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.
%I A380710 #12 Feb 19 2025 10:27:25
%S A380710 1,1,3,8,19,52,130,350,887,2386,6178,16318,42618,112632,295072,777628,
%T A380710 2039543,5379446,14139050,37212510,97869194,257724328,677880176,
%U A380710 1784741604,4694887026,12362045980,32529481476,85628088892,225332403940,593217232816,1561270271280,4109624293656,10816272052191
%N A380710 G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
%C A380710 Conjecture: for n > 0, a(n) is odd iff n is a power of 2.
%C A380710 Given radius of convergence r of series A(x), A(x) diverges at x = r, but abs(1/A(x)) at x = r equals 2 and abs(1/A(x)^2) at x = r equals 1/r, where r = 0.3799058095503261961981901830197771776983290071269961001504254947858599...
%C A380710 a(n) ~ c/r^n where 1/r = 2.63223139752362698799211..., and c = 0.3834031741009606925669633625765371168044864071774006287711316534258785...
%H A380710 Paul D. Hanna, <a href="/A380710/b380710.txt">Table of n, a(n) for n = 0..1030</a>
%F A380710 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380710 (1) A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
%F A380710 (2) A(x) = 1 / (1 - x*abs( 1/A(x)^2 )).
%F A380710 (3) abs(1/A(x)) = 2 - 1/A(x).
%F A380710 (4) abs(1/A(x)) = 1 + x*abs( 1/A(x)^2 ).
%F A380710 (5) abs(1/A(x)) = abs( abs(1/A(x))/A(x) ) + x*abs( 1/A(x)^2 )/A(x).
%F A380710 (6) abs( abs(1/A(x))/A(x) ) = 2 - 2*abs(1/A(x)) + abs(1/A(x))^2.
%F A380710 (7) abs( 1/A(x)^2 ) = A(x) * (2 - abs(1/A(x))) * (abs(1/A(x)) - 1)/x.
%F A380710 (8) A(x) = 1 + A(x)^2 * (abs(1/A(x)) - 1) * (2 - abs(1/A(x))).
%e A380710 G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 19*x^4 + 52*x^5 + 130*x^6 + 350*x^7 + 887*x^8 + 2386*x^9 + 6178*x^10 + 16318*x^11 + 42618*x^12 + ...
%e A380710 where A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
%e A380710 RELATED SERIES.
%e A380710 A(x)^2 = 1 + 2*x + 7*x^2 + 22*x^3 + 63*x^4 + 190*x^5 + 542*x^6 + 1576*x^7 + 4447*x^8 + 12702*x^9 + 35694*x^10 + ...
%e A380710 1/A(x) = 1 - x - 2*x^2 - 3*x^3 - 2*x^4 - 6*x^5 - 4*x^6 - 21*x^7 - 2*x^8 - 94*x^9 - 52*x^10 - 270*x^11 - 84*x^12 + ...
%e A380710 The absolute value of the series 1/A(x) begins
%e A380710 abs(1/A(x)) = 1 + x + 2*x^2 + 3*x^3 + 2*x^4 + 6*x^5 + 4*x^6 + 21*x^7 + 2*x^8 + 94*x^9 + 52*x^10 + 270*x^11 + 84*x^12 + ...
%e A380710 where abs(1/A(x)) = 2 - 1/A(x).
%e A380710 The absolute value of the series 1/A(x)^2 starts as
%e A380710 abs( 1/A(x)^2 ) = 1 + 2*x + 3*x^2 + 2*x^3 + 6*x^4 + 4*x^5 + 21*x^6 + 2*x^7 + 94*x^8 + 52*x^9 + 270*x^10 + 84*x^11 + 1420*x^12 + ...
%e A380710 where abs(1/A(x)) = 1 + x*abs( 1/A(x)^2 ).
%e A380710 The occurrence of signs in the expansion of 1/A(x)^2 has no obvious pattern.
%e A380710 SPECIFIC VALUES.
%e A380710 A(t) = 6 at t = 0.3521253776792082580595807444750553302449897977015...
%e A380710 A(t) = 5 at t = 0.3456635940865550514487387712620006990835608970892...
%e A380710 A(t) = 4 at t = 0.3353445841109354623507968372790782182828383144865...
%e A380710 A(t) = 3 at t = 0.3163390965835750115994353781504066116184311812558...
%e A380710 A(t) = 2 at t = 0.2703238890812296559650050596866021785482700845665...
%e A380710 A(t) = 3/2 at t = 0.21007722302555848449805443502768527123106826520...
%e A380710 A(1/3) = 3.8561630489436922241277332770003463055663996660504...
%e A380710 A(1/4) = 1.7804507530929577349684197763505149006496008002510...
%e A380710 A(1/5) = 1.4486680710862436038990844874974598495016144300066...
%e A380710 A(1/6) = 1.3133683293052424032190784618054973634892830723346...
%e A380710 A(1/7) = 1.2401905953633440750393755932922609861657646670157...
%e A380710 A(1/8) = 1.1944676144162474770850469959020275729350893069119...
%e A380710 A(1/9) = 1.1632456733394683634583953215349829285608074021805...
%e A380710 A(1/10) = 1.140597094866485485300620048216088625981556459200...
%e A380710 Let B(x) = abs(1/A(x)^2) then B(x) = (1 - 1/A(x))/x with
%e A380710 B(r) = 1/r = 2.63223139752362698799211074224388216591957118984454...
%e A380710 B(1/3) = 2.2220245975279023880827256557743784168347448472346...
%e A380710 B(1/4) = 1.7533779055380819630180398010000489157697985459653...
%e A380710 B(1/5) = 1.5485537371919237697954310652528503110387458910072...
%e A380710 B(1/6) = 1.4315938140719938763450788025656509230188193709550...
%e A380710 B(1/7) = 1.3557062711403811961362651790083206606967946565600...
%e A380710 B(1/8) = 1.3024555011399714687578982713429488443700165977339...
%o A380710 (PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + x*A*Ser(abs(Vec(1/(A^2 +x*O(x^n))))) ); polcoef(A, n)}
%o A380710 for(n=0, 40, print1(a(n), ", "))
%Y A380710 Cf. A380708, A380709.
%K A380710 nonn
%O A380710 0,3
%A A380710 _Paul D. Hanna_, Feb 18 2025