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Search: id:A002939
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| 0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306, 3540, 3782, 4032, 4290, 4556, 4830, 5112, 5402, 5700, 6006, 6320, 6642, 6972, 7310, 7656, 8010, 8372
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals.
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
a(n) = A125199(n,n) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 24 2006
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
Twice hexagonal numbers. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.
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.
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.
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FORMULA
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Sum_{n >= 0} 1/((2*n+1)*(2*n+2)) = log 2 (cf. Tijdeman).
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
a(n)=A000384(n)*2. - Omar E. Pol (info(AT)polprimos.com), May 14 2008
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]
a(n)=8*n+a(n-1)-14 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 12 2009]
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EXAMPLE
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16 17 18 19 ...
15 4 5 6 ...
14 3 0 7 ...
13 2 1 8 ...
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]
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MAPLE
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a:=n->sum(n, j=2..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007
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MATHEMATICA
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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]
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CROSSREFS
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Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.
Cf. A001107.
Cf. A016789, A017041, A017485, A125202.
Cf. A000384.
Sequence in context: A061780 A156021 A067348 this_sequence A118239 A127118 A083175
Adjacent sequences: A002936 A002937 A002938 this_sequence A002940 A002941 A002942
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KEYWORD
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nonn,nice,easy,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms and Mathematica programs Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 25 2008
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