A296295 -id:A296295 - cp's OEIS frontend

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

Clark Kimberling, Dec 10 2017

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

			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, ...)

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

Cf. A001622, A295862, A296000.

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 *)

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.

cp's OEIS Frontend

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.

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