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.

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A214826 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 4.

Original entry on oeis.org

1, 4, 4, 9, 17, 30, 56, 103, 189, 348, 640, 1177, 2165, 3982, 7324, 13471, 24777, 45572, 83820, 154169, 283561, 521550, 959280, 1764391, 3245221, 5968892, 10978504, 20192617, 37140013, 68311134, 125643764, 231094911, 425049809
Offset: 0

Views

Author

Abel Amene, Jul 29 2012

Keywords

Comments

See Comments in A214727.

Links

Crossrefs

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

Programs

  • GAP
    a:=[1,4,4];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019
  • Mathematica
    LinearRecurrence[{1,1,1},{1,4,4},33] (* Ray Chandler, Dec 08 2013 *)
  • PARI
    my(x='x+O('x^40)); Vec((1+3*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019
  • Sage
    ((1+3*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

Formula

G.f.: (1+3*x-x^2)/(1-x-x^2-x^3).
a(n) = K(n) - 2*T(n+1) + 5*T(n), where K(n) = A001644(n) and T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

A214830 a(n) = a(n-1) + a(n-2) + a(n-3), with a(0) = 1, a(1) = a(2) = 8.

Original entry on oeis.org

1, 8, 8, 17, 33, 58, 108, 199, 365, 672, 1236, 2273, 4181, 7690, 14144, 26015, 47849, 88008, 161872, 297729, 547609, 1007210, 1852548, 3407367, 6267125, 11527040, 21201532, 38995697, 71724269, 131921498, 242641464, 446287231, 820850193, 1509778888
Offset: 0

Views

Author

Abel Amene, Aug 07 2012

Keywords

Comments

See comments in A214727.

Links

Crossrefs

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

Programs

  • GAP
    a:=[1,8,8];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 24 2019
  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+7*x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 24 2019
  • Mathematica
    CoefficientList[Series[(x^2-7*x-1)/(x^3+x^2+x-1), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jun 18 2014 *)
LinearRecurrence[{1,1,1}, {1,8,8}, 40] (* G. C. Greubel, Apr 24 2019 *)
  • PARI
    my(x='x+O('x^40)); Vec((1+7*x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 24 2019
  • Sage
    ((1+7*x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 24 2019

Formula

G.f.: (1+7*x-x^2)/(1-x-x^2-x^3).
a(n) = -A000073(n) + 7*A000073(n+1) + A000073(n+2). - G. C. Greubel, Apr 24 2019

A268410 a(n) = a(n - 1) + a(n - 2) + a(n - 3) for n>2, a(0)=5, a(1)=7, a(2)=9.

Original entry on oeis.org

5, 7, 9, 21, 37, 67, 125, 229, 421, 775, 1425, 2621, 4821, 8867, 16309, 29997, 55173, 101479, 186649, 343301, 631429, 1161379, 2136109, 3928917, 7226405, 13291431, 24446753, 44964589, 82702773, 152114115, 279781477, 514598365, 946493957
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2016

Keywords

Comments

Tribonacci sequence beginning 5, 7, 9.
In general, the ordinary generating function for the recurrence relation b(n) = b(n-1) + b(n-2) + b(n-3), with n>2 and b(0)=k, b(1)=m, b(2)=q, is (k + (m-k)*x + (q-m-k)*x^2)/(1 - x - x^2 - x^3).

Links

Crossrefs

Cf. similar sequences with initial values (p,q,r): A000073 (0,0,1), A081172 (1,1,0), A001590 (0,1,0; also 1,2,3), A214899 (2,1,2), A001644 (3,1,3), A145027 (2,3,4), A000213 (1,1,1), A141036 (2,1,1), A141523 (3,1,1), A214727 (1,2,2), A214825 (1,3,3), A214826 (1,4,4), A214827 (1,5,5), A214828 (1,6,6), A214829 (1,7,7), A214830 (1,8,8), A214831 (1,9,9).

Programs

  • GAP
    a:=[5,7,9];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019
  • Magma
    I:=[5,7,9]; [n le 3 select I[n] else Self(n-1)+Self(n-2)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 04 2016
  • Mathematica
    LinearRecurrence[{1, 1, 1}, {5, 7, 9}, 40]
RecurrenceTable[{a[0]==5, a[1]==7, a[2]==9, a[n]==a[n-1]+a[n-2]+a[n-3]}, a, {n, 40}]
  • PARI
    my(x='x+O('x^40)); Vec((5+2*x-3*x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019
  • Sage
    ((5+2*x-3*x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

Formula

G.f.: (5 + 2*x - 3*x^2)/(1 - x - x^2 - x^3).
a(n) = 3*K(n) - 4*T(n+1) + 8*T(n), where K(n) = A001644(n) and T(n) =A000073(n+1). - G. C. Greubel, Apr 23 2019
Previous Showing 11-13 of 13 results.

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