%I A002939
%S A002939 0,2,12,30,56,90,132,182,240,306,380,462,552,650,756,870,992,1122,1260,
%T A002939 1406,1560,1722,1892,2070,2256,2450,2652,2862,3080,3306,3540,3782,4032,
%U A002939 4290,4556,4830,5112,5402,5700,6006,6320,6642,6972,7310,7656,8010,8372
%N A002939 2*n*(2*n-1).
%C A002939 Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 
               4 diagonals.
%C A002939 For n>1 this is the Engel expansion of cosh(1); cf. A006784 for Engel 
               expansion definition - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Mar 03 2002
%C A002939 a(n) = A125199(n,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Nov 24 2006
%C A002939 Apart from the first term, this sequence also gives the denominators 
               of the expansion of ln2=(1-1/2) + (1/3-1/4) + (1/5-1/6) + (1/7-1/
               8) + (1/9-1/10) + ... =(1/2) + (1/12) + (1/30) + (1/56) + (1/90) 
               + ... - Mohammad K. Azarian (azarian(AT)evansville.edu), Mar 21 2008
%C A002939 Twice hexagonal numbers. - Omar E. Pol (info(AT)polprimos.com), May 14 
               2008
%D A002939 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 2nd ed., 1994, p. 99.
%D A002939 A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion 
               methods: lattice model of dilute polymers, J. Statist. Phys., 67 
               (1992), 1083-1108; see Eq 4b.
%D A002939 R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 
               of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett 
               et al., Peters, 2003.
%F A002939 Sum_{n >= 0} 1/((2*n+1)*(2*n+2)) = log 2 (cf. Tijdeman).
%F A002939 Log 2 = Sum(n=1, inf.): 1/a(n) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90...; 
               = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8)...; = Sum(n=0, 
               inf.): (-1)^n/(Nn+1) with N=1 2. Log 2 = Integral(0, 1, 1/(1+x)dx) 
               - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 22 2003
%F A002939 a(n)=A000384(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
%F A002939 a(n)=3*a(n-1)-3*a(n-2)+a(n-3). G.f.: 2*x*(1+3*x)/(1-x)^3. [From R. J. 
               Mathar (mathar(AT)strw.leidenuniv.nl), Apr 23 2009]
%F A002939 a(n)=8*n+a(n-1)-14 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 12 2009]
%e A002939 16 17 18 19 ...
%e A002939 15 4 5 6 ...
%e A002939 14 3 0 7 ...
%e A002939 13 2 1 8 ...
%e A002939 For n=2, a(2)=8*2+0-14=2; n=3, a(3)=8*3+2-14=12; n=4, a(4)=8*4+12-14=30 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
%p A002939 a:=n->sum(n, j=2..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 30 2007
%t A002939 lst={};Do[AppendTo[lst, 2*n*(2*n-1)], {n, 0, 5!}];lst...and/or... s=0;
               lst={s};Do[s+=n+1;AppendTo[lst, s], {n, 1, 6!, 8}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008]
%Y A002939 Sequences from spirals: A001107, A002939, A007742, A033951, A033952, 
               A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
%Y A002939 Cf. A001107.
%Y A002939 Cf. A016789, A017041, A017485, A125202.
%Y A002939 Cf. A000384.
%Y A002939 Sequence in context: A061780 A156021 A067348 this_sequence A118239 A127118 
               A083175
%Y A002939 Adjacent sequences: A002936 A002937 A002938 this_sequence A002940 A002941 
               A002942
%K A002939 nonn,nice,easy,new
%O A002939 0,2
%A A002939 N. J. A. Sloane (njas(AT)research.att.com).
%E A002939 More terms and Mathematica programs Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Oct 25 2008