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

Take rows A005563, A061037, A061039, A061041, A061043, A061045, A061047, A061049. Principal diagonal is a(n)=A005563(0), A061037(1), A061039(2), A061041(3), A061043(4), A061045(5), A061047(6), A061049(7). Note 85 is 6-th term of numerators of sixth spectrum of hydrogen (1/36-1/n^2) due to Curtis Judson Humphreys (1898-1986) in 1953 (Humphreys series in Journal of Research of the National Bureau Standards 1953,50,1, not seen); after Lyman (1906-1914), Balmer (1885), Paschen (1908), Brackett (1922), Pfund (1924) and Hansen-Strong. [From Paul Curtz (bpcrtz(AT)free.fr), Oct 04 2008, Sep 17 2009]

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.

Index entries for sequences related to linear recurrences with constant coefficients

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

a(n)=6*n+a(n-1)-7 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

For n=2, a(2)=6*2+0-7=5; n=3, a(3)=6*3+5-7=16; n=4, a(4)=6*4+16-7=33 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]

lst={}; Do[AppendTo[lst, n*(3*n+2)], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 06 2008]

Cf. A000567.

Bisection of A001859. Cf. A049450.

Adjacent sequences: A045941 A045942 A045943 this_sequence A045945 A045946 A045947

nonn,easy,nice,new

R. K. Guy

Removed a zero in an A-number - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 22 2009