0,4

a(n) is last term in the period of continued fraction expansion of phi^n (phi being the golden number). E.g.: n=10, phi^10=[88,1,121,1,121,1,121,...] (and the period may only have 1 or 2 terms). Also a(n)=floor(phi^n)-(n+1)%2, or a(n)=A014217(n)-(n+1)%2 - Thomas Baruchel, Nov 05 2002

a(n) = A050140(Fibonacci(n)). - Thomas Baruchel, Nov 05 2002

a(n)= Lucas_number(n)-1-(-1)^n=A000032(n)-1-(-1)^n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Feb 18 2006

a(n) = resultant of the polynomials x^2-x-1 and x^(n+1)-x^n-1 for n>=1. - Richard Choulet (richardchoulet(AT)yahoo.fr), Aug 05 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

Baake, Michael; Hermisson, Joachim; Pleasants, Peter A. B.; The torus parametrization of quasiperiodic LI-classes. J. Phys. A 30 (1997), no. 9, 3029-3056.

C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22.

G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290; see Th. 7.12.

N. Garnier, O. Ramare, Fibonacci numbers and trigonometric identities, April 2006 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 26 2008]

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

G.f.: x(1+x^2)/((1-x^2)(1-x-x^2)). a(n)=a(n-1)+a(n-2)+1-(-1)^n.

Convolution of F(n) and {1, 0, 2, 0, 2, ....}. a(n)=sum{k=0..n, ((1+(-1)^k)-0^k)F(n-k)}=sum{k=0..n, F(k)((1+(-1)^(n-k))-0^(n-k))}; a(n)=2*A074331(n)-A000045(n). - Paul Barry (pbarry(AT)wit.ie), Jul 19 2004

a(n)=-(1 - ((1 + Sqrt[5])/2)^n - (-(1 + Sqrt[5])/2)^(-n) + (-1)^n). [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 26 2008]

A001350:=(1+z**2)/(z-1)/(z+1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]

(*A001350*); Clear[f, n]; f[n_] = -(1 - ((1 + Sqrt[5])/2)^n - (-(1 + Sqrt[5])/2)^(-n) + (-1)^n); Table[FullSimplify[ExpandAll[f[n]]], {n, 0, 30}] [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 26 2008]

(PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n

Cf. A031367, A098554.

Sequence in context: A076065 A066898 A118143 this_sequence A077238 A000286 A036539

Adjacent sequences: A001347 A001348 A001349 this_sequence A001351 A001352 A001353

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy

Additional comments from Michael Somos, Aug 01, 2002.