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A295964 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295964 #6 Feb 22 2018 22:43:55
%S A295964 1,4,9,18,33,58,100,168,279,459,751,1224,1990,3230,5238,8487,13745,
%T A295964 22253,36020,58296,94340,152661,247027,399715,646770,1046514,1693314,
%U A295964 2739859,4433206,7173099,11606340,18779475,30385852,49165365,79551256,128716661,208267958
%N A295964 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295964 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).
%C A295964 See A295862 for a guide to related sequences.
%H A295964 Clark Kimberling, <a href="/A295964/b295964.txt">Table of n, a(n) for n = 0..2000</a>
%H A295964 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A295964 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
%e A295964 b(3) = 6 (least "new number")
%e A295964 a(2) = a(1) + a(0) + b(2) - 1 = 9
%e A295964 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, ...)
%t A295964 a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
%t A295964 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A295964 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295964 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295964 Table[a[n], {n, 0, k}];  (* A295964 *)
%Y A295964 Cf. A001622, A000045, A295862.
%K A295964 nonn,easy
%O A295964 0,2
%A A295964 _Clark Kimberling_, Dec 08 2017