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 A215936 a(n) = -2*a(n-1) + a(n-2) for n > 2, with a(0) = a(1) = 1, a(2) = 0. 2
 1, 1, 0, 1, -2, 5, -12, 29, -70, 169, -408, 985, -2378, 5741, -13860, 33461, -80782, 195025, -470832, 1136689, -2744210, 6625109, -15994428, 38613965, -93222358, 225058681, -543339720, 1311738121, -3166815962, 7645370045, -18457556052, 44560482149 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS BINOMIAL transform is A052955. Essentially the same as A000129, A069306, A048624, A215928, A077985, and A176981. - R. J. Mathar, Sep 08 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 M. C. Firengiz, A. Dil, Generalized Euler-Seidel method for second order recurrence relations, Notes on Number Theory and Discrete Mathematics, Vol. 20, 2014, No. 4, 21-32. Index entries for linear recurrences with constant coefficients, signature (-2, 1). FORMULA G.f.: 1 / (1 - x / (1 + x / (1 + x / (1 + x)))) = (1 + 3*x + x^2) / (1 + 2*x - x^2). a(n + 3) = A077985(n). a(n) * a(n+2) - a(n+1)^2 = -(-1)^n. a(2*n + 1) = A001653(n). a(2*n + 2) = -A001542(n). a(n) = Sum_{k=0..n} A147746(n,k)*(-1)^(n-k). - Philippe Deléham, Aug 30 2012 a(n) = ((-1-sqrt(2))^(n-1) + (-1+sqrt(2))^(n-1))/2 + sqrt(2)*((-1+sqrt(2))^(n-1) - (-1 -sqrt(2))^(n-1))/4, for n > 0. - Paolo P. Lava, Oct 26 2012 G.f.: 1 + x + x^2/(1-x)  - G(0)*x^2 /(2-2*x), where G(k)= 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 10 2013 a(n) = (-1)^n a(1-n) = A000129(-1-n) if n < 0. a(n-2) = 2*a(n-1) + a(n) if n<1 or n>2. - Michael Somos, Mar 19 2019 EXAMPLE G.f. = 1 + x + x^3 - 2*x^4 + 5*x^5 - 12*x^6 + 29*x^7 - 70*x^8 + 169*x^9 - 408*x^10 + ... MATHEMATICA CoefficientList[Series[(1 + 3 x + x^2)/(1 + 2 x - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 09 2013 *) a[ n_] := With[ {m = If[ n < 1, 1 - n, n], s = If[ n < 1, (-1)^n, 1]}, s SeriesCoefficient[ x (1 + 2 x) / (1 + 2 x - x^2), {x, 0, m}]]; (* Michael Somos, Mar 19 2019 *) PROG (PARI) {a(n) = my(m=n, s=1); if(n<1, m=1-n; s=(-1)^n); s * polcoeff( x * (1 + 2*x) / (1 + 2*x - x^2) + x * O(x^m), m))}; /* Michael Somos, Mar 19 2019 */ (MAGMA) [1, 1] cat [n le 2 select (n-1) else -2*Self(n-1)+Self(n-2): n in [1..35] ]; // Vincenzo Librandi, Sep 09 2013 CROSSREFS Cf. A001542, A001653, A052955, A077985. Sequence in context: A324979 A048624 A176981 * A000129 A077985 A215928 Adjacent sequences:  A215933 A215934 A215935 * A215937 A215938 A215939 KEYWORD sign,easy AUTHOR Michael Somos, Aug 28 2012 STATUS approved

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Last modified February 26 12:33 EST 2020. Contains 332279 sequences. (Running on oeis4.)