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). Following is a guide to related sequences:
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Complementary equation: a(n) = a(n-1) + a(n-2) + b(n); initial values (a(0), a(1); b(0), b(1), b(2)):
A295862: (1,3; 2,4,5)
A295947: (2,4; 1,3,5)
A295948: (3,4; 1,2,5)
A295949: (1,2; 3,4,5)
A295950: (1,4; 2,3,5)
A295951: (2,3; 1,4,5)
A295952: (1,5; 2,3,4)
Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) + 1; initial values (a(0), a(1); b(0), b(1), b(2)):
A295953: (1,3; 2,4,5)
A295954: (2,4; 1,3,5)
A295955: (3,4; 1,2,5)
A295956: (1,2; 3,4,5)
A295957: (1,4; 2,3,5)
A295958: (2,3; 1,4,5)
A295959: (1,5; 2,3,4)
Complementary equation: a(n) = a(n-1) + a(n-2) + b(n) - 1; initial values (a(0), a(1); b(0), b(1), b(2)):
A295860: (1,3; 2,4,5)
A295961: (2,4; 1,3,5)
A295962: (3,4; 1,2,5)
A295963: (1,2; 3,4,5)
A295964: (1,4; 2,3,5)
A295965: (2,3; 1,4,5)
A295966: (1,5; 2,3,4)
LINKS
Clark Kimberling, Table of n, a(n) for n = 0..3000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
FORMULA
a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2) + f(n-2)*b(3) + ... + f(2)*b(n-1) + f(1)*b(n), where f(n) = A000045(n), the n-th Fibonacci number.
EXAMPLE
a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, so that
b(3) = 6 (least "new number");
a(2) = a(1) + a(0) + b(2) = 9;
Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, ...)
MATHEMATICA
a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
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}] (*A295862*)
Table[b[n], {n, 0, 20}] (*complement*)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 08 2017
STATUS
approved