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 A295615 Solution of the complementary equation a(n) = 2*a(n-1) - a(n-3) + b(n-1), where a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences. 2
 2, 4, 6, 15, 33, 68, 130, 237, 417, 716, 1208, 2013, 3326, 5461, 8927, 14547, 23653, 38400, 62275, 100920, 163464, 264678, 428462, 693487, 1122324, 1816215, 2938973, 4755653, 7695123, 12451307, 20146996, 32598905, 52746540, 85346122, 138093378, 223440256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295613 for a guide to related sequences. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth-rate of the Fibonacci numbers (A000045). LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 2, a(1) = 4, a(2) = 6, b(0) = 1, b(1) = 3, b(2) = 5, so that b(3) = 7 (least "new number") a(3) = 2*a(2) - a(0) + b(2) = 15 Complement: (b(n)) = (1, 3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 2; a = 4; a = 6; b = 1; b = 3; b = 5; a[n_] := a[n] = 2 a[n - 1] - a[n - 3] + b[n - 1]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 30}]  (* A295615 *) Table[b[n], {n, 0, 20}]  (* complement *) CROSSREFS Cf. A001622, A000045, A295613. Sequence in context: A192536 A227902 A217525 * A127679 A049022 A209867 Adjacent sequences:  A295612 A295613 A295614 * A295616 A295617 A295618 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 25 2017 STATUS approved

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Last modified May 10 22:15 EDT 2021. Contains 343780 sequences. (Running on oeis4.)