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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A181099 Exchange rightmost two ternary digits of n > 1; a(0)=0, a(1)=3.

Original entry on oeis.org

0, 3, 6, 1, 4, 7, 2, 5, 8, 9, 12, 15, 10, 13, 16, 11, 14, 17, 18, 21, 24, 19, 22, 25, 20, 23, 26, 27, 30, 33, 28, 31, 34, 29, 32, 35, 36, 39, 42, 37, 40, 43, 38, 41, 44, 45, 48, 51, 46, 49, 52, 47, 50, 53, 54, 57, 60, 55, 58, 61, 56, 59, 62, 63, 66, 69, 64, 67, 70, 65, 68, 71, 72
Offset: 0

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Author

Jonathan Vos Post, Oct 03 2010

Keywords

Comments

Self-inverse permutation of natural numbers: a(a(n)) = n.

Examples

			a(10) = a(101_3) = 110_3 = 12.
a(20) = a(202_3) = 220_3 = 24.
a(30) = a(1010_3) = 1001_3 = 28.

Links

Crossrefs

Cf. A007089, A080412.

Programs

  • Mathematica
    Join[{0,3,6},Table[FromDigits[Join[Drop[IntegerDigits[n,3],-2],Reverse[Take[IntegerDigits[n,3],-2]]],3],{n,3,80}]] (* Harvey P. Dale, Jun 12 2025 *)

Formula

From R. J. Mathar, Oct 12 2010: (Start)
a(n) = a(n-1) + a(n-9) - a(n-10) = 9 *floor(n/9) + 3*(n mod 3) + (floor(n/3) mod 3).
G.f.: x*(3 + 3*x - 5*x^2 + 3*x^3 + 3*x^4 - 5*x^5 + 3*x^6 + 3*x^7 + x^8) / ( (1+x+x^2)*(x^6+x^3+1)*(x-1)^2 ). (End)

Extensions

Corrected and extended by D. S. McNeil, Oct 06 2010

A269403 Expansion of x*(2 - x + 2*x^2 + x^3)/((1 - x)^3*(1 + x + x^2 + x^3)).

Original entry on oeis.org

0, 2, 3, 6, 10, 16, 21, 28, 36, 46, 55, 66, 78, 92, 105, 120, 136, 154, 171, 190, 210, 232, 253, 276, 300, 326, 351, 378, 406, 436, 465, 496, 528, 562, 595, 630, 666, 704, 741, 780, 820, 862, 903, 946, 990, 1036, 1081, 1128, 1176, 1226, 1275, 1326, 1378, 1432, 1485
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 25 2016

Keywords

Comments

Partial sums of A080412.

Examples

			a(0) = 0;
a(1) = 0 + 2 = 2;
a(2) = 0 + 2 + 1 = 3;
a(3) = 0 + 2 + 1 + 3 = 6;
a(4) = 0 + 2 + 1 + 3 + 4 = 10;
a(5) = 0 + 2 + 1 + 3 + 4 + 6 = 16;
a(6) = 0 + 2 + 1 + 3 + 4 + 6 + 5 = 21;
a(7) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 = 28;
a(8) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 = 36;
a(9) = 0 + 2 + 1 + 3 + 4 + 6 + 5 + 7 + 8 + 10 = 46, etc.

Links

Crossrefs

Cf. A001477, A080412, A116996.

Programs

  • Mathematica
    LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 2, 3, 6, 10, 16}, 55]
Table[(2 n^2 + 2 n + 2 Sin[(Pi n)/2] - (-1)^n + 1)/4, {n, 0, 54}]

Formula

G.f.: x*(2 - x + 2*x^2 + x^3)/((1 - x)^3*(1 + x + x^2 + x^3)).
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6).
a(n) = (2*n^2 + 2*n + 2*sin((Pi*n)/2) - (-1)^n + 1)/4.
Sum_{n>=1} 1/a(n) = 1.495144413654306177...
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