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 1-5 of 5 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

A232970 Expansion of (1-3*x)/(1-5*x+3*x^2+x^3).

Original entry on oeis.org

1, 2, 7, 28, 117, 494, 2091, 8856, 37513, 158906, 673135, 2851444, 12078909, 51167078, 216747219, 918155952, 3889371025, 16475640050, 69791931223, 295643364940, 1252365390981, 5305104928862, 22472785106427, 95196245354568, 403257766524697, 1708227311453354, 7236167012338111, 30652895360805796
Offset: 0

Views

Author

N. J. A. Sloane, Dec 05 2013

Keywords

Comments

For n > 2, a(n) is the number of tilings of (a 2 X (n+1) rectangle missing the top right and top left 1 X 1 cells) using 1 X 1 squares, dominoes and right trominoes. Compare with similar tiling sequences A001076 and A110679. - Greg Dresden and Yilin Zhu, Jul 10 2025

Links

Crossrefs

Cf. A001076, A110679.

Programs

  • Magma
    I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1)- 3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2017
  • Mathematica
    LinearRecurrence[{5, -3, -1}, {1, 2, 7}, 30] (* Vincenzo Librandi, Jun 24 2017 *)
CoefficientList[Series[(1-3x)/(1-5x+3x^2+x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 19 2024 *)
  • PARI
    Vec((1-3*x)/(1-5*x+3*x^2+x^3) + O(x^30)) \\ Felix Fröhlich, Apr 15 2019
  • Sage
    [(fibonacci(3*n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

Formula

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - N. J. A. Sloane, Jun 23 2017
a(n) = (Fibonacci(3*n+1) + 1)/2 = Sum_{k=0..n} Fibonacci(3*k-1). - Ehren Metcalfe, Apr 15 2019
a(2*n) = A294262(2*n); a(2*n+1) = A254627(2*n+2). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019

A254628 Triangular numbers (A000217) that are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 6, 276, 1891, 88831, 608856, 28603266, 196049701, 9210162781, 63127394826, 2965643812176, 20326825084231, 954928097357851, 6545174549727516, 307483881705415806, 2107525878187175881, 99008854981046531641, 678616787601720906126, 31880543820015277772556
Offset: 1

Views

Author

Colin Barker, Feb 03 2015

Keywords

Comments

Also hexagonal numbers (A000384) that are also centered pentagonal numbers (A005891). - Colin Barker, Feb 11 2015

Examples

			6 is in the sequence because it is the 3rd triangular number and the 2nd centered pentagonal number.

Links

Crossrefs

Cf. A000217, A005891, A254626, A254627.
Cf. A000384, A254962.

Programs

  • PARI
    Vec(-x*(x^4+5*x^3-52*x^2+5*x+1)/((x-1)*(x^2-18*x+1)*(x^2+18*x+1)) + O(x^100))

Formula

a(n) = a(n-1)+322*a(n-2)-322*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+5*x^3-52*x^2+5*x+1) / ((x-1)*(x^2-18*x+1)*(x^2+18*x+1)).

A254626 Indices of triangular numbers (A000217) that are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 3, 23, 61, 421, 1103, 7563, 19801, 135721, 355323, 2435423, 6376021, 43701901, 114413063, 784198803, 2053059121, 14071876561, 36840651123, 252509579303, 661078661101, 4531100550901, 11862575248703, 81307300336923, 212865275815561, 1459000305513721
Offset: 1

Views

Author

Colin Barker, Feb 03 2015

Keywords

Comments

Also positive integers x in the solutions to x^2 - 5*y^2 + x + 5*y - 2 = 0, the corresponding values of y being A254627.

Examples

			3 is in the sequence because the 3rd triangular number is 6, which is also the 2nd centered pentagonal number.

Links

Crossrefs

Cf. A000217, A005891, A254627, A254628.

Programs

  • Mathematica
    LinearRecurrence[{1,18,-18,-1,1},{1,3,23,61,421},30] (* Harvey P. Dale, Jun 15 2024 *)
  • PARI
    Vec(-x*(x+1)^2*(x^2+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))

Formula

a(n) = a(n-1) + 18*a(n-2) - 18*a(n-3) - a(n-4) + a(n-5).
G.f.: -x*(x+1)^2*(x^2+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).
a(n) = (-2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/4 where r = sqrt(5). - Colin Barker, Nov 25 2016

A254962 Indices of hexagonal numbers (A000384) that are also centered pentagonal numbers (A005891).

Original entry on oeis.org

1, 2, 12, 31, 211, 552, 3782, 9901, 67861, 177662, 1217712, 3188011, 21850951, 57206532, 392099402, 1026529561, 7035938281, 18420325562, 126254789652, 330539330551, 2265550275451, 5931287624352, 40653650168462, 106432637907781, 729500152756861
Offset: 1

Views

Author

Colin Barker, Feb 11 2015

Keywords

Comments

Also positive integers x in the solutions to 4*x^2 - 5*y^2 - 2*x + 5*y - 2 = 0, the corresponding values of y being A254627.

Examples

			12 is in the sequence because the 12th hexagonal number is 276, which is also the 11th centered pentagonal number.

Links

Crossrefs

Cf. A000032 (Lucas numbers), A000384, A005891, A254627, A254628.

Programs

  • PARI
    Vec(-x*(x^4+x^3-8*x^2+x+1)/((x-1)*(x^2-4*x-1)*(x^2+4*x-1)) + O(x^100))

Formula

a(n) = a(n-1)+18*a(n-2)-18*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(x^4+x^3-8*x^2+x+1) / ((x-1)*(x^2-4*x-1)*(x^2+4*x-1)).
a(n) = (2 + (2-r)^n - (-2-r)^n*(-2+r) + 2*(-2+r)^n + r*(-2+r)^n + (2+r)^n)/8 where r = sqrt(5). - Colin Barker, Nov 25 2016
a(n+2) - a(n) = A000032(3*n + 2) if n is odd, A000032(3*n + 1) if n is even. - Diego Rattaggi, May 11 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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