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 A048788 a(2n+1) = a(2n) + a(2n-1), a(2n) = 2*a(2n-1) + a(2n-2); a(n) = n for n = 0, 1. 12
 0, 1, 2, 3, 8, 11, 30, 41, 112, 153, 418, 571, 1560, 2131, 5822, 7953, 21728, 29681, 81090, 110771, 302632, 413403, 1129438, 1542841, 4215120, 5757961, 15731042, 21489003, 58709048, 80198051, 219105150, 299303201, 817711552, 1117014753 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Numerators of continued fraction convergents to sqrt(3) - 1 (A160390). See A002530 for denominators. - N. J. A. Sloane, Dec 17 2007 Convergents are 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 112/153, ... - Clark Kimberling, Sep 21 2013 A strong divisibility sequence, that is gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. - Peter Bala, Jun 06 2014 From Sarah-Marie Belcastro, Feb 15 2022: (Start) a(n) is also the number of perfect matchings of an edge-labeled 2 X (n-1) Mobius band grid graph, or equivalently the number of domino tilings of a 2 X (n-1) Mobius band grid. (The twist is on the length-n side.) a(n) is also the output of Lu and Wu's formula for the number of perfect matchings of an m X n Mobius band grid, specialized to m = 2 with the twist on the length-n side. 2*a(n) is the number of perfect matchings of an edge-labeled 2 X (n-1) projective planar grid graph, or equivalently the number of domino tilings of a 2 X (n-1) projective planar grid. (End) REFERENCES Russell Lyons, A bird's-eye view of uniform spanning trees and forests, in Microsurveys in Discrete Probability, AMS, 1998. Sarah-Marie Belcastro, Domino Tilings of 2 x n Grids (or Perfect Matchings of Grid Graphs) on Surfaces, preprint. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Marcia Edson, Scott Lewis and Omer Yayenie, The k-periodic Fibonacci sequence and an extended Binet's formula, INTEGERS 11 (2011) #A32. W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Physics Letters A, 293 (2002), 235-246. D. Panario, M. Sahin, and Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78. Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1). FORMULA G.f.: x*(1+2*x-x^2)/(1-4*x^2+x^4). - Paul Barry, Sep 18 2009 a(n) = 4*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013 a(2*n-1) = A001835(n); a(2*n) = 2*A001353(n). - Peter Bala, Jun 06 2014 From Gerry Martens, Jul 11 2015: (Start) Interspersion of 2 sequences [a1(n-1),a0(n)] for n>0: a0(n) = ((3+sqrt(3))*(2-sqrt(3))^n-((-3+sqrt(3))*(2+sqrt(3))^n))/6. a1(n) = 2*Sum_{i=1..n} a0(i). (End) a(n) = ((r + (-1)^n/r)*s^n/2^(n/2) - (1/r + (-1)^n*r)*2^(n/2)/s^n)*sqrt(6)/12, where r = 1 + sqrt(2), s = 1 + sqrt(3). - Vladimir Reshetnikov, May 11 2016 a(n) = 2*ChebyshevU(n-1, 2) if n is even and ChebyshevU(n, 2) - ChebyshevU(n-1, 2) if n in odd. - G. C. Greubel, Dec 23 2019 a(n) = -(-1)^n*a(-n) for all n in Z. - Michael Somos, Sep 17 2020 MAPLE seq( simplify( `if`(`mod`(n, 2)=0, 2*ChebyshevU((n-2)/2, 2), ChebyshevU((n-1)/2, 2) - ChebyshevU((n-3)/2, 2)) ), n=0..40); # G. C. Greubel, Dec 23 2019 MATHEMATICA Numerator[NestList[(2/(2 + #))&, 0, 40]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *) CoefficientList[Series[x(1+2x-x^2)/(1-4x^2+x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *) a0[n_]:= ((3+Sqrt)*(2-Sqrt)^n-((-3+Sqrt)*(2+Sqrt)^n))/6 // Simplify a1[n_]:= 2*Sum[a0[i], {i, 1, n}] Flatten[MapIndexed[{a1[#-1], a0[#]}&, Range]] (* Gerry Martens, Jul 10 2015 *) Round@Table[With[{r=1+Sqrt, s=1+Sqrt}, ((r + (-1)^n/r) s^n/2^(n/2) - (1/r + (-1)^n r) 2^(n/2)/s^n) Sqrt/12], {n, 0, 20}] (* or *) LinearRecurrence[ {0, 4, 0, -1}, {0, 1, 2, 3}, 40] (* Vladimir Reshetnikov, May 11 2016 *) Table[If[EvenQ[n], 2*ChebyshevU[(n-2)/2, 2], ChebyshevU[(n-1)/2, 2] - ChebyshevU[(n-3)/2, 2]], {n, 0, 40}] (* G. C. Greubel, Dec 23 2019 *) PROG (Magma) I:=[0, 1, 2, 3]; [n le 4 select I[n] else 4*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 10 2013 (PARI) main(size)=v=vector(size); v=0; v=1; v=2; v=3; for(i=5, size, v[i]=4*v[i-2] - v[i-4]); v; \\ Anders Hellström, Jul 11 2015 (PARI) a=vector(50); a=1; a=2; for(n=3, #a, if(n%2==1, a[n]=a[n-1]+a[n-2], a[n]=2*a[n-1]+a[n-2])); concat(0, a) \\ Colin Barker, Jan 30 2016 (PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 0, 4, 0]^n*[0; 1; 2; 3])[1, 1] \\ Charles R Greathouse IV, Mar 16 2017 (PARI) apply( {A048788(n)=imag((2+quadgen(12))^(n\/2)*if(bittest(n, 0), quadgen(12)-1, 2))}, [0..30]) \\ M. F. Hasler, Nov 04 2019 (PARI) {a(n) = my(s=1, m=n); if(n<0, s=-(-1)^n; m=-n); polcoeff(x*(1+2*x-x^2)/(1-4*x^2+x^4) + x*O(x^m), m)*s}; /* Michael Somos, Sep 17 2020 */ (Sage) @CachedFunction def a(n):     if (mod(n, 2)==0): return 2*chebyshev_U((n-2)/2, 2)     else: return chebyshev_U((n-1)/2, 2) - chebyshev_U((n-3)/2, 2) [a(n) for n in (0..40)] # G. C. Greubel, Dec 23 2019 (GAP) a:=[0, 1, 2, 3];; for n in [5..40] do a[n]:=4a[n-1]-a[n-2]; od; a; # G. C. Greubel, Dec 23 2019 CROSSREFS Cf. A002530, A002531. Bisections are A001835 and A052530. Sequence in context: A119064 A285113 A335632 * A239453 A143914 A041123 Adjacent sequences:  A048785 A048786 A048787 * A048789 A048790 A048791 KEYWORD nonn,easy,frac AUTHOR Robin Trew (trew(AT)hcs.harvard.edu), Dec 11 1999 EXTENSIONS Denominator of g.f. corrected by Paul Barry, Sep 18 2009 Incorrect g.f. deleted by Colin Barker, Aug 10 2012 STATUS approved

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Last modified July 2 13:15 EDT 2022. Contains 355007 sequences. (Running on oeis4.)