A329387 Solution sequence (a(n)) of the complementary equation a(n) = 4 b(n+1) - b(n), with b(0) = 1.

7, 10, 13, 16, 19, 26, 28, 35, 37, 44, 46, 53, 55, 62, 64, 67, 70, 73, 76, 83, 89, 91, 94, 97, 100, 103, 110, 116, 118, 121, 124, 127, 130, 137, 143, 145, 148, 151, 154, 157, 164, 170, 172, 175, 178, 181, 184, 191, 197, 199, 206, 208, 215, 217, 224, 226, 233

Offset

0,1

Comments

The conditions that (a(n)) and (b(n)) be increasing and complementary force the equation a(n) = 4 b(n+1) - b(n), with initial value b(0) = 1, to have a unique solution; that is, a pair of complementary sequences (a(n)) = (7,10,13,16,19,26,28,...) and (b(n)) = (1,2,3,4,6,8,9,11,12,14,...). For n = 1..30000, a(n) - 4 n ranges between -12 and 15; conjecture: {a(n) - 4 n} is unbounded below and above.

Links

Table of n, a(n) for n=0..56.

Example

b(1) >= 2; a(0) = 4*b(1) - 1 > b(1); b(1) = 2, a(0) = 7; b(2) >= 3; a(1) = 4*b(2) - 2 > b(2); b(2) = 3, a(1) = 10; etc. [edited by Georg Fischer, Sep 23 2020]

Mathematica

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
b = {1}; a = {}; h = 4;
Do[AppendTo[b, mex[Flatten[{a, b}], b[[-1]]]];
AppendTo[a, h b[[-1]] - b[[-2]]], {250}]; a
(* Peter J. C. Moses, Sep 07 2019 *)

Crossrefs

Cf. A329388, A329389, A329390.

Sequence in context: A226567 A184106 A112414 * A034810 A154487 A023390

Adjacent sequences: A329384 A329385 A329386 * A329388 A329389 A329390

Keyword

nonn

Author

Clark Kimberling, Nov 23 2019

Status

approved