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 1-2 of 2 results.

A292181 E.g.f. A(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that A'(x) = A(x) + B(x)*C(x).

Original entry on oeis.org

1, 3, 10, 45, 259, 1806, 14827, 140367, 1504576, 17972559, 236275711, 3387012720, 52572376669, 878552787927, 15729439074058, 300400031036745, 6095885898471775, 130982551821899862, 2970844882925223487, 70929401617621416243, 1778125633605205346584, 46698342082602696345555, 1282168260097348871508667, 36734284970419645262875200, 1096293296048734274708523433, 34026339905854090378353208155
Offset: 1

Views

Author

Paul D. Hanna, Sep 10 2017

Keywords

Comments

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

Examples

			E.g.f.: A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
RELATED SERIES.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Links

Crossrefs

Cf. A292182 (B), A292183 (C).

Programs

  • PARI
    {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f. A(x) and related functions B(x) and C(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
(5a) B(x) + i*A(x) = C(x) * exp( i*atan( A(x)/B(x) ) ).
(5b) A(x)/B(x) = Series_Reversion( Integral 1/( sqrt(1-x^4) * (1 + Integral 1/sqrt(1-x^4) dx) ) dx ).
Limit A292182(n)/A292181(n) = 1.
Limit A292183(n)/A292181(n) = sqrt(2).

A292183 E.g.f. C(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that C'(x) = C(x) + 2*A(x)*B(x).

Original entry on oeis.org

1, 1, 3, 13, 63, 361, 2499, 20581, 196311, 2116561, 25357563, 333765037, 4787007855, 74323701817, 1242253733619, 22243082373301, 424815246293319, 8620744969300321, 185235767397027627, 4201390722798810493, 100309092062158564959, 2514646421630798317897, 66041388198395188082595, 1813259146315114344920581, 51950114633383773360554679, 1550392693763071812557794801, 48120508780248064233484223067
Offset: 0

Views

Author

Paul D. Hanna, Sep 10 2017

Keywords

Comments

Here, the functions A(x), B(x), and C(x) are the e.g.f.s of sequences A292181, A292182, and A292183, respectively.
Another Pythagorean triple is the e.g.f.s of A289695, A193543, and A153302, which are related to the Lemniscate sine and cosine functions, sl(x) and cl(x).

Examples

			E.g.f.: C(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 63*x^4/4! + 361*x^5/5! + 2499*x^6/6! + 20581*x^7/7! + 196311*x^8/8! + 2116561*x^9/9! + 25357563*x^10/10! + 333765037*x^11/11! + 4787007855*x^12/12! + 74323701817*x^13/13! + 1242253733619*x^14/14! + 22243082373301*x^15/15! + 424815246293319*x^16/16! +...
where C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
RELATED SERIES.
A(x) = x + 3*x^2/2! + 10*x^3/3! + 45*x^4/4! + 259*x^5/5! + 1806*x^6/6! + 14827*x^7/7! + 140367*x^8/8! + 1504576*x^9/9! + 17972559*x^10/10! + 236275711*x^11/11! + 3387012720*x^12/12! + 52572376669*x^13/13! + 878552787927*x^14/14! + 15729439074058*x^15/15! + 300400031036745*x^16/16! +...
where A(x) = Integral A(x) + B(x)*C(x) dx.
B(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 35*x^4/4! + 226*x^5/5! + 1715*x^6/6! + 14701*x^7/7! + 141248*x^8/8! + 1515661*x^9/9! + 18048527*x^10/10! + 236581984*x^11/11! + 3386091821*x^12/12! + 52533799501*x^13/13! + 877993866290*x^14/14! + 15723411375931*x^15/15! + 300349139257727*x^16/16 +...
where B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
Squares of series.
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 134*x^4/4! + 1050*x^5/5! + 9158*x^6/6! + 89418*x^7/7! + 972470*x^8/8! + 11700378*x^9/9! + 154613222*x^10/10! + 2227684074*x^11/11! + 34757852054*x^12/12! + 583740365754*x^13/13! + 10497898450118*x^14/14! + 201267889853706*x^15/15! + 4097952119101814*x^16/16! +...
where A(x)^2 + B(x)^2 = C(x)^2.
B(x)^2 = 1 + 2*x + 6*x^2/2! + 26*x^3/3! + 150*x^4/4! + 1082*x^5/5! + 9222*x^6/6! + 89546*x^7/7! + 972726*x^8/8! + 11700890*x^9/9! + 154614246*x^10/10! + 2227686122*x^11/11! + 34757856150*x^12/12! + 583740373946*x^13/13! + 10497898466502*x^14/14! + 201267889886474*x^15/15! + 4097952119167350*x^16/16! +...
where B(x)^2 - A(x)^2 = exp(2*x).
C(x)^2 = 1 + 2*x + 8*x^2/2! + 44*x^3/3! + 284*x^4/4! + 2132*x^5/5! + 18380*x^6/6! + 178964*x^7/7! + 1945196*x^8/8! + 23401268*x^9/9! + 309227468*x^10/10! + 4455370196*x^11/11! + 69515708204*x^12/12! + 1167480739700*x^13/13! + 20995796916620*x^14/14! + 402535779740180*x^15/15! + 8195904238269164*x^16/16! +...
where C(x)^2 - 2*A(x)^2 = exp(2*x).

Links

Crossrefs

Cf. A292181 (A), A292182 (B).

Programs

  • PARI
    {a(n) = my(A=x,B=1,C=1); for(i=0,n, A = intformal(A + B*C + x*O(x^n));
B = 1 + intformal(B + A*C); C = 1 + intformal(C + 2*A*B)); n!*polcoeff(C,n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. C(x) and related functions A(x) and B(x) satisfy:
(1a) A(x)^2 + B(x)^2 = C(x)^2.
(1b) B(x)^2 - A(x)^2 = exp(x)^2.
(1c) C(x)^2 - 2*A(x)^2 = exp(x)^2.
(2a) A(x) = Integral A(x) + B(x)*C(x) dx.
(2b) B(x) = 1 + Integral B(x) + A(x)*C(x) dx.
(2c) C(x) = 1 + Integral C(x) + 2*A(x)*B(x) dx.
(3a) A(x) = exp(x) * sinh( Integral C(x) dx ).
(3b) B(x) = exp(x) * cosh( Integral C(x) dx ).
(3c) C(x) = exp(x) * cosh( Integral sqrt(2)*B(x) dx).
(3d) A(x) = exp(x) * sinh( Integral sqrt(2)*B(x) dx) / sqrt(2).
(4a) A(x) + B(x) = exp(x) * exp( Integral C(x) dx ).
(4b) C(x) + sqrt(2)*A(x) = exp(x) * exp( Integral sqrt(2)*B(x) dx ).
(4c) C(x) + sqrt(2)*B(x) = (1 + sqrt(2)) * exp(x) * exp( Integral sqrt(2)*A(x) dx ).
Limit A292183(n)/A292181(n) = sqrt(2).
Limit A292183(n)/A292182(n) = sqrt(2).
Showing 1-2 of 2 results.

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