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 A294533 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2) + 1, where a(0) = 1, a(1) = 2, b(0) = 3. 2
 1, 2, 7, 14, 27, 48, 84, 142, 237, 391, 641, 1046, 1703, 2766, 4487, 7272, 11779, 19072, 30873, 49968, 80865, 130858, 211749, 342634, 554412, 897076, 1451519, 2348627, 3800179, 6148840, 9949054, 16097930, 26047021, 42144989, 68192049, 110337078, 178529168 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  See A294532 for a guide to related sequences.  Conjecture:  a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).. LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, so that b(1) = 4 (least "new number") a(2)  = a(1) + a(0) + b(0) + 1 = 7 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a = 1; a = 3; b = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2] + 1; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}]  (* A294533 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A294532. Sequence in context: A227016 A268347 A210728 * A294541 A294564 A068040 Adjacent sequences:  A294530 A294531 A294532 * A294534 A294535 A294536 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 03 2017 STATUS approved

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Last modified January 28 15:20 EST 2022. Contains 350657 sequences. (Running on oeis4.)