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 A216162 Sequences A006452 and A216134 interlaced, where A216134 are the indices of the Sophie Germain triangular numbers. 10
 1, 0, 1, 1, 2, 4, 4, 9, 11, 26, 23, 55, 64, 154, 134, 323, 373, 900, 781, 1885, 2174, 5248, 4552, 10989, 12671, 30590, 26531, 64051, 73852, 178294, 154634, 373319, 430441, 1039176, 901273, 2175865, 2508794, 6056764, 5253004, 12681873, 14622323, 35301410 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS (a(2n) + a(2n - 1)) - (a(2n - 2) + a(2n - 3)) = A000129(n); n>1. It follows that sqrt(2) = lim n --> infinity ((a(2n + 2) + a(2n + 1)) - (a(2n - 2) + a(2n - 3)))/((a(2n + 2) + a(2n + 1)) - (a(2n) + a(2n - 1))). For example, for n = 5, then ((64 + 55) - (11 + 9))/((64 + 55) - (23 + 26)) = (119 - 20)/(119 - 49) = 99/70 = 1.41428571... (accurate to 5 digits). LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (0,1,0,6,0,-6,0,-1,0,1) FORMULA A006452 alternating with A216134. G.f. ( -1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9 ) / ( (x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1) ). - R. J. Mathar, Sep 08 2012 PROG (PARI) Vec((-1-x^3+5*x^4-3*x^5-2*x^6+x^7-2*x^8+x^9)/((x-1)*(1+x)*(x^4-2*x^2-1)*(x^4+2*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015 CROSSREFS Cf. A000129. For some k in n: a(2n) = A006452 (k^2 - 1 is triangular). a(2n + 1) = A216134 (T_k and 2T_k + 1 are triangular). a(2n + 1) - a(2n) = A006451 (T_k + 1 is square). a(2n + 1) + a(2n) = A124124 (T_k and (T_k - 1)/2 are triangular). a(4n + 1) + a(4n + 2) = A001108 (T_k is square). a(4n + 3) + a(4n + 4) = A001652 (T_k and 2T_k are triangular). Sum(a(n)) - 1 = A048776 for even n (the second partial summation of the Pell numbers). Sequence in context: A272196 A335057 A039887 * A114215 A292302 A151712 Adjacent sequences:  A216159 A216160 A216161 * A216163 A216164 A216165 KEYWORD nonn,easy AUTHOR Raphie Frank, Sep 07 2012 EXTENSIONS Edited by N. J. A. Sloane, May 24 2021 STATUS approved

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Last modified June 15 04:47 EDT 2021. Contains 345043 sequences. (Running on oeis4.)