A294562
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 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, 5, 10, 17, 29, 48, 80, 130, 212, 344, 558, 904, 1465, 2371, 3838, 6211, 10051, 16264, 26317, 42583, 68902, 111487, 180391, 291881, 472274, 764157, 1236433, 2000592, 3237027, 5237621, 8474650, 13712273, 22186925, 35899200, 58086127
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + b(1) - b(0) + 1 = 5
Complement: (b(n)) = (3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 18, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] + 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294562 *)
Table[b[n], {n, 0, 10}]
A294563
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n, 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, 6, 12, 24, 42, 73, 123, 205, 339, 555, 906, 1474, 2394, 3883, 6293, 10193, 16504, 26716, 43240, 69978, 113240, 183241, 296505, 479771, 776302, 1256100, 2032430, 3288559, 5321019, 8609609, 13930660, 22540302, 36470996, 59011333, 95482365
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + b(1) - b(0) + 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 16, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] - b[n - 2] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294563 *)
Table[b[n], {n, 0, 10}]
A294564
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 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, 14, 27, 50, 86, 146, 243, 401, 657, 1074, 1747, 2838, 4603, 7460, 12083, 19564, 31669, 51256, 82949, 134230, 217205, 351464, 568698, 920192, 1488921, 2409145, 3898099, 6307278, 10205412, 16512726, 26718175, 43230939, 69949153, 113180132, 183129326
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + 2*b(1) - b(0) - 1 = 7
Complement: (b(n)) = (3, 4, 5, 6, 8, 10, 11, 12, 13, 15, 16, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 2 b[n - 1] - b[n - 2] - 1;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294564 *)
Table[b[n], {n, 0, 10}]
A294565
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 2, 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, 6, 12, 25, 44, 77, 130, 217, 360, 590, 964, 1569, 2549, 4135, 6702, 10856, 17578, 28455, 46055, 74533, 120614, 195173, 315814, 511015, 826858, 1337903, 2164792, 3502727, 5667552, 9170313, 14837900, 24008249, 38846186, 62854473, 101700698, 164555211
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2) = a(1) + a(0) + 2*b(1) - b(0) - 2 = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 10, 11, 13, 14, 15, 16, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 2 b[n - 1] - b[n - 2] - 2;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}] (* A294565 *)
Table[b[n], {n, 0, 10}]
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