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 A295053 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(0) + b(1) + ... + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences. 34
 1, 2, 10, 24, 52, 101, 186, 329, 568, 962, 1608, 2662, 4377, 7162, 11679, 18999, 30855, 50051, 81124, 131415, 212802, 344505, 557621, 902467, 1460457, 2363322, 3824207, 6187988, 10012686, 16201198, 26214442, 42416233, 68631304, 111048203 (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. Guide to related sequencres: A295053: a(n) = a(n-1) + a(n-2) + b(0) + b(1) + ... + b(n-1),  a(0) = 1, a(1) = 2, b(0) = 3 A295054: a(n) = a(n-1) + a(n-2) + b(1) + b(2) + ... + b(n-1),  a(0) = 1, a(1) = 2, b(0) = 3 A295055: a(n) = a(n-2) + b(1) + b(2) + ... + b(n-1),  a(0) = 1, a(1) = 2, b(0) = 3 A295056: a(n) = 2*a(n-1) + b(n-1),  a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3 A295057: a(n) = 2*a(n-1) + b(n-1),  a(0) = 2, a(1) = 5, b(0) = 1 A295058: a(n) = 2*a(n-1) - b(n-1),  a(0) = 3, a(1) = 5, b(0) = 1 A295059: a(n) = 2*a(n-1) + b(n-2),  a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3 A295060: a(n) = 2*a(n-1) - b(n-2),  a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2 A295061: a(n) = 4*a(n-1) + b(n-1),  a(0) = 1, a(1) = 3, b(0) = 2 A295062: a(n) = 4*a(n-2) + b(n-2),  a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295063: a(n) = 4*a(n-2) + b(n-1) + b(n-2), a(0) = 1, a(1) = 3, b(0) = 2 A295064: a(n) = 8*a(n-3) + b(n-1), a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2 A295065: a(n) = 8*a(n-3) + b(n-2), a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2 A295066: a(n) = 2*a(n-2) + b(n-1), a(0) = 1, a(1) = 3, b(0) = 2 A295067: a(n) = 2*a(n-2) + b(n-2), a(0) = 1, a(1) = 3, b(0) = 2 A295068: a(n) = 2*a(n-2) - b(n-1) + n, a(0) = 3, a(1) = 4, b(0) = 1 A295069: a(n) = 2*a(n-2) - b(n-2) + n, a(0) = 3, a(1) = 4, b(0) = 1 A295070: a(n) = a(n-2) + b(n-1) + b(n-2), a(0) = 3, a(1) = 2, b(0) = 3 A295133: a(n) = 3*a(n-1) + b(n-1), a(0) = 1, a(1) = 2, b(0) = 3 A295134: a(n) = 3*a(n-1) + b(n-1) - 1, a(0) = 1, a(1) = 2, b(0) = 3 A295135: a(n) = 3*a(n-1) + b(n-1) - 2, a(0) = 1, a(1) = 2, b(0) = 3 A295136: a(n) = 3*a(n-1) + b(n-1) - 3, a(0) = 1, a(1) = 2, b(0) = 3 A295137: a(n) = 3*a(n-1) + b(n-1) - n, a(0) = 1, a(1) = 2, b(0) = 3 A295138: a(n) = 3*a(n-2) + b(n-1), a(0) = 1, a(1) = 2, b(0) = 3 A295139: a(n) = 3*a(n-1) + b(n-2), a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 A295140: a(n) = 3*a(n-1) - b(n-2) + 4, a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295141: a(n) = 2*a(n-1) + a(n-2) + b(n-2), a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 A295142: a(n) = 2*a(n-1) + a(n-2) + b(n-2), a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295143: a(n) = 2*a(n-1) + a(n-1) + b(n-1), a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 A295144: a(n) = 2*a(n-1) + a(n-2) + b(n-1), a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295145: a(n) = a(n-1) + 2*a(n-2) + b(n-2), a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 A295146: a(n) = a(n-1) + 2*a(n-2) + b(n-2), a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295147: a(n) = a(n-1) + 2*a(n-2) + b(n-1), a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 A295148: a(n) = a(n-1) + 2*a(n-2) + b(n-1), a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 LINKS Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3 b(1) = 4 (least "new number") a(2) = a(1) + a(0) + b(0) + b(1) = 10 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = a[n - 1] + a[n - 2] + Sum[b[k], {k, 0, 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, 18}]  (* A295053 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A294860. Sequence in context: A049450 A092906 A244383 * A130016 A120550 A233266 Adjacent sequences:  A295050 A295051 A295052 * A295054 A295055 A295056 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 18 2017 STATUS approved

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Last modified January 24 21:06 EST 2021. Contains 340411 sequences. (Running on oeis4.)