A. Sloane, Nov 21 2002

			1, 1, 3, 5, 3, 9/5, 29/15, 71/27, 751/261, 5095/2059, 117707/53321, ...

Cf. A076843, A076844.

a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-2)); end;

cp's OEIS Frontend

A076844 a(1) = a(2) = a(3) = 1; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

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