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A295862
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
32
1, 3, 9, 18, 34, 60, 104, 175, 291, 479, 784, 1278, 2078, 3373, 5470, 8863, 14354, 23239, 37616, 60879, 98520, 159425, 257972, 417425, 675426, 1092881, 1768338, 2861251, 4629622, 7490908, 12120566, 19611511, 31732115, 51343665, 83075820, 134419526, 217495388
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, 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