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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
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