0,2

Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) is the sequence found by reading the line from 0 in the direction 0,5,... - Floor van Lamoen (fvlamoen(AT)hotmail.com), Jul 21 2001. The spiral begins:

......16..15..14

....17..5...4...13

..18..6...0...3...12

19..7...1...2...11..26

..20..8...9...10..25

....21..22..23..24

The equations 1 + 2 = 3 and 3^2 + 4^2 = 5^2 set the stage for considering whether it is also true that 5^3 + 6^3 = 7^3 and 7^4 + 8^4 = 9^4. Reflecting on Fermat's Last Theorem or resorting to a calculator dispels any hope that either of the two equations could be correct. However, it is true that 5^3 + 6^3 + 2 = 7^3 and 7^4 + 8^4 + 64 = 9^4. More significantly, each of these equations is the first of an infinite sequence of equations featuring consecutive integers that conform to the spirit of the equations mentioned in A000384. For n>0, a(n)^3+(a(n)+1)^3 +...+(a(n)+n)^3 +2*A000217(n)^2= (a(n)+n+1)^3+...+(a(n)+2n)^3; e.g., 5^3+6^3+2*1^2=7^3; 16^3+17^3+18^3+2*3^2=19^3+20^3; see also A033954 - Charlie Marion (charliemath(AT)optonline.net), Dec 8 2007

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

O.g.f.: x*(5+x)/(1-x)^3 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 07 2008

Cf. A000567.

Bisection of A001859. Cf. A049450.

Sequence in context: A134594 A063076 A132479 this_sequence A038361 A131425 A096941

Adjacent sequences: A045941 A045942 A045943 this_sequence A045945 A045946 A045947

nonn,easy,nice

R. K. Guy