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Search: id:A005891
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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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+0
36
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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; graph; listen)
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OFFSET
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0,2
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COMMENT
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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]
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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).
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.
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
Index entries for sequences related to linear recurrences with constant coefficients
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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to centered polygonal numbers
Index entries for crystal ball sequences
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FORMULA
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a(n) = 1 + sum(5*n) - Xavier Acloque Oct 08 2003
a(n) = 5*n + a(n-1), with a(0)=1. - Vincenzo Librandi Oct 24 2009
Narayana transform (A001263) of [1, 5, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (joliverlafont(AT)gmail.com), Dec 02 2008]
a(n)=5*n+a(n-1)-5 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]
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]
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EXAMPLE
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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]
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 S. 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 Orlovsky (4vladimir(AT)gmail.com), Nov 04 2008]
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CROSSREFS
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Cf. A028895, A001844, A003215.
Cf. A004068, A006322.
Cf. A001263.
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)
Equals second row of A167546 divided by 2.
(End)
Sequence in context: A113742 A102214 A115007 this_sequence A092286 A108182 A097118
A000217: 5*A000217 + 1 [From Thomas M. Green (tgreen(AT)astound.net), Nov 16 2009]
Adjacent sequences: A005888 A005889 A005890 this_sequence A005892 A005893 A005894
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KEYWORD
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nonn,easy,new
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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More terms from Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Oct 24 2009
Formula corrected and relocated by Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 07 2009
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