A296845 -id:A296845 - cp's OEIS frontend

A296843 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n+1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 2, 9, 18, 35, 63, 109, 184, 306, 504, 825, 1345, 2187, 3551, 5758, 9330, 15110, 24463, 39597, 64085, 103708, 167820, 271556, 439405, 710991, 1150427, 1861450, 3011910, 4873394, 7885340, 12758771, 20644149, 33402959, 54047148, 87450148, 141497338

Offset: 0

Clark Kimberling, Jan 12 2018

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). See A296245 for a guide to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A001622, A296245, A296844, A296845.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
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}];  (* A296843 *)
Table[b[n], {n, 0, 20}] (* complement *)

A296844 Decimal expansion of ratio-sum for A296843; see Comments.

Original entry on oeis.org

4, 4, 6, 5, 2, 0, 5, 2, 9, 3, 4, 2, 0, 7, 5, 5, 5, 3, 6, 4, 1, 2, 6, 7, 8, 3, 7, 3, 0, 1, 3, 0, 3, 6, 0, 4, 6, 4, 1, 5, 6, 3, 1, 6, 5, 1, 8, 4, 3, 6, 4, 9, 4, 0, 1, 2, 2, 3, 2, 7, 5, 6, 0, 2, 4, 7, 3, 1, 7, 4, 6, 3, 8, 9, 3, 0, 4, 6, 8, 4, 0, 7, 6, 1, 0, 0

Offset: 1

Clark Kimberling, Jan 12 2018

Suppose that A = (a(n)), for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A296843, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.

			ratio-sum = 4.465205293420755536412678373013036046415...

Cf. A001622, A296843, A296845.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
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}];  (* A296843 *)
GoldenRatio; s = N[Sum[-g + a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A296844 *)

cp's OEIS Frontend

A296843 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n+1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

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