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Search: id:A005898
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| A005898 |
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Centered cube numbers: n^3 + (n+1)^3.
(Formerly M4616)
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+0
24
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| 1, 9, 35, 91, 189, 341, 559, 855, 1241, 1729, 2331, 3059, 3925, 4941, 6119, 7471, 9009, 10745, 12691, 14859, 17261, 19909, 22815, 25991, 29449, 33201, 37259, 41635, 46341, 51389, 56791, 62559, 68705, 75241, 82179, 89531, 97309, 105525
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Write the natural numbers in groups: 1; 2,3,4; 5,6,7,8,9; 10,11,12,13,14,15,16; ..... and add the groups, i.e. a(n)=sum(i,i=n^2-2(n-1)..n^2). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 05 2001
The numbers 1, 9, 35, 91, etc. are divisible by 1, 3, 5, 7, etc. Therefore there are no prime numbers in this list. 9 is divisible by 3 and every third number after 9 is also divisible by 3. 35 is divisible by 5 and 7 and every fifth number after 35 is also divisible by 5 and every seventh number after 35 is also divisible by 7. This pattern continues indefinitely. [From Howard Berman (howard_berman(AT)hotmail.com), Nov 07 2008]
The running sum of n^3 taken 2 at a time. [From Al Hakanson (hawkuu(AT)gmail.com), May 18 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.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (10).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
D. Zeitlin, A family of Galileo sequences, Amer. Math. Monthly 82 (1975), 819-822.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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.
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FORMULA
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a(n)=2*n^3+9*n^2+15*n+9. Offset 0. a(3)=189. [From Al Hakanson (hawkuu(AT)gmail.com), May 18 2009]
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MAPLE
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A005898:=(z+1)*(z**2+4*z+1)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
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a[n_]:=n^3; lst={}; Do[AppendTo[lst, a[n]+a[n+1]], {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 03 2009]
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PROGRAM
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sage: [i^3+(i+1)^3 for i in xrange(0, 39)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 03 2008
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CROSSREFS
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1/12*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496.
Cf. A003215, A000537, A000578 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jan 03 2009]
Sequence in context: A033566 A022275 A071398 this_sequence A034957 A002418 A118414
Adjacent sequences: A005895 A005896 A005897 this_sequence A005899 A005900 A005901
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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