A296850 -id:A296850 - cp's OEIS frontend

A296849 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(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.

1, 2, 10, 28, 73, 182, 446, 1085, 2628, 6354, 15350, 37069, 89504, 216094, 521710, 1259533, 3040796, 7341146, 17723110, 42787389, 103297912, 249383238, 602064414, 1453512093, 3509088629, 8471689381, 20452467422, 49376624257, 119205715969, 287788056229

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(2) = A014176. [corrected by Clark Kimberling, Jun 09 2018]

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

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

Cf. A014176, A296245, A296850, A296851.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 7, 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}]; (* A296849 *)
Table[b[n], {n, 0, 20}] (* complement *)
Take[u, 30]

A296851 Decimal expansion of limiting power-ratio for A296849; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 13 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 limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296849, we have g = 1+ sqrt(2). See the guide at A296469 for related sequences.

			limiting power-ratio = 2.283378414042995122799666600398678749089...

Cf. A296849, A296850.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 8, 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}]; (* A296849 *)
z = 1700; r = 1 + Sqrt[2]; h = Table[N[a[n]/r^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296851 *)

cp's OEIS Frontend

A296849 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) + b(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.

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