The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A077221 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A077221

a(0) = 0 and then alternately even and odd numbers in increasing order such that the sum of any two successive terms is a square.
(history; published version)

#72 by Michel Marcus at Mon Jan 16 08:18:02 EST 2023

STATUS

reviewed

approved

#71 by Joerg Arndt at Mon Jan 16 02:31:29 EST 2023

STATUS

proposed

reviewed

#70 by Amiram Eldar at Mon Jan 16 01:27:23 EST 2023

STATUS

editing

proposed

#69 by Amiram Eldar at Mon Jan 16 01:06:47 EST 2023

LINKS

N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>.

<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

<a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>.

FORMULA

Sum_{n>=1} 1/a(n) = Pi^2/48 + tan(Pi/(2*sqrt(2)))*Pi /(4*sqrt(2)). - Amiram Eldar, Jan 16 2023

CROSSREFS

Cf. A032528, A069129, A077222, A131925, A032528, A139098, A194274, A195040, A195041, A195042, A195142.

STATUS

approved

editing

#68 by Charles R Greathouse IV at Thu Sep 08 08:45:07 EDT 2022

PROG

(MAGMAMagma) [2*n^2 - (n mod 2): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#67 by Susanna Cuyler at Tue May 04 01:06:21 EDT 2021

STATUS

reviewed

approved

#66 by Michel Marcus at Mon May 03 23:49:01 EDT 2021

STATUS

proposed

reviewed

#65 by Jon E. Schoenfield at Mon May 03 21:47:12 EDT 2021

STATUS

editing

proposed

#64 by Jon E. Schoenfield at Mon May 03 21:47:11 EDT 2021

COMMENTS

This sequence arises from reading the line from 0, in the direction 0, 1, ... and the same line from 0, in the direction 0, 8, ..., in the square spiral whose vertices are the triangular numbers A000217. Cf. A139591, etc. - Omar E. Pol, May 03 2008

FORMULA

a(2n) = 8*n^2, a(2n+1) = 8*n(n+1) + 1.

From Ralf Stephan, Mar 31 2003: (Start)

a(n) = 2*n^2 + 4*n + 1 [+1 if n is odd] with a(0)=1. G.f.: x*(x^2+6*x+1)/(1-x)^3/(1+x). - _Ralf Stephan_, Mar 31 2003

G.f.: x*(x^2+6*x+1)/(1-x)^3/(1+x). (End)

Row sums of triangle A131925; binomial transform of (1, 7, 2, 4, -8, 16, -32, ...). - Gary W. Adamson, Jul 29 2007

a(n) = a(-n); a(n+1) = A195605(n) - (-1)^n. - Bruno Berselli, Sep 22 2011

a(n) = 2*n^2 + ((-1)^n-1)/2. - Omar E. Pol, Sep 28 2011

STATUS

approved

editing

#63 by Alois P. Heinz at Sun Feb 18 11:53:02 EST 2018

STATUS

proposed

approved