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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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...

LINKS

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

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.

EXAMPLE

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) +...

PROG

(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), ", "))

CROSSREFS

Cf. A275691, A275751, A275752.

Row sums of triangle A275670.

Sequence in context: A111099 A000632 A090245 * A006958 A036617 A007902

Adjacent sequences:  A274962 A274963 A274964 * A274966 A274967 A274968

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 16 2016

STATUS

approved

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Last modified August 15 01:47 EDT 2020. Contains 336485 sequences. (Running on oeis4.)