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A295964
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 4, 9, 18, 33, 58, 100, 168, 279, 459, 751, 1224, 1990, 3230, 5238, 8487, 13745, 22253, 36020, 58296, 94340, 152661, 247027, 399715, 646770, 1046514, 1693314, 2739859, 4433206, 7173099, 11606340, 18779475, 30385852, 49165365, 79551256, 128716661, 208267958
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
See A295862 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) - 1 = 9
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)
MATHEMATICA
a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
j = 1; While[j < 6, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A295964 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 08 2017
STATUS
approved