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Search: id:A001350
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| A001350 |
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Associated Mersenne numbers.
(Formerly M3250 N1311)
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+0
10
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| 0, 1, 1, 4, 5, 11, 16, 29, 45, 76, 121, 199, 320, 521, 841, 1364, 2205, 3571, 5776, 9349, 15125, 24476, 39601, 64079, 103680, 167761, 271441, 439204, 710645, 1149851, 1860496, 3010349, 4870845, 7881196, 12752041, 20633239, 33385280, 54018521
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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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
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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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
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MAPLE
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A001350:=(1+z**2)/(z-1)/(z+1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(PARI) a(n)=fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n
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CROSSREFS
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Cf. A031367, A098554.
Sequence in context: A076065 A066898 A118143 this_sequence A077238 A000286 A036539
Adjacent sequences: A001347 A001348 A001349 this_sequence A001351 A001352 A001353
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KEYWORD
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nonn,easy
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AUTHOR
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njas, R. K. Guy
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EXTENSIONS
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Additional comments from Michael Somos, Aug 01, 2002.
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