

A296555


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


4



1, 2, 10, 21, 42, 76, 133, 227, 380, 629, 1033, 1688, 2749, 4467, 7248, 11749, 19033, 30821, 49895, 80759, 130699, 211505, 342253, 553809, 896115, 1449979, 2346151, 3796189, 6142401, 9938653, 16081119, 26019839, 42101027, 68120937, 110222037, 178343049
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OFFSET

0,2


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


MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n  1] + a[n  2] + b[n] + n;
j = 1; While[j < 16, k = a[j]  j  1;
While[k < a[j + 1]  j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A296555 *)
Table[b[n], {n, 0, 20}] (* complement *)


CROSSREFS

Cf. A001622, A296245, A296493, A296496.
Sequence in context: A136735 A294549 A294550 * A231376 A304700 A318548
Adjacent sequences: A296552 A296553 A296554 * A296556 A296557 A296558


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Dec 19 2017


STATUS

approved



