A372533 - OEIS

%I #6 May 29 2024 06:58:06

%S 1,1,1,4,6,9,14,33,47,63,121,203,312,469,867,1482,1969,3288,6470,9376,

%T 13429,24872,42253,61377,97504,175250,277678,413675,700594,1193566,

%U 1842503,2889154,4901502,8051064,12455432,20026636,34079228,54609720,83775247,140490784,235393569

%N Expansion of g.f. A(x) satisfying A(x) = A(x^3) / A(x^2 - x^3 - x^4).

%H Paul D. Hanna, <a href="/A372533/b372533.txt">Table of n, a(n) for n = 1..2200</a>

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

%F (1) A(x) = A(x^3) / A(x^2 - x^3 - x^4).

%F (2) A(x) = A(x^9) / ( A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12) ).

%F (3) A(x) = A(x^27) / ( A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12) * A(x^18 - x^27 - x^36) ).

%F (4) A(x) = A(x^(3^n)) / Product_{k=0..n-1) A(x^(2*3^k) - x^(3*3^k) - x^(4*3^k)) for n > 0.

%F lim_{n->oo} a(n+1)/a(n) = (sqrt(5) + 1)/2, the golden ratio (A001622).

%e G.f.: A(x) = x + x^2 + x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 33*x^8 + 47*x^9 + 63*x^10 + 121*x^11 + 203*x^12 + 312*x^13 + 469*x^14 + 867*x^15 + ...

%e where A(x^3) = A(x) * A(x^2 - x^3 - x^4).

%e RELATED SERIES.

%e A(x^2 - x^3 - x^4) = x^2 - x^3 - 2*x^5 - x^7 + 5*x^8 - 11*x^9 + 14*x^10 - x^11 + 18*x^12 - 26*x^13 + 13*x^14 - 58*x^15 + 53*x^16 - 160*x^17 + 360*x^18 + ...

%e A(x^6 - x^9 - x^12) = x^6 - x^9 - 2*x^15 - x^21 + 5*x^24 - 11*x^27 + 14*x^30 - x^33 + 18*x^36 - 26*x^39 + 13*x^42 - 58*x^45 + 53*x^48 - 160*x^51 + ...

%e where A(x^9) = A(x) * A(x^2 - x^3 - x^4) * A(x^6 - x^9 - x^12).

%e SPECIFIC VALUES.

%e A(sqrt(2) - 1) = 1.

%e A(3/5) = A(27/125) / A(9/625),

%e A(1/2) = A(1/8) / A(1/16),

%e A(1/3) = A(1/27) / A(5/81),

%e where

%e A(3/5) = 19.555114187191014691309930863886185708293476404730662191...

%e A(1/2) = 2.1550667641040265957227997815239238132154727134090997030...

%e A(1/3) = 0.5842757997708565788965210514702465577564629424946531777...

%e A(1/4) = 0.3534527422903162362456108551975145834593467179842488057...

%e A(1/5) = 0.2571938792455848507418753018381908621085405095830136782...

%e A(1/8) = 0.1437811945137175769699819845041308139100698704315447556...

%e A(1/16) = 0.066717744855340897966898686726339689305046583480519537...

%o (PARI) {a(n) = my(Ax = x); for(m=1, n, Ax = truncate(Ax);

%o Ax = subst(Ax, x, x^3 +x^3*O(x^m)) / subst(Ax, x, x^2 - x^3 - x^4 +x^3*O(x^m)) ); polcoeff(Ax, n)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A372529, A001622.

%K nonn

%O 1,4

%A _Paul D. Hanna_, May 28 2024