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 1-6 of 6 results.

A106222 Coefficients of g.f. A(x) where 0 <= a(n) <= 4 for all n>1, with initial terms {1,5}, such that A(x)^(1/5) consists entirely of integer coefficients.

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, May 01 2005

Keywords

Comments

Equals the self-convolution 5th power of A106223. What is the frequency of occurrence of the nonzero digits?

Examples

			A(x) = 1 + 5*x + x^5 + 3*x^10 + x^15 + 4*x^20 + x^35 +...
A(x)^(1/5) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 +-...
A106223 = {1,1,-2,6,-21,80,-320,1326,-5637,24434,...}.

Crossrefs

Cf. A106223, A106216, A106220, A106224, A106226.

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),n)))}

Formula

A(z)=0 at z=-0.1999361633111821182995648612577212067...

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...

A196345 The 5th power of the g.f. equals the g.f. of A196344.

Original entry on oeis.org

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107542, 479196, -2157045, 9792702, -44780606, 206055345, -953305628, 4431463845, -20686696836, 96931500441, -455722376856, 2149086834269, -10162544424168, 48176923110789, -228913128188293, 1089973053510359
Offset: 0

Views

Author

Paul D. Hanna, Oct 01 2011

Keywords

Comments

A196344 is defined as the coefficients in the g.f. Q(x), where -2 <= A196344(n) <= 2 for all n>1 with initial terms {1,5}, such that Q(x)^(1/5) consists entirely of integer coefficients.
Limit a(n+1)/a(n) = -5.0015989761 6639938823 4051883169 0463138590 3476719792 3351242105 ...

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 +...
where
A(x)^5 = 1 + 5*x + x^5 - 2*x^10 + x^15 - x^20 + x^35 - 2*x^40 - x^45 - 2*x^50 + x^55 + 2*x^65 - x^70 + 2*x^80 +...+ A196344(n)*x^n +...
A196344 begins: [1,5,0,0,0,1,0,0,0,0,-2,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,...].

Links

Crossrefs

Cf. A196344, A196346, A106223.

Programs

  • PARI
    {a(n)=local(A=1+5*x); if(n==0, 1, for(j=1, n, for(k=-2, 2, t=polcoeff((A+k*x^j+x*O(x^j))^(1/5), j);
if(denominator(t)==1, A=A+k*x^j; break))); polcoeff((A+x*O(x^n))^(1/5), n))}

A352703 G.f. A(x) satisfies: A(x)^5 = A(x^5) + 5*x.

Original entry on oeis.org

1, 1, -2, 6, -21, 80, -320, 1326, -5637, 24434, -107542, 479196, -2157045, 9792702, -44780606, 206055346, -953305632, 4431463863, -20686696920, 96931500840, -455722378776, 2149086843549, -10162544469252, 48176923330632, -228913129263389, 1089973058779915
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2022

Keywords

Comments

Not the same as A106223 or A196345.

Examples

			G.f.: A(x) = 1 + x - 2*x^2 + 6*x^3 - 21*x^4 + 80*x^5 - 320*x^6 + 1326*x^7 - 5637*x^8 + 24434*x^9 - 107542*x^10 + 479196*x^11 + ...
such that A(x)^5 = A(x^5) + 5*x, as illustrated by:
A(x)^5 = 1 + 5*x + x^5 - 2*x^10 + 6*x^15 - 21*x^20 + 80*x^25 - 320*x^30 + 1326*x^35 - 5637*x^40 + 24434*x^45 - 107542*x^50 + ...

Crossrefs

Cf. A107092, A352704, A352705.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=0,n\5,
A = (subst(A,x,x^5) + 5*x + x*O(x^(5*n)))^(1/5));
polcoeff(A,n)}
for(n=0,30,print1(a(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

Views

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)))}
Showing 1-6 of 6 results.

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