%I A005891 M4112
%S A005891 1,6,16,31,51,76,106,141,181,226,276,331,391,456,526,601,681,766,856,
%T A005891 951,1051,1156,1266,1381,1501,1626,1756,1891,2031,2176,2326,2481,2641,
%U A005891 2806,2976,3151,3331,3516,3706,3901,4101,4306,4516,4731,4951,5176,5406
%N A005891 Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 
               3.3.3.4.4. planar net.
%C A005891 Equals the triangular numbers convolved with [1, 3, 1, 0, 0, 0,...]. 
               [From Gary W. Adamson & Alexander Povolotsky (qntmpkt(AT)yahoo.com), 
               May 29 2009]
%C A005891 Equals 5*(nth triangular number) + 1. [From Thomas M. Green (tgreen(AT)astound.net), 
               Nov 25, 2009] [From Thomas M. Green (tgreen(AT)astound.net), Nov 
               25 2009]
%D A005891 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005891 S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques 
               Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%D A005891 B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral 
               clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H A005891 T. D. Noe, <a href="b005891.txt">Table of n, a(n) for n=0..1000</a>
%H A005891 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005891 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005891 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005891 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CenteredPentagonalNumber.html">Link to a section of The World of 
               Mathematics.</a>
%H A005891 <a href="Sindx_Ce.html#CENTRALCUBE">Index entries for sequences related 
               to centered polygonal numbers</a>
%H A005891 <a href="Sindx_Cor.html#crystal_ball">Index entries for crystal ball 
               sequences</a>
%F A005891 a(n) = 1 + sum(5*n) - Xavier Acloque Oct 08 2003
%F A005891 a(n) = 5*n + a(n-1), with a(0)=1. - Vincenzo Librandi Oct 24 2009
%F A005891 Narayana transform (A001263) of [1, 5, 0, 0, 0,...] - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 29 2007
%F A005891 a(n)=3a(n-1)-3a(n-2)+a(n-3), a(0)=1, a(1)=6, a(2)=16 [From Jaume Oliver 
               Lafont (joliverlafont(AT)gmail.com), Dec 02 2008]
%F A005891 a(n)=5*n+a(n-1)-5 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 09 2009]
%F A005891 a(n) = 5*T(n) + 1, for n = 0, 1, 2, 3, ... and where T(n) = n*(n+1)/2 
               = nth triangular number. (Thomas M. Green, Nov. 16, 2009) [From Thomas 
               M. Green (tgreen(AT)astound.net), Nov 16 2009]
%F A005891 a(n) = 5*A000217 + 1 [From Thomas M. Green (tgreen(AT)astound.net), Nov 
               25, 2009] [From Thomas M. Green (tgreen(AT)astound.net), Nov 25 2009]
%e A005891 For n=2, a(2)=5*2+1-5=6; n=3, a(3)=5*3+6-5=16; n=4, a(4)=5*4+16-5=31 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]
%e A005891 a(2)= 5*T(2) + 1 = 5*3 + 1 = 16, a(4) = 5*T(4) + 1 = 5*10 + 1 = 51 (Thomas 
               M. Green, Nov. 16, 2009) [From Thomas M. Green (tgreen(AT)astound.net), 
               Nov 16 2009]
%p A005891 5/2*N^2+5/2*N+1;
%p A005891 A005891:=-(1+3*z+z**2)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 
               dissertation.]
%t A005891 s=1;lst={s};Do[s+=n+5;AppendTo[lst, s], {n, 0, 6!, 5}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 04 2008]
%Y A005891 Cf. A028895, A001844, A003215.
%Y A005891 Cf. A004068, A006322.
%Y A005891 Cf. A001263.
%Y A005891 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%Y A005891 Equals second row of A167546 divided by 2.
%Y A005891 (End)
%Y A005891 Sequence in context: A113742 A102214 A115007 this_sequence A092286 A108182 
               A097118
%Y A005891 A000217: 5*A000217 + 1 [From Thomas M. Green (tgreen(AT)astound.net), 
               Nov 16 2009]
%Y A005891 Adjacent sequences: A005888 A005889 A005890 this_sequence A005892 A005893 
               A005894
%K A005891 nonn,easy,new
%O A005891 0,2
%A A005891 N. J. A. Sloane (njas(AT)research.att.com).
%E A005891 More terms from Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 
               24 2009
%E A005891 Formula corrected and relocated by Johannes W. Meijer (meijgia(AT)hotmail.com), 
               Nov 07 2009