Search: id:A077221
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%I A077221
%S A077221 0,1,8,17,32,49,72,97,128,161,200,241,288,337,392,449,512,577,648,721,
               800,
%T A077221 881,968,1057,1152,1249,1352,1457,1568,1681,1800,1921,2048,2177,2312,
%U A077221 2449,2592,2737,2888,3041,3200,3361,3528,3697,3872,4049,4232
%N 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.
%C A077221 The sequence 0,1,8,17,32,49,72,97,128,... 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 (info(AT)polprimos.com), 
               May 03 2008
%C A077221 Contribution from Peter Luschny (peter(AT)luschny.de), Jul 12 2009: (Start)
%C A077221 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
%C A077221 a(k) = |(P(2,0)-(-1)^k*P(2,2*k))/2|.
%F A077221 a(2n) = 8*n^2, a(2n+1) = 8*n(n+1) +1.
%F A077221 2n^2+4n+1+[n odd]. G.f.: (x^2+6x+1)/(1-x)^3/(1+x). - Ralf Stephan (ralf(AT)ark.in-berlin.de), 
               Mar 31 2003
%F A077221 Row sums of triangle A131925; binomial transform of (1, 7, 2, 4, -8, 
               16, -32,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 29 2007
%p A077221 a := n -> 2*n^2 - (n mod 2); [From Peter Luschny (peter(AT)luschny.de), 
               Jul 12 2009]
%t A077221 a=1;lst={a};Do[b=n^2-a;AppendTo[lst,b];a=b,{n,3,6!,2}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), May 18 2009]
%Y A077221 Cf. A077222.
%Y A077221 Cf. A131925.
%Y A077221 Sequence in context: A028884 A099358 A077222 this_sequence A106648 A076980 
               A159696
%Y A077221 Adjacent sequences: A077218 A077219 A077220 this_sequence A077222 A077223 
               A077224
%K A077221 nonn
%O A077221 0,3
%A A077221 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 03 2002
%E A077221 Extended by Ralf Stephan (ralf(AT)ark.in-berlin.de), Mar 31 2003

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