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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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43
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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
(list;
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refs;
listen;
history;
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internal format)
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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,...]. [From Gary W. Adamson & Alexander R. Povolotsky, May 29 2009]
Contribution from Ant King, Jun 15 2012: (Start)
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...
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)
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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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Centered Pentagonal Number.
Index entries for sequences related to centered polygonal numbers
Index to sequences with 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)=3a(n-1)-3a(n-2)+a(n-3), a(0)=1, a(1)=6, a(2)=16 [From 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. [From Thomas M. Green (tgreen(AT)astound.net), Nov 25 2009]
a(n) = a(n-1)+5*n, with a(0)=1. [From 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]
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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) [From Thomas M. Green (tgreen(AT)astound.net), 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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s=1; lst={s}; Do[s+=n+5; AppendTo[lst, s], {n, 0, 6!, 5}]; lst [From Vladimir Joseph Stephan Orlovsky, Nov 04 2008]
FoldList[#1 + #2 &, 1, 5 Range@ 40] (* Robert G. Wilson v, Feb 02 2011 *)
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CROSSREFS
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Cf. A028895, A001844, A003215.
Cf. A004068, A006322.
Cf. A001263.
Contribution from Johannes W. Meijer, Nov 12 2009: (Start)
Equals second row of A167546 divided by 2.
(End)
Sequence in context: A113742 A102214 A115007 * A092286 A108182 A097118
Adjacent sequences: A005888 A005889 A005890 * A005892 A005893 A005894
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KEYWORD
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nonn,easy
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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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