A296284 -id:A296284 - cp's OEIS frontend

A296432 Decimal expansion of ratio-sum for A296284; see Comments.

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

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 = A296284 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 = 6.21032710946618494227967...

Cf. A001622, A296284.

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 2];
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}]; (* A296284 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296432 *)

A296459 Decimal expansion of limiting power-ratio for A296284; see Comments.

Original entry on oeis.org

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

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 = A296284 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 = 16.71583793078729559599070133315659170886...

Cf. A001622, A296284.

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*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, 15}]  (* A296284 *)
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] (* A296459 *)

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.

A296469 Decimal expansion of ratio-sum for A295862; see Comments.

Original entry on oeis.org

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

Offset: 1

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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A295862, 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.  Guide to more ratio-sums and limiting power-ratios:
****
Sequence A    ratio-sum for A     limiting power-ratio for A
A295862           A296469                 A296470
A295947           A296471                 A296472
A295948           A296473                 A296474
A295949           A296475                 A296476
A295950           A296477                 A296478
A295951           A296479                 A296480
A295952           A296481                 A296482
A295953           A296483                 A296848
A295960           A296485                 A296486
A293076           A296487                 A296488
A293358           A296489                 A296490
A294170           A296491                 A296492
A296555           A296493                 A296494
A294414           A296495                 A296496
A294541           A296497                 A296498
A294546           A296499                 A296500
A294552           A296501                 A296494
A296776           A298171                 A298172
A294553           A296503                 A296504
A296556           A296565                 A296566
A296557           A296567                 A296568
A296558           A296569                 A296570

			ratio-sum = 6.21032710946618494227967...

Cf. A001622, A296284, A296470.

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 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}]; (* A295862 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296469 *)

A296493 Decimal expansion of ratio-sum for A296555; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 19 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 = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			5.204091649313251611130187115558413050194...

Cf. A001622, A296284, A296494, A296555.

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] + 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}]; (* A296555 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296493 *)

A296491 Decimal expansion of ratio-sum for A294170; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 20 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 = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			6.358713026984299354541477968890605504302...

Cf. A001622, A294170, A296284, A296492.

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 n;
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A294170 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296491 *)

A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Dec 20 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 = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 11.22071294787201913135639932120744822352...

Cf. A001622, A294381, A296284, A296491.

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 n;
j = 1; While[j < 16, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}];  (* A294170 *)
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]   (* A296492 *)

A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 20 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 = A294414, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 8.814104632202565527925188322585412678508...

Cf. A001622, A294414, A296284, A296495.

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
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}]; (* A294414 *)
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]   (* A296496 *)

A296848 Decimal expansion of limiting power-ratio for A295953; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 06 2018

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 = A295953, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			limiting power-ratio = 7.090700687355142881167747526503371215921...

Cf. A001622, A295953, A296284, A296483.

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
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}]; (* A295953 *)
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]   (* A296848 *)

A296471 Decimal expansion of ratio-sum for A295947; see Comments.

Original entry on oeis.org

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

Offset: 1

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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A295947, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

			ratio-sum = 2.427179488056039424423653103831452251757...

Cf. A001622, A295947, A296284, A296472.

a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + 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}]; (* A295947 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296471 *)

cp's OEIS Frontend

A296432 Decimal expansion of ratio-sum for A296284; see Comments.

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A296459 Decimal expansion of limiting power-ratio for A296284; 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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A296469 Decimal expansion of ratio-sum for A295862; see Comments.

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A296493 Decimal expansion of ratio-sum for A296555; see Comments.

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A296491 Decimal expansion of ratio-sum for A294170; see Comments.

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A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

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A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

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A296848 Decimal expansion of limiting power-ratio for A295953; see Comments.

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