login
A296003
Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 2, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
3
2, 4, 10, 32, 94, 278, 824, 2440, 7228, 21408, 63406, 187800, 556234, 1647478, 4879574, 14452538, 42806168, 126785206, 375518042, 1112225982, 3294240212, 9757026674, 28898794076, 85593729210, 253515301048, 750872855508, 2223968505284, 6587048494582
OFFSET
0,1
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 2.961844324... (as in A296004). See A296000 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, so that
a(2) = a(0)*b(1) + a(1)*b(0) = 10
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
MATHEMATICA
mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
a[0] = 2; a[1] = 4; b[0] = 1; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, 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, 100}]; (* A296003 *)
t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
Take[RealDigits[Last[t], 10][[1]], 100] (* A296004 *)
CROSSREFS
Sequence in context: A243931 A005269 A070900 * A263662 A151400 A363138
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 07 2017
STATUS
approved