History of Shapiro polynomials $P_n$

 Title: Shapiro polynomials $P_n$-Definition: The Shapiro polynomials CITE{Wiki} are the polynomials $P_n(x)\in\mathbb-  Z[x]$ defined with $Q_n(x)$ by $P_0=Q_0=1$, $P_{n+1}=P_n+x^{2^n}Q_n$, and $Q_{n+1}=P_n-x^{2^n}Q_n$.+Definition: The Shapiro polynomials CITE{Wiki}, also called the Rudin-Shapiro polynomials,+  are the pair of polynomial sequences $P_n,Q_n\in\mathbb Z[x]$ defined by $P_0=Q_0=1$,+  $P_{n+1}=P_n+x^{2^n}Q_n$, and $Q_{n+1}=P_n-x^{2^n}Q_n$. This table holds $P_n$. Parameters:   n:

   comment-index: 'The table is indexed by the construction parameter $n$, not by the     degree: $P_n$ has degree $2^n-1$.'-  comment-variable: This table writes the polynomial variable as $x$. The natural-    domain for the complementary identity in CITE{formula-complementary} is the unit-    circle.+  comment-variable: This table writes the polynomial variable as $x$. Formulas:   formula-rudin-shapiro-coefficients: Equivalently, $P_n(x)=\sum_{j=0}^{2^n-1}a_jx^j$,-    where $a_j=1$ if the binary expansion of $j$ has an even number of adjacent $11$-    pairs and $a_j=-1$ otherwise.+    where $a_j=1$ if the binary expansion of $j$ has an even number of pairs of consecutive+    ones, counting overlapping pairs separately, and $a_j=-1$ otherwise.   formula-one-polynomial-recurrence: The polynomials also satisfy $P_{n+1}(x)=P_n(x^2)+xP_n(-x^2)$.+  formula-q-from-p: $Q_n(x)=(-1)^nx^{2^n-1}P_n(-1/x)$.   formula-complementary: If $d=2^n-1$, then $P_n(x)x^dP_n(1/x)+Q_n(x)x^dQ_n(1/x)=2^{n+1}x^d$.     On the unit circle this is $|P_n(z)|^2+|Q_n(z)|^2=2^{n+1}$.-  formula-specialisations: $P_n(1)=2^{\lfloor(n+1)/2\rfloor}$, and $P_n(-1)=0$ for-    odd $n$, while $P_0(-1)=1$ and $P_n(-1)=2^{n/2}$ for positive even $n$.+  formula-specialisations: $P_n(1)=2^{\lfloor(n+1)/2\rfloor}$. Also, $P_n(-1)=0$ for+    odd $n$ and $P_n(-1)=2^{n/2}$ for even $n$. Programs:   program-sage:

       from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing\n\       \nR = PolynomialRing(ZZ, \"x\")\nx = R.gen()\n\ndef shapiro_pair(n):\n    P\-      \ = R.one()\n    Q = R.one()\n    for k in range(n):\n        P, Q = P + x^(2^k)*Q,\-      \ P - x^(2^k)*Q\n    return P, Q\n\nshapiro_pair(8)[0]      # the next polynomial\+      \ = R.one()\n    Q = R.one()\n    for k in range(n):\n        P, Q = P + x**(2**k)*Q,\+      \ P - x**(2**k)*Q\n    return P, Q\n\nshapiro_pair(8)[0]      # the next polynomial\       \ after this table" Similar tables:

   rigour: exact   complete: 'no'-  complete-note: it holds every construction index $n$ with $0\leq n\leq7$+  complete-note: it holds every construction index $n$ with $0\leq n\leq7$, stopping+    before the printed polynomials reach about two thousand characters; $P_8$ has+    1,932 characters   rigour details: The generator computes $P_n$ and $Q_n$ by the exact recurrence in     $\mathbb Z[x]$. Before the draft was filled, every entry was checked against the