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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)
85
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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)
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
On the plane start with a single regular pentagon, and repeat the following procedure, "For each edge of any pentagon not already connected to an existing pentagon create a mirror image such that the mirror image does not overlap with an existing pentagon." a(n) is the number of pentagons occupying the plane after n repetitions. - Torlach Rush, Sep 14 2022
REFERENCES
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.
LINKS
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
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.
FORMULA
G.f.: (1 + 3*x + x^2)/(1 - x)^3. Simon Plouffe in his 1992 dissertation
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
Sum_{n>=0} 1/a(n) = 2*Pi /sqrt(15) *tanh(Pi/2*sqrt(3/5)) = 1.360613169863... - Ant King, Jun 15 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)
EXAMPLE
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
MAPLE
A005891 := proc(n)
1+5*n*(1+n)/2 ;
end proc:
seq(A005891(n), n=0..40) ; # R. J. Mathar, Oct 07 2021
MATHEMATICA
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 *)
Table[ j! Coefficient[Series[Exp[x]*(1 + 5 x^2/2)-1, {x, 0, 20}], x, j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 07 2023 *)
PROG
(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
CROSSREFS
Cf. A028895, A001844, A003215, A004068 (partial sums), A006322, A001263.
Partial sums of A008706.
Equals second row of A167546 divided by 2.
Sequence in context: A102214 A301679 A115007 * A108182 A244242 A092286
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Formula corrected and relocated by Johannes W. Meijer, Nov 07 2009
STATUS
approved

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Last modified March 19 01:34 EDT 2024. Contains 370952 sequences. (Running on oeis4.)