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 1-4 of 4 results.

A368627 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^3 - A(-x)^3)/2.

Original entry on oeis.org

1, 1, 3, 7, 40, 103, 723, 1941, 15060, 41382, 340657, 950061, 8132676, 22916139, 201684153, 572618987, 5145063940, 14692661910, 134152006842, 384852888898, 3559210821120, 10248531332559, 95777105998365, 276630878235275, 2607824127882204, 7551545042631558, 71714198513326425
Offset: 0

Views

Author

Paul D. Hanna, Jan 09 2024

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 40*x^4 + 103*x^5 + 723*x^6 + 1941*x^7 + 15060*x^8 + 41382*x^9 + 340657*x^10 + 950061*x^11 + 8132676*x^12 + ...
where A(x) is formed from the even bisection of A(x)^2 and the odd bisection of A(x)^3, as can be seen from the expansions
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 103*x^4 + 328*x^5 + 1941*x^6 + 6506*x^7 + 41382*x^8 + 142892*x^9 + 950061*x^10 + ...
A(x)^3 = 1 + 3*x + 12*x^2 + 40*x^3 + 198*x^4 + 723*x^5 + 3927*x^6 + 15060*x^7 + 86190*x^8 + 340657*x^9 + 2016195*x^10 + ...
so that the bisections of the above series are related by
(A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2, and
(A(x) + A(-x))/2 = 1 + x*(A(x)^3 - A(-x)^3)/2.
SPECIFIC VALUES.
The g.f. A(x) converges at the radius of convergence r, given by
A(-r) = 1 at r = (A(r) - 1)/(1 + A(r)^2) = 0.1795090246029167685576...
where A(r) = (1 + (28 + sqrt(783))^(1/3) + (28 - sqrt(783))^(1/3))/3 = 1.6956207695598620574163671... solves A(r)^3 - A(r)^2 = 2.
Other values are as follows.
A(t) = 3/2 at t = 0.1762576405478293392948378476047094214871919048852854...
with A(-t) = 0.9457634131178785046715685513829104426794138117773372...
A(1/6) = 1.39045291214794641706750008755820521981873579773148377...
A(-1/6) = 0.92547553450368274047514062093278734252641968691372863...
A(1/7) = 1.26282273990610251025800463852287012565418776197621997...
A(-1/7) = 0.91531855101291210815598364280272856428949318592006407...
A(1/8) = 1.20403758075277993770588254622742634950821058062345547...
A(-1/8) = 0.91758011120888933832570407861048171782335413914549218...

Links

Crossrefs

Cf. A368629, A368626, A368628.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A,x,-x);
A = 1 + x*(A^2 + B^2)/2 + x*(A^3 - B^3)/2 ; ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^3 - A(-x)^3)/2.
(2) A(x) = A(-x) + x*A(x)^2 + x*A(-x)^2.
(3) A(x) = 2 - A(-x) + x*A(x)^3 - x*A(-x)^3.
(4.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).

A368629 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.

Original entry on oeis.org

1, 1, 4, 9, 88, 210, 2644, 6493, 91992, 229646, 3484008, 8789562, 139443168, 354379540, 5801987316, 14824740645, 248459660984, 637465292438, 10878564788984, 28001827694446, 484778825103504, 1251132971284668, 21915195896364296, 56682787977509650, 1002570518541796720
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 88*x^4 + 210*x^5 + 2644*x^6 + 6493*x^7 + 91992*x^8 + 229646*x^9 + 3484008*x^10 + 8789562*x^11 + 139443168*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 210*x^4 + 668*x^5 + 6493*x^6 + 21538*x^7 + 229646*x^8 + 779772*x^9 + 8789562*x^10 + ...
A(x)^4 = 1 + 4*x + 22*x^2 + 88*x^3 + 605*x^4 + 2644*x^5 + 20114*x^6 + 91992*x^7 + 741154*x^8 + 3484008*x^9 + 29125100*x^10 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 210*x^5 + 6493*x^7 + 229646*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 210*x^4 + 6493*x^6 + 229646*x^8 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^4:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 88*x^4 + 2644*x^6 + 91992*x^8 + 3484008*x^10 + ...
(A(x)^4 - A(-x)^4)/2 = 4*x + 88*x^3 + 2644*x^5 + 91992*x^7 + 3484008*x^9 + ...
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071....

Links

Crossrefs

Cf. A368633, A368628, A368593, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + x*(A^4 - B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.
(2) A(x) = A(-x) + x*A(x)^2 + x*A(-x)^2.
(3) A(x) = 2 - A(-x) + x*A(x)^4 - x*A(-x)^4.
(4) A(x) = 2 - A(-x) + (A(x) - A(-x))*(A(x)^2 - A(-x)^2).
(5.a) A(x) = (1 - sqrt(1 - 4*x*A(-x) - 4*x^2*A(-x)^2)) / (2*x).
(5.b) A(-x) = (sqrt(1 + 4*x*A(x) - 4*x^2*A(x)^2) - 1) / (2*x).
(6) (A(x) + A(-x))/2 = 1/(1 - (A(x) - A(-x))^2).
(7.a) Sum_{n>=0} a(n) * (sqrt(2) - 1)^n/3^n = sqrt(2).
(7.b) Sum_{n>=0} a(n) * (1 - sqrt(2))^n/3^n = 1.

A368628 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.

Original entry on oeis.org

1, 1, 2, 14, 32, 345, 810, 10492, 24880, 356252, 848992, 12946094, 30942208, 492621678, 1179648292, 19379467704, 46468665184, 781821568212, 1876521420624, 32169136799832, 77270414837888, 1344812759618473, 3232175494812466, 56957048059132524, 136958995341531504
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = (4^k - 1)/3 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 14*x^3 + 32*x^4 + 345*x^5 + 810*x^6 + 10492*x^7 + 24880*x^8 + 356252*x^9 + 848992*x^10 + 12946094*x^11 + 30942208*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 32*x^3 + 96*x^4 + 810*x^5 + 2634*x^6 + 24880*x^7 + 84668*x^8 + 848992*x^9 + 2974649*x^10 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 84*x^3 + 345*x^4 + 2324*x^5 + 10492*x^6 + 74540*x^7 + 356252*x^8 + 2609552*x^9 + 12946094*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 32*x^4 + 810*x^6 + 24880*x^8 + 848992*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 32*x^3 + 810*x^5 + 24880*x^7 + 848992*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 14*x^3 + 345*x^5 + 10492*x^7 + 356252*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 345*x^4 + 10492*x^6 + 356252*x^8 + ...

Links

Crossrefs

Cf. A368629, A368593, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 - B^2)/2 + x*(A^4 + B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*(A(x)^4 + A(-x)^4)/2.
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*A(x)^4 + x*A(-x)^4.
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).

A368593 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).

Original entry on oeis.org

1, 1, 2, 7, 18, 78, 220, 1043, 3090, 15402, 47044, 242126, 755076, 3973820, 12580344, 67303139, 215511330, 1167556434, 3772175860, 20640707866, 67167649868, 370510806212, 1212836703304, 6735128062542, 22156120392276, 123731147310820, 408741630687656, 2293595176625340
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2024

Keywords

Comments

Conjecture: a(n) is odd when n = 2^k - 1 for k >= 0, and even elsewhere.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 18*x^4 + 78*x^5 + 220*x^6 + 1043*x^7 + 3090*x^8 + 15402*x^9 + 47044*x^10 + 242126*x^11 + 755076*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 54*x^4 + 220*x^5 + 717*x^6 + 3090*x^7 + 10562*x^8 + 47044*x^9 + 165858*x^10 + 755076*x^11 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 205*x^4 + 836*x^5 + 3178*x^6 + 13192*x^7 + 51490*x^8 + 216808*x^9 + 862588*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 18*x^4 + 220*x^6 + 3090*x^8 + 47044*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 18*x^3 + 220*x^5 + 3090*x^7 + 47044*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 7*x^3 + 78*x^5 + 1043*x^7 + 15402*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 205*x^4 + 3178*x^6 + 51490*x^8 + ...
sqrt( (A(x)^4 + A(-x)^4)/2 ) = 1 + 7*x^2 + 78*x^4 + 1043*x^6 + 15402*x^8 + ...

Links

Crossrefs

Cf. A368628, A368629, A368626, A368627.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n,
A = Vec(1 + x*(Ser(A)^2 - subst(Ser(A)^2,x,-x))/2 + x*sqrt( (Ser(A)^4 + subst(Ser(A)^4,x,-x))/2 ) +x*O(x^#A) ) );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).
(2) A(x) = 2 - A(-x) + x*A(x)^2 - x*A(-x)^2.
(3) A(x) = A(-x) + x*sqrt( (A(x)^4 + x*A(-x)^4)/2 ).
(4.a) A(x) = (1 - sqrt(1-8*x + 4*x*A(-x) + 4*x^2*A(-x)^2)) / (2*x).
(4.b) A(-x) = (sqrt(1+8*x - 4*x*A(x) + 4*x^2*A(x)^2) - 1) / (2*x).
(5) (A(x) + A(-x))/2 = 1/(1 - x*(A(x) - A(-x))).
Showing 1-4 of 4 results.

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