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 A295357 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3), where a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, and (a(n)) and (b(n)) are increasing complementary sequences. 11
 1, 3, 5, 20, 42, 83, 149, 259, 438, 730, 1204, 1973, 3219, 5237, 8504, 13792, 22350, 36200, 58612, 94878, 153559, 248509, 402143, 650730, 1052954, 1703768, 2756809, 4460667, 7217569, 11678332, 18896000, 30574434, 49470539 (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 sequences: ***** Part 1:  initial values are a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6 A295357: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3) A295358: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3) A295359: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 2*b(n-3) A295360: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - 3*b(n-3) A295361: a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) - 3*b(n-3) A295362: a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) - b(n-3) ***** Part 2: initial values as shown A295363: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4 A295364: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4 A295365: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6 A295366: a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - b(n-3); a(0) = 1, a(1) = 2, a(2) = 3, b(0) = 4, b(1) = 5, b(2) = 6 A295367: a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2); a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4 For all of these sequences, 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) = 3, a(2) = 5, b(0) = 2, b(1) = 4, b(2) = 6, so that b(3) = 7 (least "new number") a(3) = a(1) + a(0) + b(2) + b(1) + b(0) = 20 Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...) MATHEMATICA mex := First[Complement[Range[1, Max[#1] + 1], #1]] &; a[0] = 1; a[1] = 3; a[2] = 5; b[0] = 2; b[1] = 4; b[2] = 6; a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] + b[n - 3]; b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]]; z = 32; u = Table[a[n], {n, 0, z}]   (* A295357 *) v = Table[b[n], {n, 0, 10}]  (* complement *) CROSSREFS Cf. A001622, A293076, A294532. Sequence in context: A231627 A261116 A295361 * A076149 A133102 A197156 Adjacent sequences:  A295354 A295355 A295356 * A295358 A295359 A295360 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 21 2017 STATUS approved

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)