|
%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
|
