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A005891
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Centered pentagonal numbers: (5n^2+5n+2)/2; crystal ball sequence for 3.3.3.4.4. planar net.
(Formerly M4112)
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77
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1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, 3151, 3331, 3516, 3706, 3901, 4101, 4306, 4516, 4731, 4951, 5176, 5406
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OFFSET
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0,2
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COMMENTS
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Equals the triangular numbers convolved with [1, 3, 1, 0, 0, 0, ...]. - Gary W. Adamson and Alexander R. Povolotsky, May 29 2009
From Ant King, Jun 15 2012: (Start)
The limiting value of the partial sums of the reciprocals of the terms a(n) is 2*Pi/sqrt(15)*tanh(Pi/2*sqrt(3/5)) = 1.360613169863... .
a(n) == 1 (mod 5) for all n.
The digital roots of the a(n) form a purely periodic palindromic 9-cycle 1, 6, 7, 4, 6, 4, 7, 6, 1.
The units' digits of the a(n) form a purely periodic palindromic 4-cycle 1, 6, 6, 1.
(End)
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
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jan 25 2019
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Cliff Reiter, Polygonal Numbers and Fifty One Stars, Lafayette College, Easton, PA (2019).
Eric Weisstein's World of Mathematics, Centered Pentagonal Number.
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Index entries for crystal ball sequences
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FORMULA
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Narayana transform (A001263) of [1, 5, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
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
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
a(n) = a(n-1) + 5*n, with a(0)=1. - Vincenzo Librandi, Nov 18 2010
a(n) = A028895(n) + 1. - Omar E. Pol, Oct 03 2011
a(n) = 2*a(n-1) - a(n-2) + 5. - Ant King, Jun 12 2012
G.f.: (1 + 3*x + x^2)/(1 - x)^3. - Arkadiusz Wesolowski, Aug 05 2012
a(n) = A101321(5,n). - R. J. Mathar, Jul 28 2016
E.g.f.: (2 + 10*x + 5*x^2)*exp(x)/2. - Ilya Gutkovskiy, Jul 28 2016
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=0} a(n)/n! = 17*e/2.
Sum_{n>=0} (-1)^(n+1)*a(n)/n! = 3/(2*e). (End)
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EXAMPLE
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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
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MAPLE
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5/2*N^2+5/2*N+1;
A005891:=-(1+3*z+z**2)/(z-1)**3; # conjectured by Simon Plouffe in his 1992 dissertation
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MATHEMATICA
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FoldList[#1 + #2 &, 1, 5 Range@ 40] (* Robert G. Wilson v, Feb 02 2011 *)
LinearRecurrence[{3, -3, 1}, {1, 6, 16}, 50] (* Harvey P. Dale, Sep 08 2018 *)
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PROG
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(PARI) a(n)=5*n*(n+1)/2+1 \\ Charles R Greathouse IV, Mar 22 2016
(MAGMA) [5*n*(n+1)/2 + 1: n in [0..50]]; // G. C. Greubel, Nov 04 2017
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CROSSREFS
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Cf. A028895, A001844, A003215, A004068, A006322, A001263.
Equals second row of A167546 divided by 2.
Sequence in context: A102214 A301679 A115007 * A108182 A244242 A092286
Adjacent sequences: A005888 A005889 A005890 * A005892 A005893 A005894
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Formula corrected and relocated by Johannes W. Meijer, Nov 07 2009
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STATUS
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approved
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