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.

A141446 A102055(n) mod 9.

Original entry on oeis.org

1, 2, 1, 4, -4, 7, -5, 8, -5, 4, -7, 1, -5, 5, -2, 4, -1, 4, -5, 2, -8, 4, -4, 7, -5, 8, -5, 4, -7, 1, -5, 5, -2, 4, -1, 4, -5, 2, -8, 4, -4, 7, -5, 8, -5, 4, -7, 1, -5, 5, -2, 4, -1, 4, -5, 2, -8, 4, -4, 7, -5, 8, -5, 4, -7, 1, -5, 5, -2, 4, -1, 4, -5, 2, -8, 4, -4, 7, -5, 8, -5, 4, -7, 1, -5, 5
Offset: 0

Views

Author

Paul Curtz, Aug 07 2008

Keywords

Comments

We compute the positive remainder modulo 9 and subtract 9 if A102055(n) is negative.
Appears to be periodic with period length 18 after the transitional first 3 elements. (This would imply only the same 6 digits appear as found in A141425.)

Crossrefs

Cf. A141430.

Programs

  • Maple
    A102055 := proc(n) local k; if n = 0 then 1; else 1-add(A001469(k),k=1..n) ; end if; end proc:
A141446 := proc(n) local a; a := A102055(n) ; if a > 0 then a mod  9; else (a mod  9)-9; end if; end proc; # R. J. Mathar, Jul 07 2011

Formula

a(3n) + a(3n+1) + a(3n+2) = 4, 7, -2, -2, -2, 5 ever same six digits?

A133135 Third column of the inverse of the triangle of polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].

Original entry on oeis.org

1, -2, 2, -1, 1, -4, 4, 13, -13, -142, 142, 1931, -1931, -36296, 36296, 893273, -893273, -27927346, 27927346, 1081725559, -1081725559, -50861556172, 50861556172, 2854289486309, -2854289486309, -188475382997654, 188475382997654, 14467150771771043, -14467150771771043
Offset: 0

Views

Author

Paul Curtz, Sep 21 2007

Keywords

Comments

The triangle with [x^k] P(n,x) starts
1;
1, 1;
1/2, 3/2, 1;
1/2, 3/2, 2, 1;
1/2, 2, 3, 5/2, 1;
1/2, 5/2, 5, 5, 3, 1;
1/2, 3,15/2, 10,15/2, 7/2, 1;
1/2, 7/2,21/2,35/2,35/2,21/2, 4, 1;
1/2, 4, 14, 28, 35, 28, 14, 9/2, 1;
1/2, 9/2, 18, 42, 63, 63, 42, 18, 5, 1;
The sum of the rows of this triangle is A094373 (previously noticed by Paul Curtz). - Jean-François Alcover, Jul 22 2013
Apparently a(2*n) = A102055(n) and a(2*n+1) = -a(2*n) for n >= 0. - Georg Fischer, Dec 05 2022

Examples

			The inverse of the triangle of coefficients starts
1;
-1, 1;
1, -3/2, 1;
-1, 3/2, -2, 1;
1, -5/4, 2, -5/2, 1;
-1, 5/4 -1, 5/2, -3, 1;
1, -7/4, 1, 0, 3, -7/2, 1;
-1, 7/4,-4, 0, 2, 7/2,-4, 1;
1, 3/8, 4,-21/2, -2, 21/4, 4,-9/2, 1;
-1, -3/8,13, 21/2,-26,-21/4,10, 9/2,-5, 1;
and defines the sequence in its third column.
Apart from the numbers along the diagonal, the absolute values show up in pairs if read along columns. Conjectures: Starting with the third line, columns are alternatingly fractions and integers. The row sums (1, 0, 1/2, -1/2, 1/4, -1/4, 3/4, -3/4, -11/8, 11/8,..) also show up in pairs from the third row on.

Programs

  • Mathematica
    max = 28; p[0, ] = 1; p[n, x_] := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))/2; t = Table[ Coefficient[p[n, x], x, k], {n, 0, max + 2}, {k, 0, max + 2}]; a[n_] := Inverse[t][[All, 3]][[n + 3]]; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jul 22 2013 *)

Extensions

Edited and extended by R. J. Mathar, Aug 02 2008

A102054 Triangular matrix, read by rows, where T(n,k) = T(n-1,k) - [T^-1](n-1,k-1); also equals the matrix inverse of A060083 (Euler polynomials).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 4, -2, 4, 1, 1, -13, 26, -10, 5, 1, 1, 142, -229, 116, -25, 6, 1, 1, -1931, 3181, -1567, 371, -49, 7, 1, 1, 36296, -59700, 29464, -6922, 952, -84, 8, 1, 1, -893273, 1469380, -725108, 170398, -23358, 2100, -132, 9, 1, 1, 27927346, -45938639, 22669816, -5327198, 730252, -65526, 4152
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2004

Keywords

Comments

Column 1 forms A102055. Column 2 forms A102056.

Examples

			T(5,3) = -10 = T(4,3) - A060083(4,2) = 4 - 14.
T(6,2) = -229 = T(5,2) - A060083(5,1) = 26 - 255.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,1,3,1],
[1,4,-2,4,1],
[1,-13,26,-10,5,1],
[1,142,-229,116,-25,6,1],
[1,-1931,3181,-1567,371,-49,7,1],
[1,36296,-59700,29464,-6922,952,-84,8,1],...
The matrix inverse is equal to A060083:
[1],
[ -1,1],
[1,-2,1],
[ -3,5,-3,1],
[17,-28,14,-4,1],
[ -155,255,-126,30,-5,1],...

Crossrefs

Cf. A060083, A102055, A102056.

Programs

  • PARI
    {T(n,k)=local(M=matrix(n+1,n+1));M[1,1]=1;if(n>0,M[2,1]=1;M[2,2]=1); for(r=3,n+1, for(c=1,r,M[r,c]=if(c==1,M[r-1,1], if(c==r,1,M[r,c]=M[r-1,c]-((matrix(r-1,r-1,i,j,M[i,j]))^-1)[r-1,c-1])))); return(M[n+1,k+1])}

Formula

T(n, k) = T(n-1, k) - A060083(n-1, k-1), for n>0, with T(0, 0)=1.

A102056 Column 2 of A102054, the matrix inverse of A060083 (Euler polynomials).

Original entry on oeis.org

1, 3, -2, 26, -229, 3181, -59700, 1469380, -45938639, 1779367231, -83663906454, 4695118012798, -310029578255977, 23797509154723361, -2101268283286737704, 211490399673452191176, -24070116208693613274035, 3075833697885980827017235, -438528974913900528707713514
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2004

Keywords

Crossrefs

Cf. A060083, A102054, A102055.

Programs

  • PARI
    
				
    • 
				
			

		

	

	
	
	


		
			

	

		A243868
		0 followed by -(n+1)*A226158(n).
	

	
Original entry on oeis.org

	

		

			0, 0, 2, 3, 0, -5, 0, 21, 0, -153, 0, 1705, 0, -26949, 0, 573405, 0, -15802673, 0, 547591761, 0, -23302711005, 0, 1194695479813, 0, -72628776062025, 0, 5165901157067001, 0, -425013158488292213, 0
		

		
Offset: 0

	

	

		
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Author

		

			Paul Curtz, Jun 13 2014
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is A000004=0's it is of the first kind. It is of the second kind if the main diagonal is the upper diagonal multiplied by 2.


Starting from the autosequence of second kind A198631(n)/A006519(n+1),the fractional Euler numbers,we build a family of alternated sequences of second and first kind. A row is 0 followed by n+1 times the preceding one.


1, 1/2, 0, -1/4,  0, 1/2, 0, -17/8,   0, 31/2,...


0,   1, 1,    0, -1,   0, 3,     0, -17,    0, 155,... = -A226158(n)


0,   0, 2,    3,  0,  -5, 0,    21,   0, -153, 0, 1705,... = a(n).


a(n) is an autosequence of the second kind. Its difference table is:


0,     0,   2,   3,   0,  -5,   0,  21, 0, -153,...


0,     2,   1,  -3,  -5,   5,  21, -21,...


2,    -1,  -4,  -2,  10,  16, -42,...


-3,   -3,   2,  12,   6, -58,..


0,     5,  10,  -6, -64,...


5,     5, -16, -58,...


0,   -21, -42,...


-21, -21,...


0,... .


a(n) is a post Genocchi sequence.

		

	

	
	
	

		
Examples

		
			a(0)=0, a(1)=1*0=0, a(2)=2*1=2, a(3)=3*1=3, a(4)=4*0=0, a(5)=5*(-1)=-5.
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A102055, A230011, A229054.

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    Mathematica
    
				
    a[0] = a[1] = 0; a[2] = 2; a[n_] := -n*(n-1)*EulerE[n-2, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 17 2014 *)
    
				
    • 
				
			

		

	

	
	
	

		
Formula

		

			
The fourth column of the second triangle of A133135(n) (see also A140218) is


1, -5/2, 5/2, 0, 0, -21/2, 21/2,... = b(n).


c(n) = 0, 0, 0, 0, followed by 2*(-1)^n*b(n) = 0, 0, 0, 0, 2, 5, 5, 0, 0, 21, 21, -132, -132,... . Autosequence.


a(n) = c(n+2) -c(n+1).

		

	

	
	
	

		
Extensions

		

			
More terms from Jean-François Alcover, Jun 17 2014

		

	

	


		

		
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