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.

A114953 A cubic quartic recurrence.

Original entry on oeis.org

1, 1, 2, 9, 745, 413500186, 70701255783138724397185481, 353412074392865080823440901423426679423573814794711467360597541360306163522857
Offset: 0

Views

Author

Jonathan Vos Post, Feb 21 2006

Keywords

Comments

a(6) has 233 digits. This sequence is related to: A112961 "a cubic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n-1)^3 + a(n-2)^3 A112969 "a quartic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n-1)^4 + a(n-2)^4, which is the quartic (or biquadratic) analog of the Fibonacci sequence similarly to A000283 being the quadratic analog of the Fibonacci sequence. Primes in this sequence include a(n) for n = 2. Semiprimes in this sequence include a(n) for n = 3, 4, 6.

Examples

			a(2) = a(1)^3 + a(0)^4 = 1^3 + 1^4 = 2.
a(3) = a(2)^3 + a(1)^4 = 2^3 + 1^4 = 9.
a(4) = a(3)^3 + a(2)^4 = 9^3 + 2^4 = 745.
a(5) = a(4)^3 + a(3)^4 = 745^3 + 9^4 = 413500186.
a(6) = a(5)^2 + a(4)^4 = 413500186^3 + 745^4 = 70701255783138724397185481.

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == a[n-1]^3 + a[n-2]^4}, a, {n, 0, 8}] (* Vaclav Kotesovec, Dec 18 2014 *)

Formula

a(0) = a(1) = 1, for n>1 a(n) = a(n-1)^3 + a(n-2)^4.
a(n) ~ c^(3^n), where c = 1.085072477219577474852112080874481159102040272323161792230192441384737595241... . - Vaclav Kotesovec, Dec 18 2014

Extensions

Formula corrected by Vaclav Kotesovec, Dec 18 2014

A114954 A 3/2-power Fibonacci sequence.

Original entry on oeis.org

1, 1, 2, 4, 11, 45, 339, 6544, 535619, 392527477, 7777266564708, 21689055127418446258, 101009204076980364695686091211
Offset: 0

Views

Author

Jonathan Vos Post, Feb 21 2006

Keywords

Comments

This sequence is related to: A112961 "a cubic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n-1)^3 + a(n-2)^3 A112969 "a quartic Fibonacci sequence" a(1) = a(2) = 1; for n>2: a(n) = a(n-1)^4 + a(n-2)^4, which is the quartic (or biquadratic) analog of the Fibonacci sequence similarly to A000283 being the quadratic analog of the Fibonacci sequence. Primes in this sequence include a(n) for n = 2, 4. Semiprimes in this sequence include a(n) for n = 3, 6.

Examples

			a(2) = ceiling(a(0)^(3/2) + a(1)^(3/2)) = ceiling(1^1.5 + 1^1.5) = 2.
a(3) = ceiling(a(1)^(3/2) + a(2)^(3/2)) = ceiling(1^1.5 + 2^1.5) = ceiling(3.82842712) = 4.
a(4) = ceiling(2^(3/2) + 4^(3/2)) = ceiling(10.8284271) = 11.
a(5) = ceiling((4^(3/2)) + (11^(3/2))) = ceiling(44.4828727) = 45.
a(6) = ceiling((11^(3/2)) + (45^(3/2))) = ceiling(338.35205) = 339.
a(7) = ceiling((45^(3/2)) + (339^(3/2))) = ceiling(6543.52112) = 6544.

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==a[1]==1,a[n]==Ceiling[Surd[ a[n-1]^3,2]+ Surd[ a[n-2]^3, 2]]},a,{n,15}] (* Harvey P. Dale, Apr 07 2016 *)

Formula

a(0) = a(1) = 1, for n>1 a(n) = ceiling(a(n-1)^(3/2) + a(n-2)^(3/2)).

A114956 a(n) = ceiling(a(n-1)^(3/4) + a(n-2)^(3/4)), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16
Offset: 0

Views

Author

Jonathan Vos Post, Feb 21 2006

Keywords

Comments

a(17) = 16 is exactly 16^(3/4) + 16^(3/4) = 16. This is a fixed point, so a(n) = 16 for all n>14.

Examples

			a(2) = ceiling(a(0)^(3/4) + a(1)^(3/4)) = ceiling(1^(3/4) + 1^(3/4)) = 2.
a(3) = ceiling(a(1)^(3/4) + a(2)^(3/4)) = ceiling(1^(3/4) + 2^(3/4)) = ceiling(2.68179283) = 3.
a(4) = ceiling(2^(3/4) + 3^(3/4)) = ceiling(3.96129989) = 4.
a(5) = ceiling(3^(3/4) + 4^(3/4)) = ceiling(5.10793418) = 6.
a(6) = ceiling(4^(3/4) + 6^(3/4)) = ceiling(6.66208575) = 7.

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==1,a[n]==Ceiling[a[n-1]^(3/4)+ a[n-2]^(3/4)]}, a[n],{n,80}] (* Harvey P. Dale, Jul 22 2011 *)

Extensions

Edited by N. J. A. Sloane, May 20 2006

A114957 a(n) = ceiling(a(n-1)^(4/3) + a(n-2)^(4/3)), with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 4, 9, 26, 96, 517, 4589, 80409, 3546873, 544383737, 445042712531, 3398279290987133, 510914600201184438040, 4084427005585662985398294639, 6528922582874884079540382952631569851, 12202683821888699966029264978793346242448495941305
Offset: 0

Views

Author

Jonathan Vos Post, Feb 21 2006

Keywords

Examples

			a(2) = ceiling(a(0)^(4/3) + a(1)^(4/3)) = ceiling(1^(4/3) + 1^(4/3)) = 2.
a(3) = ceiling(a(1)^(4/3) + a(2)^(4/3)) = ceiling(1^(4/3) + 2^(4/3)) = ceiling(3.5198421) = 4.
a(4) = ceiling(2^(4/3) + 4^(4/3)) = ceiling(8.86944631) = 9.
a(5) = ceiling(4^(4/3) + 9^(4/3)) = ceiling(25.0703586) = 26.
a(6) = ceiling(9^(4/3) + 26^(4/3)) = ceiling(95.7456522) = 96.
a(7) = ceiling(26^(4/3) + 96^(4/3)) = ceiling(516.595167) = 517.
a(8) = ceiling(96^(4/3) + 517^(4/3)) = ceiling(4588.99022) = 4589.

Crossrefs

Cf. A000283, A112961, A112969, A114793.

Programs

  • Mathematica
    Nest[Append[#,Ceiling[Total[Take[#,-2]^(4/3)]]]&,{1,1},17]  (* Harvey P. Dale, Apr 21 2011 *)

Extensions

Corrected and extended by Harvey P. Dale, Apr 21 2011
Comments edited by Petros Hadjicostas, Nov 03 2019

A284604 Quadratic recurrence: a(n+2) = a(n+1)^2 + a(n)^2 + 1, with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 3, 11, 131, 17283, 298719251, 89233191216703091, 7962562414716697755180182566955283, 63402400208259008611705446682872670539115181497111590988296570564371
Offset: 0

Views

Author

Emanuele Munarini, Mar 30 2017

Keywords

Crossrefs

Cf. A000283.

Programs

  • Magma
    [n le 2 select 1 else Self(n-1)^2+Self(n-2)^2+1: n in [1..10]]; // Bruno Berselli, Mar 30 2017
  • Mathematica
    RecurrenceTable[{a[n + 2] == a[n + 1]^2 + a[n]^2 + 1, a[0] == 1, a[1] == 1}, a, {n, 0, 12}]
nxt[{a_,b_}]:={b,a^2+b^2+1}; NestList[nxt,{1,1},10][[;;,1]] (* Harvey P. Dale, Feb 16 2025 *)
  • Maxima
    a(n) := if (n=0 or n=1) then 1 else a(n-1)^2 + a(n-2)^2 + 1; makelist(a(n), n, 0, 12);

Formula

a(n+3) = a(n+2)^2 + a(n+2) - a(n)^2, with a(0) = a(1) = 1, a(2) = 3.
a(n) ~ c^(2^n), where c = 1.356519333072951374233963037913978335267300244021120535099185060013... - Vaclav Kotesovec, Apr 15 2017

A359502 a(n) = a(n-2)^2 + a(n-1) + 1 for n >= 2 with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 2, 4, 9, 26, 108, 785, 12450, 628676, 155631177, 395389144154, 24221458643549484, 156332599536291235925201, 586679058977510608947573592591458, 24439881677774993432884951095586547256059481860
Offset: 0

Views

Author

Andrew Wall, Jan 03 2023

Keywords

Crossrefs

Cf. A000278, A000283.

Programs

  • Mathematica
    a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n - 2]^2 + a[n - 1] + 1; Array[a, 16, 0] (* Amiram Eldar, Jan 05 2023 *)
  • Python
    # a generator
def agen_q():
    a, b = 0, 1
    while 1:
        yield a
        a, b = b, (a * a) + b + 1
f = agen_q(); print([next(f) for _ in range(20)])
Previous Showing 21-26 of 26 results.

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