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 A294116 Fibonacci sequence beginning 2, 21. 2
 2, 21, 23, 44, 67, 111, 178, 289, 467, 756, 1223, 1979, 3202, 5181, 8383, 13564, 21947, 35511, 57458, 92969, 150427, 243396, 393823, 637219, 1031042, 1668261, 2699303, 4367564, 7066867, 11434431, 18501298, 29935729, 48437027, 78372756, 126809783, 205182539, 331992322, 537174861 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES Steven Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover Publications (2008), page 24 (formula 8). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Tanya Khovanova, Recursive Sequences. Index entries for linear recurrences with constant coefficients, signature (1, 1). FORMULA G.f.: (2 + 19*x)/(1 - x - x^2). a(n) = a(n-1) + a(n-2). Let g(r,s;n) be the n-th generalized Fibonacci number with initial values r, s. We have: a(n) =     Lucas(n) + g(0,20;n), see A022354; a(n) = Fibonacci(n) + g(2,20;n), see A022372; a(n) =  2*g(1,21;n) - g(0,21;n); a(n) =     g(1,k;n) + g(1,21-k;n) for all k in Z. a(h+k) = a(h)*Fibonacci(k-1) + a(h+1)*Fibonacci(k) for all h, k in Z (see S. Vajda in References section). For h=0 and k=n: a(n) = 2*Fibonacci(n-1) + 21*Fibonacci(n). Sum_{j=0..n} a(j) = a(n+2) - 21. a(n) = (2^(-n)*((1-sqrt(5))^n*(-20+sqrt(5)) + (1+sqrt(5))^n*(20+sqrt(5)))) / sqrt(5). - Colin Barker, Oct 25 2017 MATHEMATICA LinearRecurrence[{1, 1}, {2, 21}, 40] PROG (MAGMA) a0:=2; a1:=21; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; (PARI) Vec((2 + 19*x)/(1 - x - x^2) + O(x^40)) \\ Colin Barker, Oct 25 2017 (Sage) a = BinaryRecurrenceSequence(1, 1, 2, 21) print [a(n) for n in range(38)] # Peter Luschny, Oct 25 2017 CROSSREFS Cf. A000032, A000045. Subsequence of A047201, A047592, A113763. Sequences of the type g(2,k;n): A118658 (k=0), A000032 (k=1), 2*A000045 (k=2,4), A020695 (k=3), A001060 (k=5), A022112 (k=6), A022113 (k=7), A294157 (k=8), A022114 (k=9), A022367 (k=10), A022115 (k=11), A022368 (k=12), A022116 (k=13), A022369 (k=14), A022117 (k=15), A022370 (k=16), A022118 (k=17), A022371 (k=18), A022119 (k=19), A022372 (k=20), this sequence (k=21), A022373 (k=22); A022374 (k=24); A022375 (k=26); A022376 (k=28), A190994 (k=29), A022377 (k=30); A022378 (k=32). Sequence in context: A037315 A110301 A233135 * A041513 A135053 A042565 Adjacent sequences:  A294113 A294114 A294115 * A294117 A294118 A294119 KEYWORD nonn,easy AUTHOR Bruno Berselli, Oct 23 2017 STATUS approved

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)