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).

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

1,2

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 = 1..300

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).

Example

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).

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(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

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

Sequence in context: A077002 A268224 A003704 * A236409 A348468 A000250

Adjacent sequences: A292178 A292179 A292180 * A292182 A292183 A292184

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2017

Status

approved