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 A218990 Power ceiling-floor sequence of 3+sqrt(8). 2
 6, 34, 199, 1159, 6756, 39376, 229501, 1337629, 7796274, 45440014, 264843811, 1543622851, 8996893296, 52437736924, 305629528249, 1781339432569, 10382407067166, 60513102970426, 352696210755391, 2055664161561919, 11981288758616124, 69832068390134824 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS See A214992 for a discussion of power ceiling-floor sequence and power ceiling-floor function, p3(x) = limit of a(n,x)/x^n. The present sequence is a(n,r), where r = 3+sqrt(8), and the limit p3(r) = 5.854315472394508538153482993682502287049948... REFERENCES R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.~ LINKS Clark Kimberling, Table of n, a(n) for n = 0..250 Index entries for linear recurrences with constant coefficients, signature (5,5,-1). FORMULA a(n) = floor(x*a(n-1)) if n is odd, a(n) = ceiling(x*a(n-1) if n is even, where x=3+sqrt(8) and a(0) = ceiling(x). a(n) = 5*a(n-1) + 5*a(n-2) - a(n-3). G.f.: (6 + 4*x - x^2)/(1 - 5*x - 5*x^2 + x^3). a(n) = (1/16)*(2*(-1)^n + (47-33*sqrt(2))*(3-2*sqrt(2))^n + (3+2*sqrt(2))^n*(47+33*sqrt(2))). - Colin Barker, Nov 13 2017 EXAMPLE a(0) = ceiling(r) = 6, where r = 3+sqrt(8); a(1) = floor(6*r) = 34; a(2) = ceiling(35*r) = 199. MATHEMATICA x = 3 + Sqrt[8]; z = 30; (* z = # terms in sequences *) f[x_] := Floor[x]; c[x_] := Ceiling[x]; p1[0] = f[x]; p2[0] = f[x]; p3[0] = c[x]; p4[0] = c[x]; p1[n_] := f[x*p1[n - 1]] p2[n_] := If[Mod[n, 2] == 1, c[x*p2[n - 1]], f[x*p2[n - 1]]] p3[n_] := If[Mod[n, 2] == 1, f[x*p3[n - 1]], c[x*p3[n - 1]]] p4[n_] := c[x*p4[n - 1]] t1 = Table[p1[n], {n, 0, z}] (* A001653 *) t2 = Table[p2[n], {n, 0, z}] (* A084158 *) t3 = Table[p3[n], {n, 0, z}] (* A218990 *) t4 = Table[p4[n], {n, 0, z}] (* A001109 *) LinearRecurrence[{5, 5, -1}, {6, 34, 199}, 30] (* Harvey P. Dale, Mar 21 2024 *) PROG (PARI) Vec((6 + 4*x - x^2) / ((1 + x)*(1 - 6*x + x^2)) + O(x^50)) \\ Colin Barker, Nov 13 2017 CROSSREFS Cf. A214992, A001653, A084158, A001109. Sequence in context: A273583 A126501 A370224 * A087413 A337906 A244829 Adjacent sequences: A218987 A218988 A218989 * A218991 A218992 A218993 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 11 2012 STATUS approved

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Last modified August 12 06:37 EDT 2024. Contains 375085 sequences. (Running on oeis4.)