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A014105
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Second hexagonal numbers: n(2n+1).
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67
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0, 3, 10, 21, 36, 55, 78, 105, 136, 171, 210, 253, 300, 351, 406, 465, 528, 595, 666, 741, 820, 903, 990, 1081, 1176, 1275, 1378, 1485, 1596, 1711, 1830, 1953, 2080, 2211, 2346, 2485, 2628, 2775, 2926, 3081, 3240, 3403, 3570, 3741, 3916, 4095, 4278
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OFFSET
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0,2
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COMMENTS
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Note that when starting from {a(n)}^2, equality holds between series of first n+1 and next n consecutive squares : a(n)^2+(a(n)+1)^2+...+(a(n)+n)^2 = (a(n)+n+1)^2+(a(n)+n+2)^2+...(a(n)+2n)^2, e.g. 10^2+11^2+12^2 = 13^2+14^2 - Henry Bottomley, Jan 22 2001
More generally, for n>0, let b(n,k)=a(n)+k*(4n+1).
Then b(n,k)^2+(b(n,k)+1)^2+...+(b(n,k)+n)^2=
(b(n,k)+n+1+2k)^2+...+(b(n,k)+2n+2k)^2+k^2; e.g., if n=3 and k=2,
then b(n,k)=47 and 47^2+...+50^2=55^2+...+57^2+2^2.
- Charlie Marion, Jan 01 2011
a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002
a(n) = A084849(n) - 1; A100035(a(n)+1) = 1. - Reinhard Zumkeller, Oct 31 2004
Also a(n)=3*Sum(tan^2(k*pi/(2(n+1))), k, 1, n); - Ignacio Larrosa, Apr 17 2001
If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
Sequence found by reading the line from 0, in the direction 0, 10,... and the line from 3, in the direction 3, 21,..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Nov 09 2011
a(n) is the number of positions of a domino in a pyramidal board with base 2n+1. [César Eliud Lozada, Sep 26 2012]
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LINKS
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_Reinhard Zumkeller_, Table of n, a(n) for n = 0..10000
Milan Janjic, Two Enumerative Functions
Guo-Niu Han, Enumeration of Standard Puzzles
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n)^2 = n*(a(n)+1 + a(n)+2 + ... + a(n)+2n), e.g. 10^2 = 2*(11 + 12 + 13 +14). - Charlie Marion, Jun 15 2003
G.f.: x*(3+x)/(1-x)^3. E.g.f.: exp(x)(3*x+2*x^2). a(n) = A000217(2n) = A000384(-n). - N. J. A. Sloane, Sep 13 2003
Partial sums of odd numbers 3 mod 4, i.e. 3, 3+7, 3+7+11, ... Cf. A001107. - Jon Perry, Dec 18 2004
a(n) = A126890(n,k) + A126890(n,n-k), 0<=k<=n. - Reinhard Zumkeller, Dec 30 2006
a(2*n) = A033585(n); a(3*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = a(n-1)+4*n-1 (with a(0)=0). - Vincenzo Librandi, Dec 24 2010
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MAPLE
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seq(binomial(2*n+1, 2), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 21 2007
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MATHEMATICA
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A014105[n_Integer] := n*(2*n + 1); Table[A014105[n], {n, 100}] (* Vladimir Orlovsky, Nov 16 2008 *)
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PROG
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(PARI) a(n)=n*(2*n+1)
(Haskell)
a014105 n = n * (2 * n + 1)
a014105_list = scanl (+) 0 a004767_list -- Reinhard Zumkeller, Oct 03 2012
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CROSSREFS
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Cf. A000217, A000384.
Second column of array A094416.
Cf. A100040, A100041.
Cf. A081266, A144312.
Equals A033586(n) divided by 4.
See Comments of A132124.
Second n-gonal numbers: A005449, A147875, A045944, A179986, A033954, A062728, A135705.
Sequence in context: A004194 A097590 A194141 * A146012 A027917 A038347
Adjacent sequences: A014102 A014103 A014104 * A014106 A014107 A014108
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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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Link added and minor errors corrected by Johannes W. Meijer, Feb 04 2010
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STATUS
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approved
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