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 1-7 of 7 results.

A117162 Matrix inverse of triangle A112682.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 0, -2, -1, 1, -1, 0, -2, -1, 1, 1, -1, -1, -2, -1, 1, -1, 1, -1, -1, -2, -1, 1, 0, 0, 2, -2, -1, -2, -1, 1, 0, 1, -1, 2, -2, -1, -2, -1, 1, 1, -1, 3, 0, 1, -2, -1, -2, -1, 1, -1, 1, -1, 3, 0, 1, -2, -1, -2, -1, 1, 0, 2, 2, 0, 4, -1, 1, -2, -1, -2, -1, 1, -1, 0, 2, 2, 0, 4, -1, 1, -2, -1, -2, -1, 1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

The limit of the columns (without leading zeros) is A117166, the Shift-Moebius transform of [1,0,0,0,...] (cf. A117165).

Examples

			Column 1 equals A008683 = Moebius transform of [1,0,0,0,...].
Column 2 = Moebius transform of column 1 preceded by a zero: [0,1,-1,-2,0,-1,1,0,...] = Moebius([0, 1,-1,-1,0,-1,1,-1,...]).
Column 3 = Moebius transform of column 2 preceded by a zero: [0,0,1,-1,-2,-1,-1,2,...] = Moebius([0, 0,1,-1,-2,0,-1,1,...]).
Column 4 = Moebius transform of column 3 preceded by a zero: [0,0,0,1,-1,-2,-1,-2,...] = Moebius([0, 0,0,1,-1,-2,-1,-1,...]).
Triangle begins:
1;
-1, 1;
-1,-1, 1;
0,-2,-1, 1;
-1, 0,-2,-1, 1;
1,-1,-1,-2,-1, 1;
-1, 1,-1,-1,-2,-1, 1;
0, 0, 2,-2,-1,-2,-1, 1;
0, 1,-1, 2,-2,-1,-2,-1, 1;
1,-1, 3, 0, 1,-2,-1,-2,-1, 1;
-1, 1,-1, 3, 0, 1,-2,-1,-2,-1, 1;
0, 2, 2, 0, 4,-1, 1,-2,-1,-2,-1, 1;
-1, 0, 2, 2, 0, 4,-1, 1,-2,-1,-2,-1, 1; ...

Crossrefs

Cf. A112682 (inverse), A008683 (column 1), A117163 (column 2), A117164 (column 3); A117165 (Shift-Moebius), A117170 (inverse Shift-Moebius).

Formula

Column k+1 equals the Moebius transform of column k preceded by a zero, where column k includes the k-1 zeros above the diagonal, for k>=1, starting with A008683 in column 1.

A117160 Column 1 of triangle A112682; also equals row sums of A112682 (with offset).

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 94, 210, 464, 1035, 2295, 5111, 11352, 25259, 56145, 124888, 277669, 617554, 1373201, 3053883, 6790995, 15102178, 33583784, 74684504, 166082706, 369337117, 821331578, 1826484804, 4061741926, 9032530513
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Feb 28 2006

Keywords

Comments

G.f.: A(x) = g.f. of A117169 (Shift-Moebius[1,1,1,1,...]) divided by the g.f. of A117166 (Shift-Moebius[1,0,0,0,...]) (see A117165 for the Shift-Moebius transform coefficients).
Limit_{n->oo} a(n+1)/a(n) = 2.223805416529545241557...

Crossrefs

Cf. A112682, A117165 (Shift-Moebius), A117169, A117166, A117161.

Programs

  • PARI
    {a(n)=if(n<1,0,SM=prod(i=0,n,matrix(n,n,r,c,if(r>=c, if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0)))); U=SM*vector(n,i,1)~;V=SM*vector(n,i,if(i==1,1,0))~; return(Vec(Ser(U)/Ser(V))[n]))}

A117161 Limit of columns of triangle A112682.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 73, 157, 358, 785, 1762, 3896, 8702, 19299, 42995, 95507, 212552, 472445, 1050973, 2336670, 5197036, 11556157, 25700188, 57150018, 127093805, 282627186, 628514815, 1397684691, 3108193486, 6911996344, 15370966058
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Feb 28 2006

Keywords

Comments

Equals the self-convolution inverse of A117166, where A117166 is the Shift-Moebius transform of [1,0,0,0,...] (see A117165 for the Shift-Moebius transform coefficients).
Limit_{n->oo} a(n+1)/a(n) = 2.223805416529545241557...

Links

Crossrefs

Cf. A112682, A117166 (inverse), A117165 (Shift-Moebius).

Programs

  • PARI
    {a(n)=if(n<1,0,SM=prod(i=0,n,matrix(n,n,r,c,if(r>=c, if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0)))); RV=SM*vector(n,i,if(i==1,1,0))~;return(Vec(1/Ser(RV))[n]))}

Formula

a(n) ~ c * d^n, where d = 2.22380541652954524155773588821857101036961489569496661752400123603526677062... and c = 0.26711638477321148860322250949475835248606998843344097590907725227209181124... - Vaclav Kotesovec, Feb 08 2023

A117166 Column 1 of triangle A117165 of Shift-Moebius coefficients and so equals the Shift-Moebius transform of [1,0,0,0,...].

Original entry on oeis.org

1, -1, -2, -1, -2, 1, -1, 3, 0, 4, 4, 5, 1, 7, 6, 5, 4, 6, 3, 5, 3, 2, 5, -3, -5, -1, 2, -7, -6, -13, -9, -14, -12, -22, -12, -25, -23, -26, -21, -35, -27, -38, -27, -43, -32, -47, -34, -51, -46, -52, -33, -53, -35, -58, -41, -56, -39, -54, -39, -61, -45, -53, -24, -46, -23, -44, -20, -41, -14, -32, -12, -22
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

Equals the self-convolution inverse of A117161, which is the limit of columns of triangle A112682.

Crossrefs

Cf. A117165 (triangle), A117167 (column 2), A117168 (column 3), A117169 (row sums), A117161 (inverse), A117160, A112682.

Programs

  • PARI
    {a(n)=prod(i=0,n, matrix(n,n,r,c,if(r>=c,if((r+n-i)%(c+n-i)==0,moebius((r+n-i)/(c+n-i)),0))))[ n,1]}

A117163 Column 2 of triangle A117162; equals the Moebius transform of A008683 (column 1 of A117162) preceded by a zero.

Original entry on oeis.org

0, 1, -1, -2, 0, -1, 1, 0, 1, -1, 1, 2, 0, -3, 2, 2, 0, -1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 3, 0, -1, -1, -2, 0, 0, 0, 2, 0, -2, 2, 2, 0, 2, -1, 0, -2, -2, 1, -2, -1, 0, 1, 3, 0, -1, -1, 1, 1, 0, 1, -1, 0, -1, 0, 1, 0, 2, -1, 0, 0, 3, -1, -2, 0, -2, 0, 3, -2, 1, -1, -4, -1, -1, 1, -3, 0, 1, 2, 2, 0, 2, -1, 3, 2, -1, 1, 4, 0, 1, -1, 1, 0, -1
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

The inverse Moebius transform of A117164 (column 3 of A117162) equals this sequence preceded by a zero.

Crossrefs

Cf. A117162, A112682, A008683 (column 1); A117164 (column 3).

Formula

Equals the Moebius transform of column 1 preceded by a zero:
[0,1,-1,-2,0,-1,1,0,...] = Moebius([0, 1,-1,-1,0,-1,1,-1,...]).

A117164 Column 3 of triangle A117162; equals the Moebius transform of A117163 (column 2 of A117162) preceded by a zero.

Original entry on oeis.org

0, 0, 1, -1, -2, -1, -1, 2, -1, 3, -1, 2, 2, 1, -2, 1, 2, 1, -1, 0, 1, 1, -1, -2, 2, -2, -1, 2, 3, 1, -1, -3, -2, -2, 3, -1, 2, 1, -5, 0, 2, -1, 2, 0, 4, -1, -2, -1, -1, -4, -3, 2, 3, 1, 2, -4, 1, -2, 0, 0, -1, 1, -1, 1, 1, 2, 2, 0, 0, -4, 3, -1, -2, -2, -1, 1, 5, 3, 1, -4, -3, -3, -1, -2, -3, -2, -3, 1, 2, -4, 1, 2, 3, 4, 2, 3, 4, 1, 4
Offset: 1

Views

Author

Wouter Meeussen and Paul D. Hanna, Mar 05 2006

Keywords

Comments

The inverse Moebius transform of column 4 of A117162 equals
this sequence preceded by a zero.

Crossrefs

Cf. A117162, A112682, A008683 (Moebius); A117163 (column 2).

Formula

Equals the Moebius transform of column 2 preceded by a zero:
[0,0,1,-1,-2,-1,-1,2,...] = Moebius([0, 0,1,-1,-2,0,-1,1,...]).

A162696 Trajectory of 1 under morphism taking n to sorted divisors of n+1.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 2, 3, 6, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 3, 1, 5, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 4, 1, 7, 1, 2, 1, 3, 1, 2, 1, 2, 4, 1, 2
Offset: 1

Views

Author

Franklin T. Adams-Watters, Jul 10 2009

Keywords

Comments

1 -> 1,2; 2->1,3; 3->1,2,4; ...

Examples

			1 -> 1,2 -> 1,2,1,3 -> 1,2,1,3,1,2,1,2,4 -> ...

Links

Crossrefs

Cf. A007001, A001511, A027750, A112682, A117160.

Programs

  • PARI
    v=[1,2];for(i=2,60,v=concat(v,divisors(v[i]+1)));v

Formula

a(A117160(k+1)) = k (this is the first occurrence of k in the sequence). - Rémy Sigrist, Jan 14 2023
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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