%I A045944
%S A045944 0,5,16,33,56,85,120,161,208,261,320,385,456,533,616,705,800,901,1008,
%T A045944 1121,1240,1365,1496,1633,1776,1925,2080,2241,2408,2581,2760,2945,3136,
%U A045944 3333,3536,3745,3960,4181,4408,4641,4880,5125,5376,5633,5896,6165,6440
%N A045944 Rhombic matchstick numbers: n*(3*n+2).
%C A045944 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:
%C A045944 ......16..15..14
%C A045944 ....17..5...4...13
%C A045944 ..18..6...0...3...12
%C A045944 19..7...1...2...11..26
%C A045944 ..20..8...9...10..25
%C A045944 ....21..22..23..24
%C A045944 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
%C A045944 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]
%D A045944 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.
%H A045944 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A045944 O.g.f.: x*(5+x)/(1-x)^3 . - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jan 07 2008
%F A045944 a(n)=6*n+a(n-1)-7 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A045944 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]
%t A045944 lst={};Do[AppendTo[lst, n*(3*n+2)], {n, 0, 6!}];lst [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Nov 06 2008]
%Y A045944 Cf. A000567.
%Y A045944 Bisection of A001859. Cf. A049450.
%Y A045944 Adjacent sequences: A045941 A045942 A045943 this_sequence A045945 A045946 
               A045947
%K A045944 nonn,easy,nice,new
%O A045944 0,2
%A A045944 R. K. Guy
%E A045944 Removed a zero in an A-number - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 22 2009