A200442 Expansion of 1/(1-31*x+x^2).

1, 31, 960, 29729, 920639, 28510080, 882891841, 27341136991, 846692354880, 26220121864289, 811977085438079, 25145069526716160, 778685178242762881, 24114095455998933151, 746758273957724164800, 23125392397233450175649, 716140406040279231280319

Offset: 0

Author

Bruno Berselli, Nov 18 2011

Keywords

• nonn
• easy

Comments

A Diophantine property of these numbers: (a(n+1)-a(n-1))^2 - 957*a(n)^2 = 4. (See also comment in A200441.)

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,30}. - 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 (31,-1).

Crossrefs

Cf. A029548, A097313, A200441.

Programs

Magma

/* By the closed form: */ Z:=PolynomialRing(Integers()); N:=NumberField(x^2-957); S:=[(((31+r)/2)^n-1/((31+r)/2)^n)/r: n in [1..17]]; [Integers()!S[j]: j in [1..#S]];

Mathematica

LinearRecurrence[{31, -1}, {1, 31}, 17]

Maxima

makelist(sum((-1)^k*binomial(n-k,k)*31^(n-2*k),k,0,floor(n/2)),n,0,16);

PARI

Vec(1/(1-31*x+x^2)+O(x^17))

Formula

G.f.: 1/(1-31*x+x^2).

a(n) = 31*a(n-1)-a(n-2) with a(0)=1, a(1)=31.

a(n) = -a(-n-2) = (t^(n+1)-1/t^(n+1))/(t-1/t) where t=(31+sqrt(957))/2.

a(n) = sum((-1)^k*binomial(n-k, k)*31^(n-2k), k=0..floor(n/2)).

a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*30^k. - Philippe Deléham, Feb 10 2012

Product {n >= 0} (1 + 1/a(n)) = 1/29*(29 + sqrt(957)). - Peter Bala, Dec 23 2012

Product {n >= 1} (1 - 1/a(n)) = 1/62*(29 + sqrt(957)). - Peter Bala, Dec 23 2012