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.

A110679 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 1, a(1) = 2, a(2) = 11.

Original entry on oeis.org

1, 2, 11, 44, 189, 798, 3383, 14328, 60697, 257114, 1089155, 4613732, 19544085, 82790070, 350704367, 1485607536, 6293134513, 26658145586, 112925716859, 478361013020, 2026369768941, 8583840088782, 36361730124071, 154030760585064, 652484772464329
Offset: 0

Views

Author

Creighton Dement, Aug 02 2005

Keywords

Comments

2tesseq[A*B*cyc(A)] (see program code) gives an alternative formula for A110528.
a(n) is the number of tilings of a 2 X n rectangle by using 1 X 1 squares, dominoes and right trominoes. - Roberto Tauraso, Mar 21 2017

Links

Crossrefs

Cf. A110528, A110680, A001076, A110526, A110526, A033887.

Programs

  • Magma
    [(Fibonacci(3*n+2) +(-1)^n)/2: n in [0..30]]; // G. C. Greubel, Apr 19 2019
  • Maple
    seriestolist(series((-1+x)/((x+1)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -1jesseq[A*B*cyc(A)] with A = - 'j + 'k - 'ii' - 'ij' - 'ik' and B = - .5'i - .5i' - .5'ii' + .5'jj' - .5'kk' + .5'jk' + .5'kj' - .5e
  • Mathematica
    a[n_] := (Fibonacci[3*n+2] + (-1)^n)/2; a /@ Range[0, 22] (* Giovanni Resta, Mar 21 2017 *)
  • PARI
    Vec((1 - x) / ((1 + x)*(1 - 4*x - x^2)) + O(x^30)) \\ Colin Barker, Mar 21 2017
  • PARI
    {a(n) = -(-1)^n * (fibonacci(-2 - 3*n)\2)}; /* Michael Somos, Mar 26 2017 */
  • Sage
    [(fibonacci(3*n+2) +(-1)^n)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

Formula

Program "FAMP" finds: 2*(-1^(n+1)) = A110528(n) - A001076(n+1) - 2*a(n). Program "Superseeker" finds: a(n) = A110526(n+1) - A110526(n); a(n) + a(n+1) = A033887(n+1).
a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*Fibonacci(3*k+1). - Gary Detlefs, Jan 22 2013
a(n) = (Fibonacci(3*n+2)+(-1)^n)/2. - Roberto Tauraso, Mar 21 2017
From Colin Barker, Mar 21 2017: (Start)
G.f.: (1 - x) / ((1 + x)*(1 - 4*x - x^2)).
a(n) = 3*a(n-1) + 5*a(n-2) + a(n-3) for n>2.
(End)
a(n) = -(-1)^n * A049651(-1 - n) for all n in Z. - Michael Somos, Mar 26 2017
a(2*n) = A254627(2*n+1); a(2*n+1) = A077259(2*n+1). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019
2*a(n) = A015448(n+1)+(-1)^n. - R. J. Mathar, Oct 03 2021

Extensions

Typo in program code fixed by Creighton Dement, Dec 11 2009

A110527 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

Original entry on oeis.org

0, 1, 8, 29, 128, 537, 2280, 9653, 40896, 173233, 733832, 3108557, 13168064, 55780809, 236291304, 1000946021, 4240075392, 17961247585, 76085065736, 322301510525, 1365291107840, 5783465941881, 24499154875368
Offset: 0

Views

Author

Creighton Dement, Jul 24 2005

Keywords

Comments

A048878(n) = a(n) + a(n+1). Compare with A110526.

Links

Crossrefs

Cf. A110526, A110528, A033887, A001076, A049661, A033887.

Programs

  • Maple
    seriestolist(series(-x*(1+5*x)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')], apart from initial term.
  • Mathematica
    LinearRecurrence[{3,5,1},{0,1,8},30] (* Harvey P. Dale, Feb 12 2015 *)
  • PARI
    x='x+O('x^50); concat(0, Vec(-x*(1+5*x)/((1+x)*(x^2+4*x-1)))) \\ G. C. Greubel, Aug 30 2017

Formula

G.f.: -x*(1+5*x)/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n + 3*A001076(n) - A015448(n). - Ehren Metcalfe, Nov 18 2017
a(n) = (-1)^n + 2*A110526(n) + A110679(n-2) for n >= 2. - Yomna Bakr and Greg Dresden, May 25 2024

A110528 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 1, a(1) = 10, a(2) = 37.

Original entry on oeis.org

1, 10, 37, 162, 681, 2890, 12237, 51842, 219601, 930250, 3940597, 16692642, 70711161, 299537290, 1268860317, 5374978562, 22768774561, 96450076810, 408569081797, 1730726404002, 7331474697801, 31056625195210
Offset: 0

Views

Author

Creighton Dement, Jul 24 2005

Keywords

Comments

Compare with A110526, A110527.

Links

Crossrefs

Cf. A110526, A110527.

Programs

  • Maple
    seriestolist(series(-(1+7*x+2*x^2)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')]
  • Mathematica
    LinearRecurrence[{3,5,1},{1,10,37},30] (* Harvey P. Dale, Apr 21 2016 *)
  • PARI
    x='x+O('x^50); Vec(-(1+7*x+2*x^2)/((1+x)*(x^2+4*x-1))) \\ G. C. Greubel, Aug 30 2017

Formula

G.f.: -(1 + 7*x + 2*x^2)/((1 + x)*(x^2 + 4*x - 1)).
a(n) = A001077(n+1) - (-1)^n. - Ehren Metcalfe, Nov 18 2017
Showing 1-3 of 3 results.

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