A292182 E.g.f. B(x) satisfies: A(x)^2 + B(x)^2 = C(x)^2, such that B'(x) = B(x) + A(x)*C(x).

1, 1, 2, 7, 35, 226, 1715, 14701, 141248, 1515661, 18048527, 236581984, 3386091821, 52533799501, 877993866290, 15723411375931, 300349139257727, 6095613429234730, 130983518612114231, 2970900143887175977, 70930381205350706888, 1778137090832694851161, 46698407537794612100459, 1282167191852237842607584, 36734238381564939631425737, 1096292258727541156091352361, 34026322932421876848090674594

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. B(x) and related functions A(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 ).

Limit A292181(n)/A292182(n) = 1.

Limit A292183(n)/A292182(n) = sqrt(2).

Example

E.g.f.: 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.
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.
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(B, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A292181 (A), A292183 (C).

Sequence in context: A006947 A182224 A317421 * A185054 A014307 A000154

Adjacent sequences: A292179 A292180 A292181 * A292183 A292184 A292185

Keyword

nonn

Author

Paul D. Hanna, Sep 10 2017

Status

approved