A378256 G.f. A(x) satisfies A(x) = A(x^3)/A(x^2) - x^2.

1, -1, 1, -1, 0, -1, 1, 0, 2, -1, 2, -2, 1, -2, 0, -2, 0, -1, 2, 0, 1, -1, -4, -1, -4, 2, -2, 4, 0, 5, 1, 4, -2, 0, -2, 1, 8, 4, 19, -2, 20, -14, 16, -24, 19, -28, 34, -28, 45, -34, 33, -50, 6, -56, -6, -38, -3, -8, -12, 14, -59, 24, -119, 51, -138, 115, -116, 186, -90, 211, -100, 183, -105, 148, -10, 140, 188, 92, 374, -78, 470, -351, 548, -618, 717

Offset

1,9

Links

Paul D. Hanna, Table of n, a(n) for n = 1..4300

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x) = A(x^3)/A(x^2) - x^2.

(2) A(x) = -x^2 + A(x^3)/(-x^4 + A(x^6)/(-x^8 + A(x^12)/(-x^16 + A(x^24)/(-x^32 + A(x^48)/(-x^64 + A(x^96)/(-x^128 + ...)))))), a continued fraction.

Example

G.f.: A(x) = x - x^2 + x^3 - x^4 - x^6 + x^7 + 2*x^9 - x^10 + 2*x^11 - 2*x^12 + x^13 - 2*x^14 - 2*x^16 - x^18 + 2*x^19 + x^21 - x^22 - 4*x^23 - x^24 - 4*x^25 + 2*x^26 - 2*x^27 + 4*x^28 + 5*x^30 + x^31 + 4*x^32 - 2*x^33 - 2*x^35 + x^36 + ...
where A(x) = A(x^3)/A(x^2) - x^2.
SPECIFIC VALUES.
From A(x) = A(x^3)/A(x^2) - x^2 we have
A(1/2) = A(1/8) / A(1/4) - 1/4,
A(1/3) = A(1/27) / A(1/9) - 1/9,
A(1/4) = A(1/64) / A(1/16) - 1/16,
A(1/5) = A(1/125) / A(1/25) - 1/25,
A(1/6) = A(1/216) / A(1/36) - 1/36,
A(1/7) = A(1/343) / A(1/49) - 1/49,
A(1/8) = A(1/512) / A(1/64) - 1/64,
A(1/9) = A(1/729) / A(1/81) - 1/81,
where the specific values are as follows.
A(4/5) = 0.30923006090770859539731645485195402642444158131471...	
A(3/4) = 0.32891889249989327585881604453996382030884420026703...
A(2/3) = 0.32918051535512872777334887291138963012500689244768...
A(1/2) = 0.30807456963099643527933755965509929029590798821815...
A(1/3) = 0.24609144941732309549220831942169160497153547826516...
A(1/4) = 0.19904268485708382282652149202417356411854765570428...
A(1/5) = 0.16634975484625682676514479579249983559141497188569...
A(1/6) = 0.14272923891897737412397032134141464563253205856737...
A(1/7) = 0.12494069966095791135584820669189770792863975170565...
A(1/8) = 0.11108066068981510563403181798551106481000347893598...
A(1/9) = 0.09998309075221596797979995346417619188703574043821...
A(1/16) = 0.05882257598488413441703770424401558267078743230019...
A(1/25) = 0.03846143606835460816368358623635505438959504021096...
A(1/27) = 0.03571421602623835467065750082074808783436886434216...
A(1/36) = 0.02702701048921966330868811706085303465321986555076...
A(1/49) = 0.01999999645989753033919945070685437782674085909299...
A(1/64) = 0.01538461445329641823807684791822619943879542133372...
A(1/81) = 0.01219512166442285943887178676705913122273301682700...
A(1/125) = 0.00793650790373995341937875714718968589811241970188...
A(1/216) = 0.00460829492874875375815548342904719622768096145884...
A(1/343) = 0.00290697674397541206844007649139289143319748504823...
A(1/512) = 0.00194931773876300129273108556005106994060376579037...
A(1/729) = 0.00136986301369377320124923630586164361864578607526...

Prog

(PARI) \\ From formula: A(x) = A(x^3)/A(x^2) - x^2 \\
{a(n) = my(A=[0, 1]); for(m=1, n, A=concat(A, 0);
A[#A] = polcoef( subst(Ser(A), x, x^3)/subst(Ser(A), x, x^2) - Ser(A) - x^2, #A-1) ); A[n+1]}
for(n=1, 80, print1(a(n), ", "))

Crossrefs

Sequence in context: A334506 A098199 A309474 * A022828 A129406 A123018

Adjacent sequences: A378253 A378254 A378255 * A378260 A378261 A378263

Keyword

sign,new

Author

Paul D. Hanna, Nov 24 2024

Status

approved