OFFSET
1,2
COMMENTS
a(n) = Sum_{k>=2} z(k)*v(k)^n where v(k) is the real positive solution to x^k = x + 1 (i.e., the k-th Pisot-Vijayaraghavan number) and z(k) is the real positive root of a polynomial P(k,x) with integer coefficients of degree k.
In particular a(n) is asymptotic to (1/10)*(5 + sqrt(5))*phi^n where phi is the golden ratio.
First P(k,x) are P(2,x) = 5x^2 - 5x - 1; P(3,x) = 23x^3 - 23x^2 + 8x - 1; P(4) = 283x^4 - 283x^3 + 105x^2 - 17x + 1; P(5) = 2869x^5 - 2869x^4 + 1154x^3 - 234x^2 + 24x - 1.
a(n) is the number of compositions of n into almost equal parts. It means the difference between the largest part and the smallest part is at most 1. For example, there are 6 compositions of 4 into almost equal parts, (4), (2,2), (2,1,1), (1,2,1), (1,1,2), (1,1,1,1). - Ran Pan, Oct 16 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
MATHEMATICA
Table[Sum[Binomial[k, Mod[n, k]], {k, n}], {n, 37}] (* Michael De Vlieger, Jul 30 2017 *)
PROG
(PARI) a(n)=sum(k=1, n, binomial(k, n%k))
(Magma) [&+[Binomial(k, n mod k): k in [1..n]]: n in [1..40]]; // Vincenzo Librandi, Jul 31 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Aug 20 2002
EXTENSIONS
Name from Benoit Cloitre, May 03 2003
STATUS
approved