

A295955


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


2



3, 4, 13, 24, 45, 78, 133, 222, 367, 602, 984, 1602, 2603, 4223, 6845, 11088, 17954, 29064, 47041, 76129, 123196, 199352, 322576, 521957, 844563, 1366551, 2211146, 3577730, 5788910, 9366675, 15155621, 24522333, 39677992, 64200364, 103878396, 168078801
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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 A295862 for a guide to related sequences.


LINKS

Clark Kimberling, Table of n, a(n) for n = 0..2000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

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


MATHEMATICA

a[0] = 3; a[1] = 4; b[0] = 1; b[1] = 2; b[2] = 5;
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}]; (* A295955 *)


CROSSREFS

Cf. A001622, A000045, A295862.
Sequence in context: A182691 A026700 A187775 * A151521 A142860 A111954
Adjacent sequences: A295952 A295953 A295954 * A295956 A295957 A295958


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Dec 08 2017


EXTENSIONS

Definition corrected by Georg Fischer, Sep 27 2020


STATUS

approved



