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 A294368 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4. 2
 1, 3, 11, 23, 45, 81, 141, 239, 399, 660, 1083, 1769, 2880, 4679, 7591, 12304, 19931, 32273, 52244, 84559, 136848, 221454, 358351, 579856, 938260, 1518171, 2456488, 3974718, 6431267, 10406048, 16837380, 27243495, 44080944, 71324510, 115405527, 186730112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The initial values of each sequence in the following guide are a(0) = 1, a(2) = 3, b(0) = 2, b(1) = 4: Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.  See A293358 for a guide to related sequences. LINKS Robert Israel, Table of n, a(n) for n = 0..4775 Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that a(2)  = a(1) + a(0) + b(1) + 3 = 11; b(2) is the first positive integer not already seen, namely 5. Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...) MAPLE A[0]:= 1: B[0]:= 2: A[1]:= 3: B[1]:= 4: Av:= {\$5..200}: for n from 2 to 100 do   A[n]:= A[n-1]+A[n-2]+B[n-1]+n+1;   B[n]:= min(Av minus {A[n]});   Av:= Av minus {A[n], B[n]}; od: seq(A[i], i=0..100); # Robert Israel, Oct 29 2017 MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + 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, 40}]  (* A294368 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622 (golden ratio), A293765. Sequence in context: A342174 A159791 A078723 * A296556 A141187 A107138 Adjacent sequences:  A294365 A294366 A294367 * A294369 A294370 A294371 KEYWORD nonn,easy AUTHOR Clark Kimberling, Oct 29 2017 EXTENSIONS Example clarified by Robert Israel, Oct 29 2017 STATUS approved

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Last modified May 22 13:12 EDT 2022. Contains 353950 sequences. (Running on oeis4.)