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 A355325 Upper midsequence of the Fibonacci numbers (1,2,3,5,8,...) and Lucas numbers (1,3,4,7,11,...); see Comments. 2
 1, 3, 4, 6, 10, 16, 25, 41, 66, 106, 172, 278, 449, 727, 1176, 1902, 3078, 4980, 8057, 13037, 21094, 34130, 55224, 89354, 144577, 233931, 378508, 612438, 990946, 1603384, 2594329, 4197713, 6792042, 10989754, 17781796, 28771550, 46553345, 75324895, 121878240 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Suppose that s = (s(n)) and t = (t(n)) are integer sequences. The lower midsequence, m = m(s,t), of s and t is defined by m(n) = floor((s(n) + t(n))/2). The upper midsequence, M = M(s,t), is defined by M(n) = ceiling((s(n) + t(n))/2). Here, s(n) = F(n+2) and t(n) = L(n+1), for n >= 0, where F = A000045 (Fibonacci numbers) and L = A000032 (Lucas numbers). LINKS Table of n, a(n) for n=0..38. Index entries for linear recurrences with constant coefficients, signature (1,1,1,-1,-1). FORMULA a(n) = ceiling((A000045(n+2) + A000032(n+1))/2). a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4) - a(n-5) for n >= 5. G.f.: (1 + 2 x - 2 x^3 - 2 x^4)/(1 - x - x^2 - x^3 + x^4 + x^5). G.f.: ((1 + 2 x - 2 x^3 - 2 x^4)/((-1 + x) (-1 + x + x^2) (1 + x + x^2))). a(n) = (10 + 3*((5 - 4*sqrt(5))*(1 - sqrt(5))^n + (1 + sqrt(5))^n*(5 + 4*sqrt(5)))/2^n - 10*cos(2*n*Pi/3))/30. - Stefano Spezia, Jul 17 2022 EXAMPLE a(0) = 1 = ceiling((1+1)/2); a(1) = 3 = ceiling((2+3)/2); a(2) = 4 = ceiling((3+4)/2). The Fibonacci and Lucas numbers are interspersed: 1 < 2 < 3 < 4 < 5 < 7 < 8 < 11 < 13 < 18 < 21 < 29 < ... The midsequences m and M intersperse the ordered union of the Fibonacci and Lucas sequences, A116470, as indicated by the following table: F m M L 1 1 1 1 2 2 3 3 3 3 4 4 5 6 6 7 8 9 10 11 13 15 16 18 21 25 25 29 MATHEMATICA Table[Floor[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}] (* A355324 *) Table[Ceiling[(LucasL[n + 1] + Fibonacci[n + 2])/2], {n, 0, 50}] (* A355325 *) CROSSREFS Cf. A000032, A000045, A116470, A355324. Sequence in context: A310004 A214289 A310005 * A068922 A032408 A347567 Adjacent sequences: A355322 A355323 A355324 * A355326 A355327 A355328 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jul 16 2022 STATUS approved

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Last modified September 25 13:00 EDT 2023. Contains 365647 sequences. (Running on oeis4.)