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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A332491 a(n) = 2*a(n-1) + a(n-3), where a(0) = 3, a(1) = 1, a(2) = 2.

Original entry on oeis.org

3, 1, 2, 7, 15, 32, 71, 157, 346, 763, 1683, 3712, 8187, 18057, 39826, 87839, 193735, 427296, 942431, 2078597, 4584490, 10111411, 22301419, 49187328, 108486067, 239273553, 527734434, 1163954935, 2567183423, 5662101280, 12488157495, 27543498413
Offset: 0

Views

Author

Michael Tulskikh, Feb 13 2020

Keywords

Programs

  • Mathematica
    LinearRecurrence[{2,0,1},{3,1,2},40] (* Harvey P. Dale, Apr 20 2025 *)

Formula

a(n) = 2*a(n-1) + a(n-3).
a(n) = A052980(n) + 2*A008998(n-3).
a(n) = A008998(n-1) + 3*A008998(n-3).
G.f.: (5x-3)/(x^3+2*x-1).

A345448 Number of tilings of a 2 X n rectangle with dominoes and long L-shaped 4-minoes.

Original entry on oeis.org

1, 1, 2, 7, 15, 32, 79, 185, 422, 987, 2307, 5352, 12451, 29005, 67478, 156991, 365391, 850304, 1978615, 4604465, 10715078, 24934611, 58024779, 135028632, 314222011, 731218981, 1701605078, 3959769367, 9214694391, 21443322032, 49900304047, 116121942377
Offset: 0

Views

Author

Greg Dresden and Yiwen Zhang, Jun 19 2021

Keywords

Examples

			For n = 3 the a(3)=7 tilings are:
._____.  ._____.  ._____.  ._____.
| |___|  |___| |  |  ___|  |___  |
|_____|  |_____|  |_|___|  |___|_|
._____.  ._____.  ._____.
|___| |  | |___|  | | | |
|___|_|  |_|___|  |_|_|_|
		

Crossrefs

Cf. A052980.

Programs

  • Mathematica
    LinearRecurrence[{1, 1, 4, 2}, {1, 1, 2, 7}, 40]

Formula

a(n) = a(n-1) + a(n-2) + 4*a(n-3) + 2*a(n-4).
Sum_{j=0..n} a(n) = (1/7)(a(n+4) - a(n+2) - 5*a(n+1) - 1).
G.f.: 1/(1 - x - x^2 - 4*x^3 - 2*x^4). - Stefano Spezia, Jun 19 2021
a(n) = F(n+1) + 2*Sum_{j=3..n} a(n-j)*F(j) for F(i) = A000045(i) the i-th Fibonacci number. - Greg Dresden, Nov 10 2024

A387619 Number of ways to tile a 2 X n strip with squares, dominoes, and (straight and bent) trominoes.

Original entry on oeis.org

1, 2, 11, 51, 235, 1092, 5064, 23489, 108954, 505377, 2344171, 10873339, 50435526, 233943074, 1085135139, 5033353844, 23347000765, 108294084141, 502317568673, 2329978980834, 10807509809918, 50130181109907, 232526743191648, 1078565548781339, 5002881075314417
Offset: 0

Views

Author

Greg Dresden and Daeheon Shin, Sep 03 2025

Keywords

Comments

Compare to A030186 which counts tilings with just squares and dominoes, and to A052980 which counts tilings with just dominos and bent trominos.

Crossrefs

Programs

  • Mathematica
    LinearRecurrence[{3, 6, 7, 2, 0, -1}, {1, 2, 11, 51, 235, 1092}, 40]

Formula

a(n) = 3*a(n-1) + 6*a(n-2) + 7*a(n-3) + 2*a(n-4) - a(n-6).
G.f.: (1 - x - x^2 - x^3)/(1 - 3*x - 6*x^2 - 7*x^3 - 2*x^4 + x^6).
Previous Showing 21-23 of 23 results.