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.

Previous Showing 21-26 of 26 results.

A374003 Decimal expansion of the positive real root of x^8 - x^7 - x^6 + x^2 - 1.

Original entry on oeis.org

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

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Author

Paolo Xausa, Jun 25 2024

Keywords

Comments

Tenth smallest Pisot-Vijayaraghavan number.

Examples

			1.5736789683935169887742514186293214678127040615079...

Links

Crossrefs

Cf. A060006, A086106, A228777, A092526, A293508, A293509, A293557, A374002, A293506.

Programs

  • Mathematica
    First[RealDigits[Root[#^8 - #^7 - #^6 + #^2 - 1 &, 2], 10, 100]]

A381124 Numerators of convergents to the supergolden ratio.

Original entry on oeis.org

1, 3, 19, 22, 85, 447, 1873, 41653, 43526, 85179, 384242, 469421, 1323084, 111608477, 112931561, 450403160, 563334721, 3830411486, 4393746207, 17011650107, 21405396314, 209660216933, 231065613247, 440725830180, 671791443427, 1112517273607, 21809619641960, 66541376199487
Offset: 0

Views

Author

Eric W. Weisstein, Feb 14 2025

Keywords

Links

Crossrefs

Cf. A092526 (decimal digits), A369346 (continued fraction).
Cf. A381125 (denominators of convergents).

Programs

  • Mathematica
    Numerator[Convergents[Root[-1 - #^2 + #^3&, 1, 0], 20]]

A381125 Denominators of the convergents to the supergolden ratio.

Original entry on oeis.org

1, 2, 13, 15, 58, 305, 1278, 28421, 29699, 58120, 262179, 320299, 902777, 76153567, 77056344, 307322599, 384378943, 2613596257, 2997975200, 11607521857, 14605497057, 143056995370, 157662492427, 300719487797, 458381980224, 759101468021, 14881309872623
Offset: 0

Views

Author

Eric W. Weisstein, Feb 14 2025

Keywords

Links

Crossrefs

Cf. A092526 (decimal digits), A369346 (continued fraction).
Cf. A381124 (numerators of convergents).

Programs

  • Mathematica
    Denominator[Convergents[Root[-1 - #^2 + #^3&, 1, 0], 20]]

A138251 Beatty sequence of the positive root of x^3 - x^2 - 1.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102
Offset: 1

Views

Author

Clark Kimberling, Mar 09 2008

Keywords

Comments

First differs from A110118 at 73rd term.

Crossrefs

Cf. A092526, A110118, A138253.

Formula

a(n)=Floor(r*n), where r=1.46557123187676...; see A092526 for more decimal places.

A306276 a(0) = a(1) = a(2) = a(3) = 1; thereafter a(n) = a(n-2) + a(n-3) + a(n-4).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 5, 7, 11, 15, 23, 33, 49, 71, 105, 153, 225, 329, 483, 707, 1037, 1519, 2227, 3263, 4783, 7009, 10273, 15055, 22065, 32337, 47393, 69457, 101795, 149187, 218645, 320439, 469627, 688271, 1008711, 1478337, 2166609, 3175319, 4653657, 6820265
Offset: 0

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Author

Joseph Damico, Feb 02 2019

Keywords

Comments

The characteristic equation of this sequence is x^4 = x^2 + x + 1. The characteristic equation of A000930 is x^3 = x^2 + 1 [1], which can be rewritten as x^4 = x^3 + x [2]. By substituting the value of x^3 from equation [1] in equation [2], we get x^4 = (x^2 + 1) + x, which is the characteristic equation for this sequence. Hence the ratio a(n+1)/a(n) has the same limit as the A000930 sequence does, about 1.465571231.

Links

Crossrefs

Cf. A000930, A092526.

Programs

  • Mathematica
    Nest[Append[#, Total@ #[[-4 ;; -2]] ] &, {1, 1, 1, 1}, 40] (* or *)
CoefficientList[Series[(x^3 - x - 1)/(x^4 + x^3 + x^2 - 1), {x, 0, 43}], x] (* Michael De Vlieger, Feb 09 2019 *)

Formula

G.f.: (x^3 - x - 1)/(x^4 + x^3 + x^2 - 1).
a(n) = a(n-2) + a(n-3) + a(n-4) for n >= 4, a(n) = 1 for n < 4.
Lim_{n->infinity} a(n+1)/a(n) = A092526.

A344388 Decimal expansion of a constant related to the asymptotics of A048634.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Aug 16 2021

Keywords

Comments

This constant is a very close to A201506.
Conjecture: It is equal to the limit of column "h^2" in the Table 1 in reference by Wright and Trefethen, p. 336.

Examples

			1.05747359610293071458836136901117212325956834040149469519600889340841418929257...

Links

Crossrefs

Cf. A048634, A092526, A201506.

Programs

  • Mathematica
    A092526 = 1/3 + 2/(3*(116 + 12*Sqrt[93])^(1/3)) + (1/6)*(116 + 12*Sqrt[93])^(1/3); terms = 500; b = ConstantArray[0, terms]; b[[7]] = N[Log[2], 1000]; b[[8]] = N[Log[3], 1000]; b[[9]] = N[Log[5], 1000]; Quiet[Do[b[[n]] = b[[n-1]] + b[[n-3]] - Sum[Exp[k*(b[[n-2]] - b[[n-1]] - b[[n-3]])]/k*(-1)^k, {k, 1, 1000}], {n, 10, terms}]; Exp[Table[N[b[[n]]/A092526^n, 110], {n, Length[b] - 20, Length[b]}]]]

Formula

Equals exp(limit_{n->infinity} log(A048634(n)) / A092526^n ).
Previous Showing 21-26 of 26 results.

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