

A296276


Solution of the complementary equation a(n) = a(n1) + a(n2) + b(n1)*b(n), where a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.


2



2, 4, 21, 55, 118, 229, 419, 738, 1267, 2137, 3560, 5879, 9649, 15768, 25689, 41763, 67794, 109937, 178171, 288614, 467337, 756551, 1224538, 1981791, 3207085, 5189688, 8397643, 13588261, 21986896, 35576213, 57564231, 93141634, 150707125, 243850091, 394558622
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OFFSET

0,1


COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n1) > (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.


LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + b(1)*b(2) = 21
Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)


MATHEMATICA

a[0] = 2; a[1] = 4; b[0] = 1; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n  1] + a[n  2] + b[n  1] b[n];
j = 1; While[j < 10, 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}]; (* A296276 *)
Table[b[n], {n, 0, 20}] (* complement *)


CROSSREFS

Cf. A001622, A296245.
Sequence in context: A058522 A292534 A122736 * A092458 A071779 A107388
Adjacent sequences: A296273 A296274 A296275 * A296277 A296278 A296279


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Dec 13 2017


STATUS

approved



