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 A294532 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3. 35
 1, 2, 6, 12, 23, 42, 73, 124, 207, 342, 562, 918, 1495, 2429, 3941, 6388, 10348, 16756, 27125, 43903, 71052, 114980, 186058, 301065, 487151, 788245, 1275426, 2063702, 3339160, 5402895, 8742089, 14145019, 22887144, 37032200, 59919382, 96951621, 156871043 (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, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3: a(n) = a(n-1) + a(n-2) + b(n-2) A294532 a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533 a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534 a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535 a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536 a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537 a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538 a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539 a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540 a(n) = a(n-1) + a(n-2) + b(n-1) A294541 a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542 a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543 a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544 a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545 a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546 a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547 a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548 a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557 a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558 a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294559 a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294560 a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294561 a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562 a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563 a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1 A294564 a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 3 A294565 Conjecture: for every sequence listed here, 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(0) + a(1) + b(0) = 6 Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; b[0] = 2; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; Table[a[n], {n, 0, 40}] (* A294532 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A001622, A293076, A294413. Sequence in context: A062476 A192703 A192969 * A323950 A291014 A257479 Adjacent sequences: A294529 A294530 A294531 * A294533 A294534 A294535 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 03 2017 STATUS approved

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