A295138
Solution of the complementary equation a(n) = 3*a(n-2) + b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 2, 7, 11, 27, 41, 90, 133, 282, 412, 860, 1251, 2596, 3770, 7806, 11329, 23438, 34008, 70336, 102047, 211032, 306166, 633122, 918526, 1899395, 2755608, 5698216, 8266856, 17094681, 24800602, 51284078, 74401842
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
a(2) =3*a(0) + b(1) = 7
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... )
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mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 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}] (* A295138 *)
Table[b[n], {n, 0, 10}]
A295139
Solution of the complementary equation a(n) = 3*a(n-2) + b(n-2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 2, 6, 10, 23, 37, 77, 120, 242, 372, 739, 1130, 2232, 3406, 6713, 10236, 20158, 30728, 60495, 92206, 181509, 276643, 544553, 829956, 1633687, 2489897, 4901091, 7469722, 14703305, 22409199, 44109949, 67227632, 132329883
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) =3*a(0) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, ...)
-
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] = 3 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}] (* A295139 *)
Table[b[n], {n, 0, 10}]
A295140
Solution of the complementary equation a(n) = 3*a(n-2) - b(n-2) + 4, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
Original entry on oeis.org
1, 3, 5, 9, 13, 24, 35, 66, 98, 190, 284, 559, 840, 1664, 2506, 4977, 7502, 14914, 22488, 44723, 67443, 134147, 202306, 402417, 606893, 1207225, 1820652, 3621647, 5461927, 10864911, 16385749, 32594700, 49157213, 97784065, 147471603, 293352158
Offset: 0
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) =3*a(0) - b(0) + 4 = 5
Complement: (b(n)) = (2, 4, 6, 7, 8, 10, 11, 12, 14, 15, 16, ...)
-
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] = 3 a[n - 2] + b[n - 2] + 4;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A295140 *)
Table[b[n], {n, 0, 10}]
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