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.

0, 1, 8, 17, 32, 49, 72, 97, 128, 161, 200, 241, 288, 337, 392, 449, 512, 577, 648, 721, 800, 881, 968, 1057, 1152, 1249, 1352, 1457, 1568, 1681, 1800, 1921, 2048, 2177, 2312, 2449, 2592, 2737, 2888, 3041, 3200, 3361, 3528, 3697, 3872, 4049, 4232

Offset: 0

Author

Amarnath Murthy, Nov 03 2002

Keywords

• nonn
• easy

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

The general formula for alternating sums of powers of odd integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,0)-(-1)^k*P(n,2*k))/2. Here n=2, thus a(k) = |(P(2,0)-(-1)^k*P(2,2*k))/2|. - Peter Luschny, Jul 12 2009

Axis perpendicular to A046092 in the square spiral whose vertices are the triangular numbers A000217. See the comment above. - Omar E. Pol, Sep 14 2011

Column 8 of A195040. - Omar E. Pol, Sep 28 2011

Concentric octagonal numbers. A139098 and A069129 interleaved. - Omar E. Pol, Sep 17 2011

Subsequence of A194274. - Bruno Berselli, Sep 22 2011

Partial sums of A047522. - Reinhard Zumkeller, Jan 07 2012

Alternating sum of the first n odd squares in decreasing order, n >= 1. Also number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton. The rules are: on the infinite square grid, start with all cells OFF, so a(0) = 0. Turn a single cell to the ON state, so a(1) = 1. At each subsequent step, the neighbor cells of each cell of the old generation are turned ON, and the cells of the old generation are turned OFF. Here "neighbor" refers to the eight adjacent cells of each ON cell. See example. - Omar E. Pol, Feb 16 2014

Examples

			From _Omar E. Pol_, Feb 16 2014: (Start)
Illustration of initial terms as a cellular automaton:
.
.                                   O O O O O O O
.                     O O O O O     O           O
.           O O O     O       O     O   O O O   O
.     O     O   O     O   O   O     O   O   O   O
.           O O O     O       O     O   O O O   O
.                     O O O O O     O           O
.                                   O O O O O O O
.
.     1       8           17              32
.
(End)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Bruno Berselli, An origin of A077221 (illustration) (see Pol's comment).
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
• Index entries for sequences related to cellular automata.

Crossrefs

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

Programs

Haskell

a077221 n = a077221_list !! n
a077221_list = scanl (+) 0 a047522_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

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

Maple

a := n -> 2*n^2 - (n mod 2); # Peter Luschny, Jul 12 2009

Mathematica

a=1;lst={a};Do[b=n^2-a;AppendTo[lst,b];a=b,{n,3,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, May 18 2009 *)

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). (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

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

Extensions

Extended by Ralf Stephan, Mar 31 2003