A020714 a(n) = 5 * 2^n.

5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 0

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

Same as Pisot sequences E(5,10), L(5,10), P(5,10), T(5,10). See A008776 for definitions of Pisot sequences.

The first differences are the sequence itself. - Alexandre Wajnberg & Eric Angelini, Sep 07 2005

5 times powers of 2. - Omar E. Pol, Dec 16 2008

Subsequence of A051916. - Reinhard Zumkeller, Mar 20 2010

With the addition of "2, 3," at the beginning, this sequence gives terms (n + 3) through the first term greater than 2^n, for n odd, of the negabinary Keith sequence for 2^n, thus proving that with the exception of 2 itself, no odd-indexed power of 2 is a negabinary Keith number (see A188381). - Alonso del Arte, Feb 02 2012

Let b(0) = 5 and b(n+1) = the smallest number not in the sequence such that b(n+1) - Product_{i=0..n} b(i) divides b(n+1) - Sum_{i=0..n} b(i). Then b(n+2) = a(n) for n > 0. - Derek Orr, Jan 15 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..238
• John Elias, Illustration: Alternating Tetrahedrons of Tetrahedrons
• Tanya Khovanova, Recursive Sequences.
• Petro Kosobutskyy, Anastasiia Yedyharova, and Taras Slobodzyan, From Newton's binomial and Pascal's triangle to Collatz's problem, Comp. Des. Sys., Theor. Practice (2023) Vol. 5, No. 1, 121-127.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1003.
• Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016. See Table 1.
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Row sums of (4, 1)-Pascal triangle A093561.

Row sums of (9, 1)-Pascal triangle A093644.

Row sums of (1, 4)-Pascal triangle A095666 (with leading 4).

Cf. A000079, A001045, A008776, A051916, A118416, A173786, A188381.

Programs

Magma

[5*2^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

Mathematica

Table[5*2^n, {n, 0, 31}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *)
NestList[2#&,5,40] (* Harvey P. Dale, Mar 13 2022 *)

PARI

a(n)=5<Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 5*2^n. a(n) = 2*a(n-1).

G.f.: 5/(1-2*x).

If m is a term greater than 5 of this sequence then m = 5*phi(phi(m)). - Farideh Firoozbakht, Aug 16 2005

a(n) = A118416(n+1,3) for n>2. - Reinhard Zumkeller, Apr 27 2006

a(n) = A000079(n)*5. - Omar E. Pol, Dec 16 2008

a(n) = A173786(n+2,n) for n > 1. - Reinhard Zumkeller, Feb 28 2010

a(n) = A001045(n+4) - A001045(n). - Paul Curtz, Nov 08 2012

Sum_{n>=1} 1/a(n) = 2/5. - Amiram Eldar, Oct 28 2020

E.g.f.: 5*exp(2*x). - Stefano Spezia, May 15 2021