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A295959
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 5, 11, 23, 42, 74, 126, 211, 350, 575, 940, 1531, 2488, 4037, 6544, 10601, 17166, 27789, 44978, 72792, 117796, 190615, 308439, 499083, 807552, 1306666, 2114250, 3420949, 5535233, 8956217, 14491486, 23447740, 37939264, 61387043, 99326347, 160713431
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) = 5, b(0) = 2, b(1) = 3, b(2) = 4
b(3) = 6 (least "new number")
a(2) = a(1) + a(0) + b(2) + 1 = 11
Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)
MATHEMATICA
a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;
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}]; (* A295959 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 08 2017
STATUS
approved