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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). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

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).

EXAMPLE

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).

PROG

(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), ", "))

CROSSREFS

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

Sequence in context: A107097 A202837 A180111 * A006923 A283667 A011272

Adjacent sequences:  A292180 A292181 A292182 * A292184 A292185 A292186

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 10 2017

STATUS

approved

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Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)