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A001350
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Associated Mersenne numbers.
(Formerly M3250 N1311)
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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; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - Michael Somos, Feb 12 2012
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REFERENCES
| Baake, Michael; Hermisson, Joachim; Pleasants, Peter A. B.; The torus parametrization of quasiperiodic LI-classes. J. Phys. A 30 (1997), no. 9, 3029-3056.
N. Garnier and O. Ramare, Fibonacci numbers and trigonometric identities, April 2006 [From Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Nov 26 2008]
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. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
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
| G.f.: x*(1+x^2)/((1-x^2)*(1-x-x^2)).
a(n) = a(n-1) + a(n-2) + 1 -(-1)^n. a(-n) = (-1)^n * a(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, 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, Nov 26 2008]
a(n) = n*sum(sum(binomial(i,n-k-i)*binomial(k+i-1,k-1),i,ceiling((n-k)/2),n-k)/k*(-1)^(k+1),k,1,n), n>0. [From Kruchinin Vladimir, Sep 03 2010]
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EXAMPLE
| x + x^2 + 4*x^3 + 5*x^4 + 11*x^5 + 16*x^6 + 29*x^7 + 45*x^8 + 76*x^9 + ...
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MAPLE
| A001350:=(1+z**2)/(z-1)/(z+1)/(z**2+z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| (*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, Nov 26 2008]
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PROG
| (PARI) {a(n) = fibonacci(n+1) + fibonacci(n-1) - 1 - (-1)^n}
(PARI) {a(n) = local(w = quadgen(5)); simplify( -(w^n - 1) * ((-1/w)^n - 1))} /* Michael Somos, Feb 12 2012 */
(MAGMA) [Floor(-(1 - ((1 + Sqrt(5))/2)^n - (-(1 + Sqrt(5))/2)^(-n) + (-1)^n)): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| Cf. A031367, A098554.
Sequence in context: A176115 A066898 A118143 * A077238 A185507 A000286
Adjacent sequences: A001347 A001348 A001349 * A001351 A001352 A001353
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. K. Guy
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EXTENSIONS
| Additional comments from Michael Somos, Aug 01 2002.
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