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 A193530 Expansion of (1 - 2*x - 2*x^2 + 3*x^3 + x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)). 1
 1, 1, 2, 3, 7, 13, 31, 66, 159, 363, 876, 2065, 4985, 11915, 28765, 69156, 166957, 402373, 971414, 2343519, 5657755, 13654969, 32966011, 79577190, 192116331, 463786191, 1119678912, 2703086893, 6525829037, 15754607063, 38034986041, 91824246216, 221683340569, 535190123593, 1292063254826 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This sequence was initially confused with A003120, but they are different sequences. The g.f. used here as the definition was found by Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63). Index entries for linear recurrences with constant coefficients, signature (3,1,-7,3,-1,1,1). FORMULA a(n) = 1 + A005409(floor((n+3)/2)) + A107769(n). From G. C. Greubel, May 21 2021: (Start) a(n) = (1 + A001333(n) + A135153(n+2))/4. a(n) = (2 + Q(n) + 2*(1+(-1)^n)*Pell((n+2)/2) + 2*(1-(-1)^n)*Pell((n+1)/2))/8. a(2*n) = (2 + Q(2*n) + 4*Pell(n+1))/8. a(2*n+1) = (2 + Q(2*n+1) + 4*Pell(n+1))/8, where Pell(n) = A000129(n), and Q(n) = A002203. (End) MAPLE f:=n->if n mod 2 = 0 then (1/4)*(A001333(n-2)+A001333((n-2)/2)+A001333((n-4)/2)+1) else (1/4)*(A001333(n-2)+A001333((n-1)/2)+A001333((n-3)/2)+1); fi; # produces the sequence with a different offset MATHEMATICA LinearRecurrence[{3, 1, -7, 3, -1, 1, 1}, {1, 1, 2, 3, 7, 13, 31}, 40] (* Vincenzo Librandi, Aug 28 2016 *) Table[(2 +LucasL[n, 2] +2*(1+(-1)^n)*Fibonacci[(n+2)/2, 2] + 2*(1-(-1)^n)*Fibonacci[(n+1)/2, 2])/8, {n, 0, 40}] (* G. C. Greubel, May 21 2021 *) PROG (MAGMA) m:=40; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-2*x-2*x^2 +3*x^3+x^5)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)) )); // Vincenzo Librandi, Aug 28 2016 (Sage) @CachedFunction def Pell(n): return n if (n<2) else 2*Pell(n-1) + Pell(n-2) def A193530(n): return (1 + Pell(n+1) - Pell(n) + (1 + (-1)^n)*Pell((n+2)/2) + (1-(-1)^n)*Pell((n+1)/2) )/4 [A193530(n) for n in (0..40)] # G. C. Greubel, May 21 2021 CROSSREFS Cf. A000129, A001333, A002203, A135153. Sequence in context: A124430 A002013 A171416 * A003120 A032131 A324844 Adjacent sequences:  A193527 A193528 A193529 * A193531 A193532 A193533 KEYWORD nonn,easy AUTHOR F. Chapoton and N. J. A. Sloane, Jul 29 2011 STATUS approved

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Last modified June 16 08:46 EDT 2021. Contains 345056 sequences. (Running on oeis4.)