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A077221
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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.
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26
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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
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0,3
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COMMENTS
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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
Contribution from Peter Luschny, Jul 12 2009: (Start)
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|. (End)
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
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]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Bruno Berselli, An origin of A077221 (illustration) (see Pol's comment).
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
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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)/(1-x)^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)^n-1)/2. - Omar E. Pol, Sep 28 2011
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MAPLE
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a := n -> 2*n^2 - (n mod 2); [From Peter Luschny, Jul 12 2009]
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MATHEMATICA
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a=1; lst={a}; Do[b=n^2-a; AppendTo[lst, b]; a=b, {n, 3, 6!, 2}]; lst [From Vladimir Orlovsky, May 18 2009]
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PROG
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(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
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CROSSREFS
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Cf. A077222, A131925, A032528, A195041, A195042, A195142.
Column 8 of A195040. - Omar E. Pol, Sep 28 2011
Sequence in context: A028884 A099358 A077222 * A106648 A076980 A209376
Adjacent sequences: A077218 A077219 A077220 * A077222 A077223 A077224
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KEYWORD
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nonn,easy
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 03 2002
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EXTENSIONS
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Extended by Ralf Stephan, Mar 31 2003
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STATUS
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approved
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