A036799 a(n) = 2 + 2^(n+1)*(n-1).

0, 2, 10, 34, 98, 258, 642, 1538, 3586, 8194, 18434, 40962, 90114, 196610, 425986, 917506, 1966082, 4194306, 8912898, 18874370, 39845890, 83886082, 176160770, 369098754, 771751938, 1610612738, 3355443202, 6979321858, 14495514626, 30064771074, 62277025794

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

This sequence is a part of a class of sequences of the type: a(n) = Sum_{i=0..n} (C^i)*(i^k). This sequence has C=2, k=1. Sequence A036800 has C=2, k=2. Suppose C >= 2, k >= 1 are integers. What is the general closed form for a(n)? - Ctibor O. Zizka, Feb 07 2008

Partial sums of A036289. - Vladimir Joseph Stephan Orlovsky, Jul 09 2011

a(n) is the number of swaps needed in the worst case, when successively inserting 2^(n+1) - 1 keys into an initially empty binary heap (thus creating a tree with n+1 full levels). - Rudy van Vliet, Nov 09 2015

a(n) is also the total path length of the complete binary tree of height n, with nodes at depths 0,...,n. Total path length is defined to be the sum of depths over all nodes. - F. Skerman, Jul 02 2017

For n >= 1, every number greater than or equal to a(n-1) can be written as a sum of (not necessarily distinct) numbers of the form 2^n - 2^k with 0 <= k < n. However, a(n-1) - 1 cannot be written in this way. See problem N1 from the 2014 International Mathematics Olympiad Shortlist. - Dylan Nelson, Jun 02 2023

References

• M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
• Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 3, 4, 9.
• Problem Selection Committee for the 2014 IMO, Shortlisted Problems with Solutions for the 55th International Mathematical Olympiad. See pp. 9, 68-70.
• Stanislav Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
• Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Crossrefs

Cf. A000337, A036289, A036800, A232599, A232600, A232601, A232602.

Programs

Haskell

a036799 n = (n-1)*2^(n+1) + 2  -- Reinhard Zumkeller, May 24 2012

Magma

[2+2^(n+1)*(n-1) : n in [0..40]]; // Wesley Ivan Hurt, Nov 12 2015

Maple

A036799:=n->2+2^(n+1)*(n-1): seq(A036799(n), n=0..40); # Wesley Ivan Hurt, Nov 12 2015

Mathematica

Accumulate[Table[n*2^n, {n, 0, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 09 2011 *)

PARI

a(n)=2+(n-1)<<(n+1) \\ Charles R Greathouse IV, Sep 28 2015

PARI

concat(0, Vec(2*x/((1-x)*(1-2*x)^2) + O(x^40))) \\ Altug Alkan, Nov 09 2015

Sage

[2^(n+1)*(n-1) +2 for n in (0..40)] # G. C. Greubel, Mar 29 2021

Formula

a(n) = (n-1) * 2^(n+1) + 2.

a(n) = 2 * A000337(n).

a(n) = Sum_{k=1..n} k*2^k. - Benoit Cloitre, Oct 25 2002

G.f.: 2*x/((1-x)*(1-2*x)^2). - Colin Barker, Apr 30 2012

a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3) for n > 2. - Wesley Ivan Hurt, Nov 12 2015

a(n) = Sum_{k=0..n} Sum_{i=0..n} k * binomial(k,i). - Wesley Ivan Hurt, Sep 21 2017

E.g.f.: 2*exp(x) - 2*(1-2*x)*exp(2*x). - G. C. Greubel, Mar 29 2021