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 A230602 a(n) = Lucas(2^n - 2). 3
 2, 3, 18, 843, 1860498, 9062201101803, 215002084978043708894524818, 121020968315000050139390193037122554865361969834971243, 38343921554607207587938114587587818441864732465057252794474861753545122655196096751375348482086938743684498 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Let phi := 1/2*(1 + sqrt(5)) denote the golden ratio. This sequence, apart from the initial term, gives an Engel expansion of 1 to the base phi^2 (see A230601 for a definition of this term). The associated Engel series expansion of 1 to the base phi^2 begins 1 = phi^2/3 + phi^4/(3*18) + phi^6/(3*18*843) + phi^8/(3*18*843*1860498) + .... This result can be extended in two ways. Firstly, the sequence Lucas(2*k*(2^n - 1)) for k = 1,2,3,... is an Engel expansion of 1 to the base phi^(2*k). Secondly, for n = 1,2,3,... the sequence [a(n),a(n+1),a(n+2),...] is an Engel expansion of phi^(4 - 2^n) to the base phi^2. Some example are given below. LINKS Wikipedia, Engel Expansion FORMULA a(n) = A000032(2^n-2) = phi^(2^n-2) + (1/phi)^(2^n-2), where phi := 1/2*(1 + sqrt(5)). Recurrence equation: a(1) = 2, a(2) = 3 and a(n) = floor(phi^2*a(n-1)^2) - 5 for n >= 3. EXAMPLE Engel series expansion of phi^(4 - 2^n) to the base phi^2 for n = 1 to 5. n = 1: phi^2 = phi^2/2 + phi^4/(2*3) + phi^6/(2*3*18) + phi^8/(2*3*18*843) + ... n = 2: 1 = phi^2/3 + phi^4/(3*18) + phi^6/(3*18*843) + phi^8/(3*18*843*1860498) + ... n = 3: 1/phi^4 = phi^2/18 + phi^4/(18*843) + phi^6/(18*843*1860498) + ... n = 4: 1/phi^12 = phi^2/843 + phi^4/(843*1860498) + phi^6/(843*1860498*9062201101803) + ... n = 5: 1/phi^28 = phi^2/1860498 + phi^4/(1860498*9062201101803) + ... MATHEMATICA Table[LucasL[2^n - 2], {n, 1, 10}] CROSSREFS Cf. A000032, A001622, A192223, A230600, A230601. Sequence in context: A107095 A102939 A073983 * A269343 A292991 A139319 Adjacent sequences:  A230599 A230600 A230601 * A230603 A230604 A230605 KEYWORD nonn,easy AUTHOR Peter Bala, Oct 28 2013 STATUS approved

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Last modified May 26 23:51 EDT 2019. Contains 323597 sequences. (Running on oeis4.)