

A295146


Solution of the complementary equation a(n) = a(n1) + 2*a(n2) + b(n2), 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, 7, 17, 36, 76, 156, 317, 639, 1284, 2574, 5155, 10317, 20642, 41292, 82594, 165197, 330405, 660820, 1321652, 2643315, 5286643, 10573298, 21146610, 42293233, 84586481, 169172976
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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



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(0) = 7
Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, ...)


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[ n  1] + 2 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}] (* A295146 *)
Table[b[n], {n, 0, 10}]


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



