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.

A380710 G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).

Original entry on oeis.org

1, 1, 3, 8, 19, 52, 130, 350, 887, 2386, 6178, 16318, 42618, 112632, 295072, 777628, 2039543, 5379446, 14139050, 37212510, 97869194, 257724328, 677880176, 1784741604, 4694887026, 12362045980, 32529481476, 85628088892, 225332403940, 593217232816, 1561270271280, 4109624293656, 10816272052191
Offset: 0

Views

Author

Paul D. Hanna, Feb 18 2025

Keywords

Comments

Conjecture: for n > 0, a(n) is odd iff n is a power of 2.
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...
a(n) ~ c/r^n where 1/r = 2.63223139752362698799211..., and c = 0.3834031741009606925669633625765371168044864071774006287711316534258785...

Examples

			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 + ...
where A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
RELATED SERIES.
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 + ...
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 + ...
The absolute value of the series 1/A(x) begins
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 + ...
where abs(1/A(x)) = 2 - 1/A(x).
The absolute value of the series 1/A(x)^2 starts as
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 + ...
where abs(1/A(x)) = 1 + x*abs( 1/A(x)^2 ).
The occurrence of signs in the expansion of 1/A(x)^2 has no obvious pattern.
SPECIFIC VALUES.
A(t) = 6 at t = 0.3521253776792082580595807444750553302449897977015...
A(t) = 5 at t = 0.3456635940865550514487387712620006990835608970892...
A(t) = 4 at t = 0.3353445841109354623507968372790782182828383144865...
A(t) = 3 at t = 0.3163390965835750115994353781504066116184311812558...
A(t) = 2 at t = 0.2703238890812296559650050596866021785482700845665...
A(t) = 3/2 at t = 0.21007722302555848449805443502768527123106826520...
A(1/3) = 3.8561630489436922241277332770003463055663996660504...
A(1/4) = 1.7804507530929577349684197763505149006496008002510...
A(1/5) = 1.4486680710862436038990844874974598495016144300066...
A(1/6) = 1.3133683293052424032190784618054973634892830723346...
A(1/7) = 1.2401905953633440750393755932922609861657646670157...
A(1/8) = 1.1944676144162474770850469959020275729350893069119...
A(1/9) = 1.1632456733394683634583953215349829285608074021805...
A(1/10) = 1.140597094866485485300620048216088625981556459200...
Let B(x) = abs(1/A(x)^2) then B(x) = (1 - 1/A(x))/x with
B(r) = 1/r = 2.63223139752362698799211074224388216591957118984454...
B(1/3) = 2.2220245975279023880827256557743784168347448472346...
B(1/4) = 1.7533779055380819630180398010000489157697985459653...
B(1/5) = 1.5485537371919237697954310652528503110387458910072...
B(1/6) = 1.4315938140719938763450788025656509230188193709550...
B(1/7) = 1.3557062711403811961362651790083206606967946565600...
B(1/8) = 1.3024555011399714687578982713429488443700165977339...

Links

Crossrefs

Cf. A380708, A380709.

Programs

  • 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)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).
(2) A(x) = 1 / (1 - x*abs( 1/A(x)^2 )).
(3) abs(1/A(x)) = 2 - 1/A(x).
(4) abs(1/A(x)) = 1 + x*abs( 1/A(x)^2 ).
(5) abs(1/A(x)) = abs( abs(1/A(x))/A(x) ) + x*abs( 1/A(x)^2 )/A(x).
(6) abs( abs(1/A(x))/A(x) ) = 2 - 2*abs(1/A(x)) + abs(1/A(x))^2.
(7) abs( 1/A(x)^2 ) = A(x) * (2 - abs(1/A(x))) * (abs(1/A(x)) - 1)/x.
(8) A(x) = 1 + A(x)^2 * (abs(1/A(x)) - 1) * (2 - abs(1/A(x))).

A380711 G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).

Original entry on oeis.org

1, 1, 4, 13, 32, 147, 460, 1436, 5662, 17287, 60644, 209377, 688370, 2391256, 8105590, 27102666, 92744010, 312994179, 1067043874, 3659563265, 12430287670, 42225015449, 143808001426, 487301478188, 1658050374982, 5637187122368, 19153301908756, 65251831433398, 222042679730222, 755372323224172
Offset: 0

Views

Author

Paul D. Hanna, Feb 18 2025

Keywords

Comments

Conjecture: a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) for n >= 0.
Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 are the fibbinary numbers. (This is equivalent to the prior conjecture.)
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)^3) at x = r equals 1/r, where r = 0.29395473764961622547646584308431424060367446826992230069820567167994719...
a(n) ~ c/r^n where 1/r = 3.4018842764560748576..., and c = 0.2869732827715974104746811524073635455389390484876881563355879896659794...

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + 460*x^6 + 1436*x^7 + 5662*x^8 + 17287*x^9 + 60644*x^10 + 209377*x^11 + 688370*x^12 + ...
where A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 34*x^3 + 106*x^4 + 462*x^5 + 1639*x^6 + 5800*x^7 + 22722*x^8 + 78754*x^9 + 289543*x^10 + ...
1/A(x) = 1 - x - 3*x^2 - 6*x^3 - x^4 - 51*x^5 - 84*x^6 - 42*x^7 - 891*x^8 - 627*x^9 - 2373*x^10 + ...
The absolute value of the series 1/A(x) begins
abs(1/A(x)) = 1 + x + 3*x^2 + 6*x^3 + x^4 + 51*x^5 + 84*x^6 + 42*x^7 + 891*x^8 + 627*x^9 + 2373*x^10 + 7848*x^11 + 15624*x^12 + ...
where abs(1/A(x)) = 2 - 1/A(x).
The absolute value of the series 1/A(x)^3 starts as
abs( 1/A(x)^3 ) = 1 + 3*x + 6*x^2 + x^3 + 51*x^4 + 84*x^5 + 42*x^6 + 891*x^7 + 627*x^8 + 2373*x^9 + 7848*x^10 + 15624*x^11 + ...
where abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
The occurrence of signs in the expansion of 1/A(x)^3 has no obvious pattern.
SPECIFIC VALUES.
A(t) = 6 at t = 0.2776546403334668208899822312116577117579321589899...
A(t) = 5 at t = 0.27378228956266390389083139456755472304789559095846856286...
A(t) = 4 at t = 0.26751987468975853019031596683845328283581047906415763868...
A(t) = 3 at t = 0.25570653476578627566868647080655632304757429284241743094...
A(t) = 2 at t = 0.22541634177918528190705637551445570310188162066848813268...
A(t) = 3/2 at t = 0.181930310644869474243648515956090159019218115295765171...
A(1/4) = 2.7078534198843535187257007342533795310245294411311514375...
A(1/5) = 1.6527957689077139045813143038292189120779186108157811947...
A(1/6) = 1.4039503414912111190464124769746901176157597824012670753...
A(1/7) = 1.2919470482512907310654123055517832107265014355362879392...
A(1/8) = 1.2281933933225341024142993760196501004863261649342668152...
A(1/9) = 1.1870632020801295908616256565906659737605022656656501307...
A(1/10) = 1.158356714849802903775203606108124940003741201462273033...
Let B(x) = abs(1/A(x)^3) then B(x) = (1 - 1/A(x))/x with
B(r) = 1/r = 3.4018842764560748576093421240750088532575559256068992507649...
B(1/4) = 2.5228151676796334019994272154634466465512689587018470209...
B(1/5) = 1.9748228461981367841496533002109555899836788562358152424...
B(1/6) = 1.7263445702594598152254927641049269014234569252848538480...
B(1/7) = 1.5818212832524217283358477460759387507949300014779006395...
B(1/8) = 1.4863678281494130346658952345106428057380981383649536590...

Links

Crossrefs

Cf. A380708, A380709, A380710.

Programs

  • PARI
    {a(n) = my(A=1); for(i=1, n, A = 1 + x*A*Ser(abs(Vec(1/(A^3 +x*O(x^n))))) ); polcoef(H=A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).
(2) A(x) = 1 / (1 - x*abs( 1/A(x)^3 )).
(3) abs(1/A(x)) = 2 - 1/A(x).
(4) abs(1/A(x)) = 1 + x*abs( 1/A(x)^3 ).
(5) abs(1/A(x)) = abs( abs(1/A(x))/A(x) ) + x*abs( 1/A(x)^3 )/A(x).
(6) abs( abs(1/A(x))/A(x) ) = 2 - 2*abs(1/A(x)) + abs(1/A(x))^2.
(7) abs( 1/A(x)^3 ) = A(x) * (2 - abs(1/A(x))) * (abs(1/A(x)) - 1)/x.
(8) A(x) = 1 + A(x)^2 * (abs(1/A(x)) - 1) * (2 - abs(1/A(x))).
Showing 1-2 of 2 results.

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