A296292 -id:A296292 - cp's OEIS frontend

A296434 Decimal expansion of ratio-sum for A296292; see Comments.

8, 0, 1, 2, 9, 6, 8, 9, 0, 3, 0, 9, 5, 6, 6, 1, 4, 7, 2, 5, 1, 5, 5, 4, 1, 4, 9, 9, 4, 1, 6, 3, 7, 7, 2, 7, 3, 1, 9, 8, 3, 2, 6, 4, 4, 4, 4, 1, 6, 2, 6, 7, 6, 9, 3, 1, 5, 1, 4, 1, 5, 0, 8, 2, 0, 5, 3, 7, 5, 1, 2, 3, 9, 1, 3, 8, 9, 6, 8, 4, 6, 5, 4, 7, 4, 2

Offset: 1

Clark Kimberling, Dec 15 2017

Suppose that A = {a(n)}, for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A296292 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			Ratio-sum = 8.012968903095661472515541...

Cf. A001622, A296292.

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] + n*b[n];
j = 1; While[j < 13, 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}]; (* A296292 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296434 *)

A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.

Original entry on oeis.org

2, 1, 7, 4, 1, 3, 0, 7, 3, 5, 5, 2, 3, 5, 5, 8, 7, 3, 5, 5, 8, 1, 4, 9, 8, 5, 8, 5, 9, 0, 8, 9, 1, 5, 8, 5, 6, 8, 9, 6, 3, 3, 2, 1, 7, 2, 8, 0, 7, 1, 9, 6, 3, 7, 5, 6, 3, 3, 6, 9, 0, 1, 3, 3, 8, 3, 5, 5, 4, 4, 6, 2, 2, 8, 6, 5, 5, 8, 3, 9, 8, 9, 6, 2, 9, 6

Offset: 2

Clark Kimberling, Dec 18 2017

Suppose that A = (a(n)), for n >=0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296292 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

			limiting power-ratio = 21.74130735523558735581498585908915856896...

Cf. A001622, A296292.

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] + n*b[n];
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, 15}]  (* A296292 *)
z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296461 *)

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.

Original entry on oeis.org

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).
*****
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

			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.

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A296434 Decimal expansion of ratio-sum for A296292; see Comments.

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A296461 Decimal expansion of limiting power-ratio for A296292; see Comments.

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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.

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