A294869
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 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, 8, 20, 39, 66, 103, 151, 211, 284, 371, 473, 591, 726, 879, 1051, 1243, 1457, 1694, 1955, 2241, 2553, 2892, 3259, 3655, 4081, 4538, 5027, 5549, 6105, 6696, 7323, 7987, 8689, 9430, 10212, 11036, 11903, 12814, 13770, 14772, 15821, 16918, 18064, 19260
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) + 1 = 8
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 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}] (* A294869 *)
Table[b[n], {n, 0, 10}]
A294870
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 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, 9, 23, 45, 76, 117, 170, 236, 316, 411, 522, 650, 796, 961, 1146, 1352, 1580, 1831, 2106, 2407, 2735, 3091, 3476, 3891, 4337, 4815, 5326, 5871, 6451, 7067, 7720, 8411, 9141, 9911, 10722, 11575, 12471, 13411, 14396, 15427, 16506, 17634, 18812, 20041
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) + 2 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, ...)
m
-
ex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 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}] (* A294870 *)
Table[b[n], {n, 0, 10}]
A294871
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 3, 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, 10, 26, 51, 86, 132, 190, 262, 349, 452, 572, 710, 867, 1044, 1242, 1462, 1705, 1972, 2264, 2582, 2927, 3300, 3703, 4137, 4603, 5102, 5635, 6203, 6807, 7448, 8127, 8845, 9603, 10402, 11243, 12127, 13055, 14028, 15047, 16113, 17227, 18390, 19603, 20867
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) + 3 = 10
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + 3;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A294871 *)
Table[b[n], {n, 0, 10}]
A294872
Solution of the complementary equation a(n) = 2*a(n-1) - a(n-2) + b(n-1) + 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, 9, 24, 49, 86, 137, 205, 292, 400, 531, 687, 870, 1082, 1325, 1601, 1912, 2260, 2647, 3075, 3546, 4063, 4628, 5243, 5910, 6631, 7408, 8243, 9138, 10095, 11116, 12203, 13358, 14583, 15880, 17251, 18698, 20223, 21828, 23515, 25286, 27143, 29088, 31123
Offset: 0
a(0) = 1, a(1) = 2, b(0) = 3
b(1) = 4 (least "new number")
a(2) = 2*a(1) - a(0) + b(1) + 2 = 9
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
-
mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 1] - a[n - 2] + b[n - 1] + n;
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 18}] (* A294872 *)
Table[b[n], {n, 0, 10}]
A295998
Solution of the complementary equation a(n) = 2*a(n-2) + b(n-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, 5, 8, 16, 23, 41, 56, 93, 124, 199, 262, 413, 541, 844, 1101, 1708, 2223, 3438, 4470, 6901, 8966, 13829, 17960, 27687, 35950, 55405, 71932, 110843, 143898, 221721, 287832, 443479, 575702, 886997, 1151444, 1774036, 2302931, 3548116, 4605907, 7096278
Offset: 0
-
mex[t_] := NestWhile[# + 1 &, 1, MemberQ[t, #] &];
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 2 a[n - 2] + b[n - 2]; (* A295998 *)
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 100}];
Table[b[n], {n, 0, 30}]
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