A296245 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2, 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, 28, 66, 143, 273, 497, 870, 1488, 2502, 4159, 6857, 11241, 18354, 29884, 48562, 78807, 127769, 207017, 335270, 542816, 878662, 1422103, 2301441, 3724273, 6026555, 9751728, 15779244, 25531996, 41312329, 66845481, 108159035, 175005812, 283166216

Offset

0,2

Comments

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).

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Guide to related sequences, each determined by a complementary equation and initial values (a(0),a(1); b(0),b(1),b(2)):

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n)^2,

Initial values (1,2; 3,4,5): A296245

Initial values (1,3; 2,4,5): A296246

Initial values (1,4; 2,3,5): A296247

Initial values (2,3; 1,4,5): A296248

Initial values (2,4; 1,3,5): A296249

Initial values (3,4; 1,2,5): A296250

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)^2,

Initial values (1,2; 3,4): A296251

Initial values (1,3; 2,4): A296252

Initial values (1,4; 2,3): A296253

Initial values (2,3; 1,4): A296254

Initial values (2,4; 1,3): A296255

Initial values (3,4; 1,2): A296256

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)^2,

Initial values (1,2; 3): A296257

Initial values (1,3; 2): A296258

Initial values (2,3; 2): A296259

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2),

Initial values (1,2; 3,4): A295367

Initial values (1,3; 2,4): A295363

Initial values (1,4; 2,3): A296262

Initial values (2,3; 1,4): A296263

Initial values (2,4; 1,3): A296264

Initial values (3,4; 1,2): A296265

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-2),

Initial values (1,2; 3,4,5): A296266

Initial values (1,3; 2,4,5): A296267

Initial values (1,4; 2,3,5): A296268

Initial values (2,3; 1,4,5): A296269

Initial values (2,4; 1,3,5): A296270

Initial values (3,4; 1,2,5): A296271

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-1),

Initial values (1,2; 3,4,5): A296272

Initial values (1,3; 2,4,5): A296273

Initial values (1,4; 2,3,5): A296274

Initial values (2,3; 1,4,5): A296275

Initial values (2,4; 1,3,5): A296276

Initial values (3,4; 1,2,5): A296277

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Complementary equation a(n) = a(n-1) + a(n-2) + b(n)*b(n-1)*b(n-2),

Initial values (1,2; 3,4,5): A296278

Initial values (1,3; 2,4,5): A296279

Initial values (1,4; 2,3,5): A296280

Initial values (2,3; 1,4,5): A296281

Initial values (2,4; 1,3,5): A296282

Initial values (3,4; 1,2,5): A296283

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Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2),

Initial values (1,2; 3): A296284

Initial values (1,2; 4): A296285

Initial values (1,3; 2): A296286

Initial values (2,3; 1): A296287

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Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1),

Initial values (1,2; 3,4): A296288

Initial values (1,3; 2,4): A296289

Initial values (1,4; 2,3): A296290

Initial values (2,3; 1,4): A296291

Initial values (2,4; 1,3): A296292

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Complementary equation a(n) = a(n-1) + a(n-2) + n*b(n),

Initial values (1,2; 3,4,5): A296293

Initial values (1,3; 2,4,5): A296294

Initial values (1,4; 2,3,5): A296295

Initial values (2,3; 1,4,5): A296296

Initial values (2,4; 1,3,5): A296297

Links

Clark Kimberling, Table of n, a(n) for n = 0..1000

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Formula

a(n) = H + R, where H = f(n-1)*a(0) + f(n)*a(1) and R = f(n-1)*b(2)^2 + f(n-2)*b(3)^2 + ... + f(2)*b(n-1)^2 + f(1)*b(n)^2, where f(n) = A000045(n), the n-th Fibonacci number.

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 = 28
Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

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]^2;
j = 1; While[j < 12, 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}]  (* A296245 *)
Table[b[n], {n, 0, 20}] (* complement *)

Crossrefs

Cf. A001622, A295862, A296000.

Sequence in context: A339990 A336464 A245801 * A156471 A329595 A138964

Adjacent sequences: A296242 A296243 A296244 * A296246 A296247 A296248

Keyword

nonn,easy

Author

Clark Kimberling, Dec 10 2017

Status

approved