0,2

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)lhsystems.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

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.

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

16 17 18 19 ...

15 4 5 6 ...

14 3 0 7 ...

13 2 1 8 ...

a:=n->sum(n, j=2..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2007

Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, A002943, A033996, A033988.

Cf. A001107.

Cf. A016789, A017041, A017485, A125202.

Adjacent sequences: A002936 A002937 A002938 this_sequence A002940 A002941 A002942

Sequence in context: A034318 A061780 A067348 this_sequence A118239 A127118 A083175

nonn,nice,easy

njas