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.

A156257 Digit of runs of length 2 in the Kolakoski sequence A000002: a(n) = A000002(A078649(n)).

Original entry on oeis.org

2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2
Offset: 1

Views

Author

Benoit Cloitre, Feb 07 2009

Keywords

Comments

Often equal to A074292 (at the beginning), but not always (see comments in A074292). First differences between the two sequences are at n = 47, 48, 56, 57, 128, 129, 137, 139, 147, 148,176, 177,... (see A248345 = A156257 - A074292). - Jean-Christophe Hervé, Oct 11 2014
As in the Kolakoski sequence, runs in this sequence are of length 1 or 2: a run XX in this sequence implies YXXYX in OK for the first X, and this cannot be continued by a single Y (because XYXYX is not possible), thus we have YXXYXXY, which can be continued by YXXYXXYY or by YXXYXXYXYY, but not by YXXYXXYXX (because this would imply an impossible 21212 in OK). However, words of the form YXYXY appear in this sequence, but they don't in A000002. - Jean-Christophe Hervé, Oct 12 2014
Applying Lenormand's "raboter" transformation (see A318921) to A000002 leads to this sequence. - Rémy Sigrist, Nov 11 2020

Examples

			Kolakoski sequence begins (1),(2,2),(1,1),(2),(1),(2,2),(1),(2,2), so this one begins 2,1,2,2.

Links

Crossrefs

Cf. A000002, A074292, A318921.

Programs

Formula

a(n) = A000002(A078649(n)) = A000002(A078649(n)+1).
Strictly positive terms of (A000002(n)-1)*(mod(n-1, 2)+1). - Jean-Christophe Hervé, Oct 11 2014
Strictly positive terms of (1-abs(A000002(n+1)-A000002(n)))*A000002(n). - Jean-Christophe Hervé, Oct 11 2014

Extensions

Definition revised by Jean-Christophe Hervé, Oct 11 2014

A247236 Triangle read by rows: T(n,k) appears in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} T(n,k)*(x+k)^k.

Original entry on oeis.org

1, -1, 2, -1, -10, 3, -1, 26, -33, 4, -1, -54, 207, -76, 5, -1, 96, -993, 824, -145, 6, -1, -156, 4047, -6736, 2375, -246, 7, -1, 236, -14769, 46184, -28985, 5634, -385, 8, -1, -340, 49743, -280408, 293575, -95166, 11711, -568, 9, -1, 470, -157617, 1556672, -2609465, 1322334, -260449, 22112, -801, 10
Offset: 0

Views

Author

Derek Orr, Nov 27 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = T(n,0)*(x+0)^0 + T(n,1)*(x+1)^1 + T(n,2)*(x+2)^2 + ... + T(n,n)*(x+n)^n, for n >= 0.

Examples

			From _Wolfdieter Lang_, Jan 12 2015: (Start)
The triangle T(n,k) starts:
n\k 0    1       2       3        4       5       6     7    8  9 ...
0:  1
1: -1    2
2: -1  -10       3
3: -1   26     -33       4
4: -1  -54     207     -76        5
5: -1   96    -993     824     -145       6
6: -1 -156    4047   -6736     2375    -246       7
7: -1  236  -14769   46184   -28985    5634    -385     8
8: -1 -340   49743 -280408   293575  -95166   11711  -568    9
9: -1  470 -157617 1556672 -2609465 1322334 -260449 22112 -801 10
... Reformatted.
---------------------------------------------------------------------
n=3: 1 + 2*x + 3*x^2 + 4*x^3 = -1*(x+0)^0 + 26*(x+1)^1 - 33*(x+2)^2 + 4*(x+3)^3. (End)

Crossrefs

Cf. A248345, A000027, A002717, A085490.

Programs

  • PARI
    T(n,k)=(k+1)-sum(i=k+1,n,i^(i-k)*binomial(i,k)*T(n,i))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

Formula

T(n,n) = n+1 = A000027(n+1), n >= 0.
T(n,1) = ((-1)^n*(1-4*n^3-10*n^2-4*n)-1)/8 = 2*(-1)^(n+1)*A002717(n), for n >= 1.
T(n,n-1) = n - n^2 - n^3 (A085490), for n >= 1.
T(n,n-2) = (n^5-2*n^4-3*n^3+6*n^2-2)/2, for n >= 2.

Extensions

Edited. - Wolfdieter Lang, Jan 12 2015

A246788 Triangle read by rows: T(n,k) is the coefficient A_k in the transformation Sum_{k=0..n} (k+1)*x^k = Sum_{k=0..n} A_k*(x+2)^k.

Original entry on oeis.org

1, -3, 2, 9, -10, 3, -23, 38, -21, 4, 57, -122, 99, -36, 5, -135, 358, -381, 204, -55, 6, 313, -986, 1299, -916, 365, -78, 7, -711, 2598, -4077, 3564, -1875, 594, -105, 8, 1593, -6618, 12051, -12564, 8205, -3438, 903, -136, 9, -3527, 16422, -34029, 41196, -32115, 16722, -5817, 1304, -171, 10
Offset: 0

Views

Author

Derek Orr, Nov 15 2014

Keywords

Comments

Consider the transformation 1 + 2x + 3x^2 + 4x^3 + ... + (n+1)*x^n = A_0*(x+2)^0 + A_1*(x+2)^1 + A_2*(x+2)^2 + ... + A_n*(x+2)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0.

Examples

			1;
-3,        2;
9,       -10,      3;
-23,      38,    -21,      4;
57,     -122,     99,    -36,      5;
-135,    358,   -381,    204,    -55,     6;
313,    -986,   1299,   -916,    365,   -78,     7;
-711,   2598,  -4077,   3564,  -1875,   594,  -105,    8;
1593,  -6618,  12051, -12564,   8205, -3438,   903, -136,    9;
-3527, 16422, -34029,  41196, -32115, 16722, -5817, 1304, -171, 10;

Crossrefs

Cf. A248345, A045883, A014105, A001057.

Programs

  • PARI
    T(n,k) = (k+1)*sum(i=0,n-k,(-2)^i*binomial(i+k+1,k+1))
for(n=0,10,for(k=0,n,print1(T(n,k),", ")))

Formula

T(n,0) = ((6*n+8)*(-2)^n+1)/9, for n >= 0.
T(n,n-1) = -n*(2*n+1), for n >= 1.
Row n sums to A001057(n+1).
Showing 1-3 of 3 results.

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