Search: id:A001350
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%I A001350 M3250 N1311
%S A001350 0,1,1,4,5,11,16,29,45,76,121,199,320,521,841,1364,2205,3571,5776,9349,
%T A001350 15125,24476,39601,64079,103680,167761,271441,439204,710645,1149851,
%U A001350 1860496,3010349,4870845,7881196,12752041,20633239,33385280,54018521
%N A001350 Associated Mersenne numbers.
%C A001350 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
%C A001350 a(n) = A050140(Fibonacci(n)). - Thomas Baruchel, Nov 05 2002
%C A001350 a(n)= Lucas_number(n)-1-(-1)^n=A000032(n)-1-(-1)^n. - Hieronymus Fischer 
               (Hieronymus.Fischer(AT)gmx.de), Feb 18 2006
%C A001350 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
%D A001350 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001350 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001350 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.
%D A001350 Baake, Michael; Hermisson, Joachim; Pleasants, Peter A. B.; The torus 
               parametrization of quasiperiodic LI-classes. J. Phys. A 30 (1997), 
               no. 9, 3029-3056.
%D A001350 C. B. Haselgrove, Associated Mersenne numbers, Eureka, 11 (1949), 19-22.
%D A001350 G. I. Lehrer and G. B. Segal, Homology stability for classical regular 
               semisimple varieties, Math. Zeit., 236 (2001), 251-290; see Th. 7.12.
%D A001350 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]
%H A001350 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A001350 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A001350 G.f.: x(1+x^2)/((1-x^2)(1-x-x^2)). a(n)=a(n-1)+a(n-2)+1-(-1)^n.
%F A001350 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
%F A001350 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]
%p A001350 A001350:=(1+z**2)/(z-1)/(z+1)/(z**2+z-1); [Conjectured by S. Plouffe 
               in his 1992 dissertation.]
%t A001350 (*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]
%o A001350 (PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n
%Y A001350 Cf. A031367, A098554.
%Y A001350 Sequence in context: A076065 A066898 A118143 this_sequence A077238 A000286 
               A036539
%Y A001350 Adjacent sequences: A001347 A001348 A001349 this_sequence A001351 A001352 
               A001353
%K A001350 nonn,easy
%O A001350 0,4
%A A001350 N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
%E A001350 Additional comments from Michael Somos, Aug 01, 2002.

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