A296776 -id:A296776

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A298171

Decimal expansion of ratio-sum for A296776; see Comments.

+20
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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 = A298171, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

ratio-sum = 5.611562057656225477032443456092579480982...

MATHEMATICA

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] + 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}]; (* A296776 *)

g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]

Take[RealDigits[s, 10][[1]], 100] (* A298171 *)

CROSSREFS

Cf. A001622, A296462, A296776.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

A298172

Decimal expansion of limiting power-ratio for A296776; see Comments.

+20
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,3

COMMENTS

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 = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.

LINKS

Table of n, a(n) for n=2..87.

EXAMPLE

limiting power-ratio = 11.65837748505646668170868260629394947392...

MATHEMATICA

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] + 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}]; (* A296776 *)

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] (* A298172 *)

CROSSREFS

Cf. A001622, A296462, A296776.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Feb 09 2018

STATUS

approved

A296469

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

+10
44

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

ratio-sum = 6.21032710946618494227967...

MATHEMATICA

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 *)

CROSSREFS

Cf. A001622, A296284, A296470.

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Dec 18 2017

STATUS

approved

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