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A232544 G.f. is the limit of n-th degree polynomial P(n,x) = P(n-1,x) + a(n)*x^n that yields the least sum of squares of coefficients in P(n,x)^n, where a(n) = {-1,+1}, starting with a(0) = a(1) = 1. 0
1, 1, -1, 1, 1, 1, -1, 1, -1, -1, -1, -1, 1, 1, -1, 1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, -1, -1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0

COMMENTS

It appears that if the value of a(n) results in the least sum of squares of coefficients in P(n,x)^n, then it also results in the least sum of squares of coefficients in P(n,x)^k for all k>n.

LINKS

Table of n, a(n) for n=0..89.

EXAMPLE

G.f.: A(x) = 1 + x - x^2 + x^3 + x^4 + x^5 - x^6 + x^7 - x^8 - x^9 - x^10 - x^11 + x^12 + x^13 - x^14 + x^15 - x^16 - x^17 - x^18 - x^19 + x^20 +...

Illustrate generating method.

Given P(1,x) = 1+x, to determine a(2), we compare the coefficients in (P(1,x) + x^2)^2 to (P(1,x) - x^2)^2 to see which has the least sum of squares of coefficients:

(1+x+x^2)^2 = 1 + 2*x + 3*x^2 + 2*x^3 + x^4;

(1+x-x^2)^2 = 1 + 2*x - x^2 - 2*x^3 + x^4;

the sum of squares of coefficients is 19 and 11, respectively; thus a(2) = -1 since it yields the least sum of squares.

Then since P(2,x) = 1+x-x^2, to determine a(3), we compare the coefficients in (P(2,x) + x^3)^3 to (P(2,x) - x^3)^3 to see which has the least sum of squares of coefficients:

(1+x-x^2+x^3)^3 = 1 + 3*x - 2*x^3 + 6*x^4 - 4*x^6 + 6*x^7 - 3*x^8 + x^9;

(1+x-x^2-x^3)^3 = 1 + 3*x - 8*x^3 - 6*x^4 + 6*x^5 + 8*x^6 - 3*x^8 - x^9;

the sum of squares of coefficients is 112 and 220, respectively; thus a(3) = +1 since it yields the least sum of squares. (Should they ever equal, choose +1 as the new term.)

Continuing in this way generates all the terms of this sequence.

PROG

(PARI) {A=[1, 1]; print1("1, 1, "); for(i=1, 60,

A=concat(A, y); P=truncate(Ser(A));

SUMSQ=Vec(P^(#A))*Vec(P^(#A))~;

SNEG=subst(SUMSQ, y, -1); SPOS=subst(SUMSQ, y, 1);

if(SNEG>=SPOS, t=1, t=-1); A[#A]=t; print1(t, ", "); )}

CROSSREFS

Sequence in context: A033999 A000012 A216430 * A162511 A157895 A063747

Adjacent sequences:  A232541 A232542 A232543 * A232545 A232546 A232547

KEYWORD

sign

AUTHOR

Paul D. Hanna, Nov 25 2013

STATUS

approved

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Last modified February 21 15:11 EST 2019. Contains 320374 sequences. (Running on oeis4.)