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A005891 Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.
(Formerly M4112)
70

%I M4112

%S 1,6,16,31,51,76,106,141,181,226,276,331,391,456,526,601,681,766,856,

%T 951,1051,1156,1266,1381,1501,1626,1756,1891,2031,2176,2326,2481,2641,

%U 2806,2976,3151,3331,3516,3706,3901,4101,4306,4516,4731,4951,5176,5406

%N Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.

%C Equals the triangular numbers convolved with [1, 3, 1, 0, 0, 0,...]. - _Gary W. Adamson_ & _Alexander R. Povolotsky_, May 29 2009

%C From _Ant King_, Jun 15 2012: (Start)

%C The limiting value of the partial sums of the reciprocals of the a(n) is 2*Pi/sqrt(15)*tanh(Pi/2*sqrt(3/5)) = 1.360613169863... .

%C a(n) == 1 (mod 5) for all n.

%C The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 6, 7, 4, 6, 4, 7, 6, 1.

%C The units' digits of the a(n) form a purely periodic palindromic 4-cycle 1, 6, 6, 1.

%C (End)

%C Binomial transform of (1, 5, 5, 0, 0, 0,...) and second partial sum of (1, 4, 5, 5, 5,...). - _Gary W. Adamson_, Sep 09 2015

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.

%H T. D. Noe, <a href="/A005891/b005891.txt">Table of n, a(n) for n = 0..1000</a>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPentagonalNumber.html">Centered Pentagonal Number.</a>

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%H <a href="/index/Cor#crystal_ball">Index entries for crystal ball sequences</a>

%F Narayana transform (A001263) of [1, 5, 0, 0, 0,...]. - _Gary W. Adamson_, Dec 29 2007

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), a(0)=1, a(1)=6, a(2)=16. - _Jaume Oliver Lafont_, Dec 02 2008

%F a(n) = 5*A000217(n) + 1 = 5*T(n) + 1, for n = 0, 1, 2, 3, ... and where T(n) = n*(n+1)/2 = n-th triangular number. - _Thomas M. Green_, Nov 25 2009

%F a(n) = a(n-1) + 5*n, with a(0)=1. - _Vincenzo Librandi_, Nov 18 2010

%F a(n) = A028895(n) + 1. - _Omar E. Pol_, Oct 03 2011

%F a(n) = 2*a(n-1) - a(n-2) + 5. - _Ant King_, Jun 12 2012

%F G.f.: (1 + 3*x + x^2)/(1 - x)^3. - _Arkadiusz Wesolowski_, Aug 05 2012

%F a(n) = A101321(5,n). - _R. J. Mathar_, Jul 28 2016

%F E.g.f.: (2 + 10*x + 5*x^2)*exp(x)/2. - _Ilya Gutkovskiy_, Jul 28 2016

%e 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

%p 5/2*N^2+5/2*N+1;

%p A005891:=-(1+3*z+z**2)/(z-1)**3; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t s=1;lst={s};Do[s+=n+5;AppendTo[lst, s], {n, 0, 6!, 5}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 04 2008 *)

%t FoldList[#1 + #2 &, 1, 5 Range@ 40] (* _Robert G. Wilson v_, Feb 02 2011 *)

%t LinearRecurrence[{3,-3,1},{1,6,16},50] (* _Harvey P. Dale_, Sep 08 2018 *)

%o (PARI) a(n)=5*n*(n+1)/2+1 \\ _Charles R Greathouse IV_, Mar 22 2016

%o (MAGMA) [5*n*(n+1)/2 + 1: n in [0..50]]; // _G. C. Greubel_, Nov 04 2017

%Y Cf. A028895, A001844, A003215, A004068, A006322, A001263.

%Y Equals second row of A167546 divided by 2.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Formula corrected and relocated by _Johannes W. Meijer_, Nov 07 2009

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Last modified October 19 22:28 EDT 2018. Contains 316378 sequences. (Running on oeis4.)