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 1-8 of 8 results.

A106216 Coefficients of g.f. A(x) where 0 <= a(n) <= 2 for all n>1, with initial terms {1,3}, such that A(x)^(1/3) consists entirely of integer coefficients.

Original entry on oeis.org

1, 3, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

The self-convolution cube-root equals A106219. Positions of 1's is given by A106217. Positions of 2's is given by A106218. What is the frequency of occurrence of the 1's and 2's?

Examples

			A(x)^(1/3) = 1 + 1x - 1x^2 + 2x^3 - 4x^4 + 9x^5 - 21x^6 + 53x^6 -+...

Crossrefs

Cf. A106217, A106218, A106219, A083953.

Programs

  • PARI
    {a(n)=local(A=1+3*x);if(n==0,1,if(n==1,3, for(j=2,n, for(k=0,2,t=polcoeff((A+k*x^j+x*O(x^j))^(1/3),j); if(denominator(t)==1,A=A+k*x^j;break)));polcoeff(A,n)))}

Formula

A(z)=0 at z=-0.322846893915891638743032676733152456643928599...

A106223 Self-convolution 5th power equals A106222, which consists entirely of digits {0,1,2,3,4} after the initial terms {1,5}.

Original entry on oeis.org

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107541, 479192, -2157027, 9792618, -44780207, 206053429, -953296364, 4431418833, -20686477329, 96930426941, -455717114981, 2149060994827, -10162417338993, 48176297258115, -228910042632050, 1089957826522693, -5199911987465160
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Examples

			A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 +-...
A(x)^5 = 1 + 5*x + x^5 + 3*x^10 + x^15 + 4*x^20 + x^35 +...
A106222 = {1,5,0,0,0,1,0,0,0,0,3,0,0,0,0,1,0,0,0,0,4,...}.

Crossrefs

Cf. A106222, A106219, A106221, A106225.

Programs

  • PARI
    {a(n)=local(A=1+5*x);if(n==0,1, for(j=1,n, for(k=0,4,t=polcoeff((A+k*x^j+x*O(x^j))^(1/5),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff((A+x*O(x^n))^(1/5),n)))}

Formula

Limit a(n+1)/a(n) = -5.001596426773442826534115368782519...

A106221 Self-convolution 4th power equals A106220, which consists entirely of digits {0,1,2,3} after the initial terms {1,4}.

Original entry on oeis.org

1, 1, -1, 2, -4, 10, -26, 71, -199, 569, -1652, 4855, -14413, 43153, -130143, 394967, -1205268, 3695771, -11381215, 35183209, -109138163, 339599993, -1059702401, 3315256789, -10396158911, 32671424776, -102879610571, 324557399534, -1025643986057, 3246330348415, -10290418283163
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Examples

			A(x) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 10*x^5 - 26*x^6 + 71*x^7 -+...
A(x)^4 = 1 + 4*x + 2*x^2 + 3*x^4 + 2*x^6 + x^8 + 2*x^14 +...
A106220 = {1,4,2,0,3,0,2,0,1,0,0,0,0,0,2,0,0,0,2,...}.

Crossrefs

Cf. A106220, A106219, A106223, A106225.

Programs

  • PARI
    {a(n)=local(A=1+4*x);if(n==0,1, for(j=1,n, for(k=0,3,t=polcoeff((A+k*x^j+x*O(x^j))^(1/4),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff((A+x*O(x^n))^(1/4),n)))}

Formula

Limit a(n+1)/a(n) = -3.30697774878897620974321728382452592372871...

A106225 Self-convolution 6th power equals A106224, which consists entirely of digits {0,1,2,3,4,5} after the initial terms {1,6}.

Original entry on oeis.org

1, 1, -2, 7, -27, 114, -506, 2322, -10919, 52316, -254369, 1251563, -6218656, 31153743, -157167147, 797682007, -4069817562, 20860266354, -107358128720, 554533772363, -2873667741743, 14935575580894, -77833224795929, 406595414780038, -2128748177726089, 11167899337858904
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Examples

			A(x) = 1 + x - 2*x^2 + 7*x^3 - 27*x^4 + 114*x^5 - 506*x^6 +-...
A(x)^6 = 1 + 6*x + 3*x^2 + 2*x^3 + 3*x^4 + 3*x^8 + 4*x^9 +...
A106224 = {1,6,3,2,3,0,0,0,3,4,3,0,0,0,3,2,0,0,0,0,3,2,...}.

Crossrefs

Cf. A106224, A106219, A106221, A106223, A106227.

Programs

  • PARI
    {a(n)=local(A=1+6*x);if(n==0,1, for(j=1,n, for(k=0,5,t=polcoeff((A+k*x^j+x*O(x^j))^(1/6),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff((A+x*O(x^n))^(1/6),n)))}

Formula

Limit a(n+1)/a(n) = -5.502856676359094846755190514140489974645...

A106217 Positions of 1's in A106216.

Original entry on oeis.org

0, 3, 24, 30, 48, 57, 63, 72, 81, 99, 105, 114, 120, 129, 132, 135, 141, 147, 156, 159, 168, 177, 180, 195, 198, 201, 204, 207, 210, 222, 243, 249, 252, 261, 267, 279, 282, 285, 297, 309, 312, 315, 327, 333, 342, 351, 375, 387, 393, 399, 402, 408, 411, 414
Offset: 0

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Crossrefs

Cf. A106216, A106218, A106219.

Programs

  • PARI
    
				
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Formula

		

			
a(n) = 0 (mod 3) for all n.

		

	

	
	


		
			

	

		A106218
		Positions of 2's in A106216.
	

	
Original entry on oeis.org

	

		

			6, 9, 12, 21, 27, 33, 69, 75, 84, 87, 93, 96, 111, 123, 126, 162, 165, 192, 225, 228, 231, 234, 237, 240, 264, 270, 273, 276, 288, 300, 306, 318, 321, 339, 345, 354, 357, 360, 378, 381, 384, 390, 417, 426, 429, 432, 438, 441, 447, 453, 468, 471, 474, 477, 483
		

		
Offset: 0

	

	

		
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Author

		

			Paul D. Hanna, May 01 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	
	
	
	

		
Crossrefs

		

			
Cf. A106216, A106217, A106219.

		

	

	
	
	

		
Programs

		

			
    
				
				
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Formula

		

			
a(n) = 0 (mod 3) for all n.

		

	

	
	


		
			

	

		A106227
		Self-convolution 7th power equals A106226, which consists entirely of digits {0,1,2,3,4,5,6} after the initial terms {1,7}.
	

	
Original entry on oeis.org

	

		

			1, 1, -3, 13, -65, 351, -1989, 11650, -69900, 427167, -2648438, 16612947, -105215448, 671760933, -4318468133, 27926126547, -181520036139, 1185220461607, -7769787811032, 51117085986564, -337373170566291, 2233091754693676, -14819626688607761, 98582852441111688
		

		
Offset: 0

	

	

		
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Author

		

			Paul D. Hanna, May 01 2005
		

	

	

		
Keywords

		

			
		

	

	
	
	

		
Examples

		
			A(x) = 1 + x - 3*x^2 + 13*x^3 - 65*x^4 + 351*x^5 - 1989*x^6 +-...
A(x)^7 = 1 + 7*x + x^7 + 4*x^14 + 6*x^21 + 5*x^28 + x^35 + 6*x^42 +...
A106226 = {1,7,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,6,...}.
		

	

	
	
	
	
	

		
Crossrefs

		

			
Cf. A106226, A106219, A106221, A106223, A106225.

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    PARI
    
				
    {a(n)=local(A=1+7*x);if(n==0,1, for(j=1,n, for(k=0,6,t=polcoeff((A+k*x^j+x*O(x^j))^(1/7),j); if(denominator(t)==1,A=A+k*x^j;break))); return(polcoeff((A+x*O(x^n))^(1/7),n)))}
    
				
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		A196307
		The cube of the g.f. equals the g.f. of A196306.
	

	
Original entry on oeis.org

	

		

			1, 1, -1, 2, -4, 9, -22, 55, -142, 375, -1009, 2753, -7599, 21178, -59509, 168401, -479477, 1372536, -3947678, 11402376, -33059314, 96177750, -280671373, 821379083, -2409938978, 7087502564, -20889306810, 61691675424, -182531101523, 541000651928, -1606046079955, 4774977156350
		

		
Offset: 0

	

	

		
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Author

		

			Paul D. Hanna, Oct 01 2011
		

	

	

		
Keywords

		

			
		

	

	
	

		
Comments

		

			
A196306 is defined as the Coefficients in the g.f. C(x), where -1 <= A196306(n) <= 1 for all n>1 with initial terms {1,3}, such that C(x)^(1/3) consists entirely of integer coefficients.


Limit a(n+1)/a(n) = -3.1069369226 1299813830 3346689095 3281527516 0860761416 4775926338 8951561634 ...

		

	

	
	
	

		
Examples

		
			G.f.: A(x) = 1 + x - x^2 + 2*x^3 - 4*x^4 + 9*x^5 - 22*x^6 + 55*x^7 - 142*x^8 + 375*x^9 - 1009*x^10 + 2753*x^11 - 7599*x^12 +...
where
A(x)^3 = 1 + 3*x + x^3 - x^6 - x^9 - x^12 - x^18 + x^21 - x^24 - x^30 - x^33 + x^39 - x^42 - x^45 + x^48 +...+ A196306(n)*x^n +...
A196306 begins: [1,3,0,1,0,0,-1,0,0,-1,0,0,-1,0,0,0,0,0,-1,0,0,1,0,0,-1,...].
		

	

	
	
	
	

		
Links

		

			
		

	

	
	
	

		
Crossrefs

		

			
Cf. A196306, A196308, A106219 (variant).

		

	

	
	
	

		
Programs

		

			
    
				
				
  • 
				
    PARI
    
				
    {a(n)=local(A=1+3*x); if(n==0, 1, for(j=1, n, for(k=-1, 1, t=polcoeff((A+k*x^j+x*O(x^j))^(1/3), j);
if(denominator(t)==1, A=A+k*x^j; break))); polcoeff((A+x*O(x^n))^(1/3), n))}
    
				
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		Showing 1-8 of 8 results.
		
	


		

			
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