cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A296569 Decimal expansion of ratio-sum for A294558; see Comments.

Original entry on oeis.org

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

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Author

Clark Kimberling, Jan 06 2018

Keywords

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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Examples

			ratio-sum = 2.820339618753543689299555329148441281565...

Crossrefs

Cf. A001622, A294558, A296469, A296570.

Programs

  • 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 - 1] + b[n - 2] - 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}]; (* A294558 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296569 *)

A296570 Decimal expansion of limiting power-ratio for A294558; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 07 2018

Keywords

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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Examples

			limiting power-ratio = 4.090372669767607839097909012745956501942...

Crossrefs

Cf. A001622, A294558, A296469, A296569.

Programs

  • 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 - 1] + b[n - 2] - 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}]; (* A294558 *)
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]   (* A296570 *)

A294532 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.

Original entry on oeis.org

1, 2, 6, 12, 23, 42, 73, 124, 207, 342, 562, 918, 1495, 2429, 3941, 6388, 10348, 16756, 27125, 43903, 71052, 114980, 186058, 301065, 487151, 788245, 1275426, 2063702, 3339160, 5402895, 8742089, 14145019, 22887144, 37032200, 59919382, 96951621, 156871043
Offset: 0

Views

Author

Clark Kimberling, Nov 03 2017

Keywords

Comments

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values, which, for the sequences in the following guide, are a(0) = 1, a(1) = 2, b(0) = 3:
a(n) = a(n-1) + a(n-2) + b(n-2) A294532
a(n) = a(n-1) + a(n-2) + b(n-2) + 1 A294533
a(n) = a(n-1) + a(n-2) + b(n-2) + 2 A294534
a(n) = a(n-1) + a(n-2) + b(n-2) + 3 A294535
a(n) = a(n-1) + a(n-2) + b(n-2) - 1 A294536
a(n) = a(n-1) + a(n-2) + b(n-2) + n A294537
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n A294538
a(n) = a(n-1) + a(n-2) + b(n-2) + n - 1 A294539
a(n) = a(n-1) + a(n-2) + b(n-2) + 2n - 1 A294540
a(n) = a(n-1) + a(n-2) + b(n-1) A294541
a(n) = a(n-1) + a(n-2) + b(n-1) + 1 A294542
a(n) = a(n-1) + a(n-2) + b(n-1) + 2 A294543
a(n) = a(n-1) + a(n-2) + b(n-1) + 3 A294544
a(n) = a(n-1) + a(n-2) + b(n-1) - 1 A294545
a(n) = a(n-1) + a(n-2) + b(n-1) + n A294546
a(n) = a(n-1) + a(n-2) + b(n-1) + 2n A294547
a(n) = a(n-1) + a(n-2) + b(n-1) + n - 1 A294548
a(n) = a(n-1) + a(n-2) + b(n-1) + n + 1 A294549
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) A294550
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 1 A294551
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n A294552
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n A294553
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2 A294554
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 3 A294555
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n + 1 A294556
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + n - 1 A294557
a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) + 2n A294558
a(n) = a(n-1) + a(n-2) + b(n-1) + 2*b(n-2) A294559
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + 2*b(n-2) A294560
a(n) = a(n-1) + a(n-2) + 2*b(n-1) + b(n-2) A294561
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + 1 A294562
a(n) = a(n-1) + a(n-2) + b(n-1) - b(n-2) + n A294563
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 1 A294564
a(n) = a(n-1) + a(n-2) + 2*b(n-1) - b(n-2) - 3 A294565
Conjecture: for every sequence listed here, a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

Examples

			a(0) = 1, a(1) = 2, b(0) = 3, so that
b(1) = 4 (least "new number")
a(2)  = a(0) + a(1) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, ...)

Links

Crossrefs

Cf. A001622, A293076, A294413.

Programs

  • Mathematica
    mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 3; b[0] = 2;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
Table[a[n], {n, 0, 40}]  (* A294532 *)
Table[b[n], {n, 0, 10}]
Showing 1-3 of 3 results.

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