[seqfan] Re: A269254, A034807 and Chebyshev polynomials

[Brad Klee]

After: http://list.seqfan.eu/pipermail/seqfan/2017-October/018063.html

Verification of induction base cases when n = T_3(j), j>2, ( Cubic
Chebyshev )
===============================================
Define the initial conditions as j-polynomials, with offset zero,

Z_0(n) = -1 + Sum_{i=0}^n T_i( T_3( j ) ) ;               n=0,1,2 . . . 8;

X_0(n) = -1 + Sum_{i=0}^n T_i( j );                          n=3*k and
n=3*k+2; k=0,1,2;
X_0(n) = -1 + Sum_{i=0}^{(n-1)/3} T_{3*i}( j );      n =3*k+1; k=0,1,2;

Y_0(n) = Z_0(n) / X_0(n);                                             n
=0,1,2 . . . 8.

These definitions must and do satisfy the X recurrence up to index 8:

X_0(n) = T_3(j)*X_0(n-3) - X_0(n-6),                 n=6,7,8 ;

Explicitly the initial condition vectors are,

Z_0 = {1, 1 - 3 j + j^3, -1 - 3 j + 9 j^2 + j^3 - 6 j^4 + j^6, -1 + 6 j +
  9 j^2 - 29 j^3 - 6 j^4 + 27 j^5 + j^6 - 9 j^7 + j^9,
 1 + 6 j - 27 j^2 - 29 j^3 + 99 j^4 + 27 j^5 - 111 j^6 - 9 j^7 +
  54 j^8 + j^9 - 12 j^10 + j^12,
 1 - 9 j - 27 j^2 + 111 j^3 + 99 j^4 - 351 j^5 - 111 j^6 + 441 j^7 +
  54 j^8 - 274 j^9 - 12 j^10 + 90 j^11 + j^12 - 15 j^13 + j^15, -1 -
  9 j + 54 j^2 + 111 j^3 - 441 j^4 - 351 j^5 + 1275 j^6 + 441 j^7 -
  1728 j^8 - 274 j^9 + 1275 j^10 + 90 j^11 - 545 j^12 - 15 j^13 +
  135 j^14 + j^15 - 18 j^16 + j^18, -1 + 12 j + 54 j^2 - 274 j^3 -
  441 j^4 + 1728 j^5 + 1275 j^6 - 4707 j^7 - 1728 j^8 + 6733 j^9 +
  1275 j^10 - 5643 j^11 - 545 j^12 + 2925 j^13 + 135 j^14 -
  951 j^15 - 18 j^16 + 189 j^17 + j^18 - 21 j^19 + j^21,
 1 + 12 j - 90 j^2 - 274 j^3 + 1275 j^4 + 1728 j^5 - 6733 j^6 -
  4707 j^7 + 17577 j^8 + 6733 j^9 - 26181 j^10 - 5643 j^11 +
  24207 j^12 + 2925 j^13 - 14553 j^14 - 951 j^15 + 5796 j^16 +
  189 j^17 - 1519 j^18 - 21 j^19 + 252 j^20 + j^21 - 24 j^22 + j^24}

X_0 = {1, 1, -1 + j + j^2, -1 - 2 j + j^2 + j^3, 1 - 3 j + j^3,
 1 + 3 j - 3 j^2 - 4 j^3 + j^4 + j^5, -1 + 3 j + 6 j^2 - 4 j^3 -
  5 j^4 + j^5 + j^6, -1 - 3 j + 9 j^2 + j^3 - 6 j^4 + j^6,
 1 - 4 j - 10 j^2 + 10 j^3 + 15 j^4 - 6 j^5 - 7 j^6 + j^7 + j^8}

Y_0 = {1, 1 - 3 j + j^3, 1 + 4 j - 4 j^2 - j^3 + j^4,
 1 - 8 j + 8 j^2 + 6 j^3 - 6 j^4 - j^5 + j^6,
 1 + 9 j - 30 j^3 + 27 j^5 - 9 j^7 + j^9,
 1 - 12 j + 12 j^2 + 43 j^3 - 43 j^4 - 34 j^5 + 34 j^6 + 10 j^7 -
  10 j^8 - j^9 + j^10,
 1 + 12 j - 12 j^2 - 79 j^3 + 79 j^4 + 103 j^5 - 103 j^6 - 53 j^7 +
  53 j^8 + 12 j^9 - 12 j^10 - j^11 + j^12,
 1 - 15 j + 140 j^3 - 378 j^5 + 450 j^7 - 275 j^9 + 90 j^11 -
  15 j^13 + j^15,
 1 + 16 j - 16 j^2 - 188 j^3 + 188 j^4 + 526 j^5 - 526 j^6 -
  596 j^7 + 596 j^8 + 339 j^9 - 339 j^10 - 103 j^11 + 103 j^12 +
  16 j^13 - 16 j^14 - j^15 + j^16}

Then define bilinear basis vectors,

B_i = {X_0(6+i)*Y_0(6+i),X_0(6+i)*Y_0(3+i),X_0(6+i)*Y_0(i),X_0(3+i
)*Y_0(6+i),X_0(3+i)*Y_0(3+i),X_0(3+i)*Y_0(i)}

With the earlier coefficient vectors v_{k}, again it's easy to verify that,

B_{i} dot v_{k} = 0;               i=0,1,2, k = 0,1;
===> B_{i} dot v_{2} = 0,   i=0,1,2.

For each of three subsequences, these j-polynomials satisfy the base case.

===============================================

A Mathematica algorithm for verifying the complete induction proof
evaluates in less than one tenth of one second.
( cf. second post of http://community.wolfram.com/groups/-/m/t/1210746 )
This essentially reproves the theorem earlier claimed by Andrew Hone,
( cf. http://list.seqfan.eu/pipermail/seqfan/2017-October/018035.html ),
with m^2 (m + 3) - 1 = 1 - 3 j + j^3 under index shift m = j - 1.

Also worth mentioning that definitions above for X(n) and Z(n) appear to be
valid for all values of n.

"What is in a name?" To my eyes, A269254 at first /was/ something of an
unseen rose. Now we have a budding sight, which combines simple linear
algebra and the Chebyshev polynomials. Yet a thorny question remains. Are
the Chebyshev cases the only cases with interesting factorization?

Regards,

Brad