OFFSET
0,2
COMMENTS
Chebyshev polynomials S(n, 37).
A Diophantine property of these numbers: (a(n+1)-a(n-1))^2 - 1365*a(n)^2 = 4. - Bruno Berselli, Feb 09 2012
a(n) equals the number of 01-avoiding words of length n on alphabet {0,1,...,36}. - Milan Janjic, Jan 26 2015
LINKS
Bruno Berselli, Table of n, a(n) for n = 0..500
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (37,-1).
FORMULA
G.f.: 1/(1 - 37*x + x^2).
a(n) = Sum_{k=0..n} A049310(n,k)*37^k.
a(n) = 37*a(n-1) - a(n-2), n>=1; a(0)=1, a(1) = 37, a(-1) = 0.
a(n) = -a(-n-2) = (t^(n+1)-1/t^(n+1))/(t-1/t) where t=(37+sqrt(1365))/2. - Bruno Berselli, Feb 09 2012
a(n) = Sum_{k=0..n} A101950(n,k)*36^k. - Philippe Deléham, Feb 10 2012
Product {n >= 0} (1 + 1/a(n)) = 1/35*(35 + sqrt(1365)). - Peter Bala, Dec 23 2012
Product {n >= 1} (1 - 1/a(n)) = 1/74*(35 + sqrt(1365)). - Peter Bala, Dec 23 2012
MATHEMATICA
CoefficientList[Series[1/(1-37x+x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[ {37, -1}, {1, 37}, 20] (* Harvey P. Dale, Dec 13 2017 *)
PROG
(PARI) Vec(1/(1-37*x+x^2)+O(x^15))
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=NumberField(x^2-1365); S:=[(((37+r)/2)^n-1/((37+r)/2)^n)/r: n in [1..15]]; [Integers()!S[j]: j in [1..#S]];
(Maxima) makelist(sum((-1)^k*binomial(n-k, k)*37^(n-2*k), k, 0, floor(n/2)), n, 0, 14);
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Feb 09 2012
STATUS
approved