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A295148 Solution of the complementary equation a(n) = a(n-1) + 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences. 5
1, 3, 9, 20, 44, 91, 187, 379, 764, 1534, 3075, 6157, 12322, 24652, 49313, 98635, 197280, 394571, 789153, 1578318, 3156648, 6313309, 12626631, 25253276, 50506566, 101013147, 202026309 (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. See A295053 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
FORMULA
a(n+1)/a(n) -> 2.
EXAMPLE
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = a(1) + 2*a(0) + b(1) = 9
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
MATHEMATICA
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[ n - 1] + 2 a[n - 2] + b[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}] (* A295148 *)
Table[b[n], {n, 0, 10}]
CROSSREFS
Sequence in context: A011796 A164680 A210634 * A364535 A176163 A203861
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 19 2017
STATUS
approved

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Last modified May 23 15:34 EDT 2024. Contains 372763 sequences. (Running on oeis4.)