

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.


31



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
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OFFSET

0,3


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 SwissKnife 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 nth stage in simple 2dimensional 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


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 sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,0,2,1).


FORMULA

a(2n) = 8*n^2, a(2n+1) = 8*n(n+1)+1.
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)/(1x)^3/(1+x).  Ralf Stephan, Mar 31 2003
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)^n1)/2.  Omar E. Pol, Sep 28 2011


EXAMPLE

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)


MAPLE

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


MATHEMATICA

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


PROG

(MAGMA) [2*n^2  (n mod 2): n in [0..50]]; // Vincenzo Librandi, Sep 22 2011
(Haskell)
a077221 n = a077221_list !! n
a077221_list = scanl (+) 0 a047522_list
 Reinhard Zumkeller, Jan 07 2012


CROSSREFS

Cf. A077222, A131925, A032528, A195041, A195042, A195142.
Sequence in context: A247117 A099358 A077222 * A226601 A106648 A209376
Adjacent sequences: A077218 A077219 A077220 * A077222 A077223 A077224


KEYWORD

nonn,easy


AUTHOR

Amarnath Murthy, Nov 03 2002


EXTENSIONS

Extended by Ralf Stephan, Mar 31 2003


STATUS

approved



