A296502 -id:A296502 - cp's OEIS frontend

A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.

1, 3, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 a(n) + b(n);
where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A296502, A297149.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)

A297149 Solution (c(n)) of the system of 3 complementary equations in Comments.

Original entry on oeis.org

4, 11, 26, 46, 69, 95, 124, 158, 196, 239, 286, 336, 389, 445, 504, 566, 631, 699, 770, 844, 923, 1006, 1092, 1181, 1273, 1370, 1471, 1575, 1682, 1792, 1905, 2021, 2140, 2262, 2387, 2515, 2646, 2780, 2919, 3062, 3208, 3357, 3509, 3664, 3824, 3988, 4155, 4325

Offset: 0

Clark Kimberling, Apr 24 2018

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2, c(0) = 4:
a(n) = least new;
b(n) = a(n-1)+c(n-1);
c(n) = 2 a(n) + b(n);
where "least new k" means the least positive integer not yet placed. The sequences a,b,c partition the positive integers.

			n:   0   1    2    3    4    5    6    7   8     9
a:   1   3    6    7    8    9   10   12  13    15
b:   2   5   14   32   53   77  104  134  170  209
c:   4  11   26   46   69   95  124  158  196  239

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A299634, A296484, A296502.

z = 300;
mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a = {1}; b = {2}; c = {4}; n = 1;
Do[{n++, AppendTo[a, mex[Flatten[{a, b, c}], 1]],
   AppendTo[b, a[[n - 1]] + c[[n - 1]]],
   AppendTo[c, 2 Last[a] + Last[b]]}, {z}];
Take[a, 100]  (* A296484 *)
Take[b, 100]  (* A296502 *)
Take[c, 100]  (* A297149 *)

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A296484 Solution (a(n)) of the system of 3 complementary equations in Comments.

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