login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A072951 a(n) = Sum_{k=1..n} binomial(k, n mod k). 3

%I #27 Sep 08 2022 08:45:06

%S 1,2,4,6,11,15,27,39,63,100,159,247,403,641,1023,1644,2653,4264,6872,

%T 11081,17895,28899,46680,75420,121918,197113,318728,515420,833592,

%U 1348309,2181022,3528144,5707568,9233629,14938481,24168531,39102324

%N a(n) = Sum_{k=1..n} binomial(k, n mod k).

%C 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.

%C In particular a(n) is asymptotic to (1/10)*(5 + sqrt(5))*phi^n where phi is the golden ratio.

%C 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.

%C 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

%H Vincenzo Librandi, <a href="/A072951/b072951.txt">Table of n, a(n) for n = 1..1000</a>

%t Table[Sum[Binomial[k, Mod[n, k]], {k, n}], {n, 37}] (* _Michael De Vlieger_, Jul 30 2017 *)

%o (PARI) a(n)=sum(k=1,n,binomial(k,n%k))

%o (Magma) [&+[Binomial(k, n mod k): k in [1..n]]: n in [1..40]]; // _Vincenzo Librandi_, Jul 31 2017

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Aug 20 2002

%E Name from _Benoit Cloitre_, May 03 2003

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)