A274965 G.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2) -...- x^n)^(1/2) -..., an infinite series of nested square roots.

1, 1, 2, 4, 9, 20, 46, 104, 238, 540, 1228, 2780, 6289, 14180, 31924, 71688, 160694, 359452, 802642, 1788988, 3980916, 8844064, 19618506, 43455324, 96121164, 212331796, 468445180, 1032216460, 2271818652, 4994434788, 10968013396, 24061103888, 52730956193, 115449870424, 252530306764, 551873275488, 1204991320660, 2628810554176, 5730295148952, 12480957518212, 27163290056278

Offset: 0

Author

Paul D. Hanna, Jul 16 2016

Keywords

• nonn

Comments

Odd terms occur at positions k*2^(k-1) for k>=0.

Limit a(n+1)/a(n) = 2, and A(x) diverges at x=1/2.

A(-1/2) = 1.0891636602638152861240865158090054430536947422594419370337760...

A(2/5) = 4.27983467184471084235872646732512184377478311914374590...

A(1/3) = 2.15485192359458408375371476779655861137906655796801630...

A(x) = 2 at x = 0.32026273178798900824351068844199852911740930864617900985902...

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 104*x^7 + 238*x^8 + 540*x^9 + 1228*x^10 +...
Illustration of the definition.
R1 = (A(x) - x)^(1/2);
R2 = ((A(x) - x)^(1/2) - x^2)^(1/2);
R3 = (((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2);
R4 = ((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2);
R5 = (((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2); ...
where the above series begin:
R1 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 36*x^7 + 78*x^8 + 168*x^9 + 364*x^10 + 786*x^11 + 1700*x^12 +...
R2 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 16*x^7 + 33*x^8 + 68*x^9 + 142*x^10 + 296*x^11 + 620*x^12 + 1296*x^13 +...
R3 = 1 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 16*x^8 + 32*x^9 + 65*x^10 + 132*x^11 + 270*x^12 + 552*x^13 + 1132*x^14 +...
R4 = 1 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 32*x^10 + 64*x^11 + 129*x^12 + 260*x^13 + 526*x^14 + 1064*x^15 +...
R5 = 1 + x^6 + 2*x^7 + 4*x^8 + 8*x^9 + 16*x^10 + 32*x^11 + 64*x^12 + 128*x^13 + 257*x^14 + 516*x^15 + 1038*x^16 +...
etc., so that 1 is obtained as a limit.
GENERATING METHOD.
The g.f. of this sequence can be obtained as a limit, as n grows, of the following process: start with 1 + x^n, then square the result and add x^(n-1), then square the result and add x^(n-2), then continue in this way until you reach x^1; this process is illustrated at n=6 as follows:
S6 = 1 + x^6,
S5 = S6^2 + x^5 = 1 + x^5 + 2*x^6 + x^12,
S4 = S5^2 + x^4 = 1 + x^4 + 2*x^5 + 4*x^6 + x^10 + 4*x^11 + 6*x^12 + 2*x^17 +...,
S3 = S4^2 + x^3 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + x^8 + 4*x^9 + 14*x^10 +...,
S2 = S3^2 + x^2 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 4*x^7 + 14*x^8 + 40*x^9 + 76*x^10 +...,
S1 = S2^2 + x = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 40*x^7 + 110*x^8 + 220*x^9 + 396*x^10 +...,
which matches the g.f. A(x) up to x^6.
RELATED SERIES.
Note that the bisections are self-convolutions of integer sequences:
sqrt( (A(x) + A(-x))/2 ) = 1 + x^2 + 4*x^4 + 19*x^6 + 92*x^8 + 446*x^10 + 2150*x^12 + 10280*x^14 + 48761*x^16 + 229558*x^18 + 1073278*x^20 + 4986624*x^22 + 23037102*x^24 + 105877968*x^26 + 484337300*x^28 +...+ A275751(n)*x^(2*n) +...
sqrt( x*(A(x) - A(-x))/2 ) = x + 2*x^3 + 8*x^5 + 36*x^7 + 166*x^9 + 770*x^11 + 3574*x^13 + 16560*x^15 + 76516*x^17 + 352498*x^19 + 1619014*x^21 + 7414134*x^23 + 33855996*x^25 + 154181234*x^27 + 700333366*x^29 +...+ A275752(n)*x^(2*n+1) +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1030

Crossrefs

Cf. A275691, A275751, A275752.

Row sums of triangle A275670.

Programs

PARI

{a(n) = my(A=1 +x*O(x^n)); for(k=0,n, A = A^2 + x^(n+1-k)); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f.: A(x) = G(x,1), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.

G.f.: A(x) = F(x)^2 + x, where F(x) is the g.f. of A275691.