A296245 - OEIS

%I #10 Dec 14 2017 14:29:39

%S 1,2,28,66,143,273,497,870,1488,2502,4159,6857,11241,18354,29884,

%T 48562,78807,127769,207017,335270,542816,878662,1422103,2301441,

%U 3724273,6026555,9751728,15779244,25531996,41312329,66845481,108159035,175005812,283166216

%N 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.

%C 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 *****

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

%C *****

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

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

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

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

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

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

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

%C *****

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

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

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

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

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

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

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

%C *****

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

%C Initial values (1,2; 3):  A296257

%C Initial values (1,3; 2):  A296258

%C Initial values (2,3; 2):  A296259

%C *****

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

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

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

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

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

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

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

%C *****

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

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

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

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

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

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

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

%C *****

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

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

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

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

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

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

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

%C *****

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

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

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

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

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

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

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

%C *****

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

%C Initial values (1,2; 3):  A296284

%C Initial values (1,2; 4):  A296285

%C Initial values (1,3; 2):  A296286

%C Initial values (2,3; 1):  A296287

%C *****

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

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

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

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

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

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

%C *****

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

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

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

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

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

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

%H Clark Kimberling, <a href="/A296245/b296245.txt">Table of n, a(n) for n = 0..1000</a>

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.

%F 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.

%e a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5;

%e a(2) = a(0) + a(1) + b(2)^2 = 28

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

%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2;

%t j = 1; While[j < 12, k = a[j] - j - 1;

%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

%t Table[a[n], {n, 0, k}]  (* A296245 *)

%t Table[b[n], {n, 0, 20}] (* complement *)

%Y Cf. A001622, A295862, A296000.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Dec 10 2017