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 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. 64
 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 (list; graph; refs; listen; history; text; internal format)
 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). ***** Guide to related sequences, each determined by a complementary equation and initial values (a(0),a(1); b(0),b(1),b(2)): ***** 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 ***** 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 ***** 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 ***** 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 ***** 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 ***** 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 ***** 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 ***** 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 ***** 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 ***** 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

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Last modified January 24 19:12 EST 2021. Contains 340411 sequences. (Running on oeis4.)