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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
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
Sequence in context: A232739 A342712 A006697 * A183920 A079717 A247179
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.)