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 A271972 Expansion of (1 + 3*x)/(1 - 4*x + 7*x^2). 0
 1, 7, 21, 35, -7, -273, -1043, -2261, -1743, 8855, 47621, 128499, 180649, -176897, -1972131, -6650245, -12796063, -4632537, 71042293, 316596931, 769091673, 860188175, -1942889011, -13792873269, -41571269999, -69734967113, 12059021541, 536380855955, 2061110273033, 4489775100447 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Satisfies of recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=1, b(0)=2. More generally, for the recurrence relations system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = b(n-1) - 2*a(n-1), a(0)=k, b(0)=m solution is a(n) = ((2 + i*sqrt(3))^n*((sqrt(3) - i)*k - 2*i*m) + (2 - i*sqrt(3))^n*((sqrt(3) + i)*k + 2*i*m))/(2*sqrt(3)), b(n) = ((2 - i*sqrt(3))^n*((sqrt(3) - i)*m - 2*i*k) + (2 + i*sqrt(3))^n*(2*i*k + (sqrt(3) + i)*m))/(2*sqrt(3)), where i is the imaginary unit. Convolution of A169585 and A168175. LINKS Table of n, a(n) for n=0..29. Index entries for linear recurrences with constant coefficients, signature (4,-7) FORMULA O.g.f.: (1 + 3*x)/(1 - 4*x + 7*x^2). E.g.f.: (5*sqrt(3)*sin(sqrt(3)*x) + 3*cos(sqrt(3)*x))*exp(2*x)/3. a(n) = 4*a(n-1) - 7*a(n-2). a(n) = ((2 + i*sqrt(3))^n*(-5*i + sqrt(3)) + (2 - i*sqrt(3))^n*(5*i + sqrt(3)))/(2*sqrt(3)), where i is the imaginary unit. a(n) = 7^(n/2)*((5/sqrt(3))*sin(c)+cos(c)) with c = n*arctan(sqrt(3)/2). - Peter Luschny, Jul 21 2016 MAPLE a:=series((1+3*x)/(1-4*x+7*x^2), x=0, 30): seq(coeff(a, x, n), n=0..29); # Paolo P. Lava, Mar 27 2019 MATHEMATICA LinearRecurrence[{4, -7}, {1, 7}, 30] PROG (PARI) Vec((1+3*x)/(1-4*x+7*x^2) + O(x^99)) \\ Altug Alkan, Jul 13 2016 CROSSREFS Cf. A168175, A169585. Sequence in context: A032639 A264619 A015729 * A001485 A230210 A087111 Adjacent sequences: A271969 A271970 A271971 * A271973 A271974 A271975 KEYWORD sign,easy AUTHOR Ilya Gutkovskiy, Jul 13 2016 STATUS approved

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Last modified November 28 19:28 EST 2023. Contains 367419 sequences. (Running on oeis4.)