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 A294860 Solution of the equation a(n) = a(n-2) + b(n-2), where a( ) and b( ) are increasing sequences of positive integers such that every positive integer is in one of them and only one term is in both. 16
 1, 2, 4, 6, 9, 13, 17, 23, 28, 35, 42, 50, 58, 68, 77, 88, 98, 110, 122, 135, 148, 162, 177, 192, 208, 224, 241, 258, 277, 295, 315, 334, 355, 375, 398, 419, 443, 465, 490, 513, 539, 564, 591, 617, 645, 672, 701, 729, 760, 789, 821, 851, 884, 915, 949, 981 (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. The initial values sequences in the following guide are a(0) = 1, a(1) = 2, b(0) = 3. A294860: a(n) = a(n-2) + b(n-2); not quite complementary A022939: a(n) = a(n-2) + b(n-2); offset 1, complementary A294861: a(n) = a(n-2) + b(n-2) + 1 A294862: a(n) = a(n-2) + b(n-2) + 2 A294863: a(n) = a(n-2) + b(n-2) + 3 A294864: a(n) = a(n-2) + b(n-2) + n A294865: a(n) = a(n-2) + 2*b(n-2) A294866: a(n) = 2*a(n-1) - a(n-2) + b(n-1) A294867: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 1 A294868: a(n) = 2*a(n-1) - a(n-2) + b(n-1) - 2 A294869: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 1 A294870: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 2 A294871: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3 A294872: a(n) = 2*a(n-1) - a(n-2) + b(n-1) + n A022942: a(n) = a(n-2) + b(n-1); offset 1 A295998: a(n) = 2*a(n-2) + b(n-2) LINKS Table of n, a(n) for n=0..55. Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13. EXAMPLE a(0) = 1, a(1) = 2, b(0) = 3, so that a(2) = 4 (b(n)) = (3,4,5,7,8,10,11,12,14,15,...) 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 - 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, 18}] (* A294860 *) Table[b[n], {n, 0, 10}] CROSSREFS Cf. A294861, A294864, A294865. Sequence in context: A232739 A342712 A006697 * A183920 A079717 A247179 Adjacent sequences: A294857 A294858 A294859 * A294861 A294862 A294863 KEYWORD nonn,easy AUTHOR Clark Kimberling, Nov 16 2017 EXTENSIONS Edited by Clark Kimberling, Dec 02 2017 STATUS approved

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Last modified April 23 13:04 EDT 2024. Contains 371913 sequences. (Running on oeis4.)