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A014105 Second hexagonal numbers: n*(2n+1). 92
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Also a(n) = 3*sum(tan^2(k*pi/(2(n+1))), k=1..n). - Ignacio Larrosa Cañestro, Apr 17 2001

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

Partial sums of odd numbers 3 mod 4, that is, 3, 3+7, 3+7+11, ... See A001107. - Jon Perry, Dec 18 2004

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

More generally (see the first comment), 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

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

Differences of row sums of two consecutive rows of triangle A120070, i.e., first differences of A016061. - J. M. Bergot, Jun 14 2013

a(n)*Pi is the total length of half circle spiral after n rotations. See illustration in links. - Kival Ngaokrajang, Nov 05 2013

a(n) = A242342(2*n+1). - Reinhard Zumkeller, May 11 2014

For n>1 a(n)=sum[C(n-2+k,n-2)*(C(n+2-k,n) {0<=k<=2}]. - J. M. Bergot, Jun 14 2014

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

Milan Janjic, Two Enumerative Functions

Kival Ngaokrajang, Illustration of half circle spiral

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

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

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

a(n) = sum( (-1)^k*k^2, k=0..2*n ). - Bruno Berselli, Aug 29 2013

EXAMPLE

For n=6, a(6) = 0^2-1^2+2^2-3^2+4^2-5^2+6^2-7^2+8^2-9^2+10^2-11^2+12^2 = 78. - Bruno Berselli, Aug 29 2013

MAPLE

seq(binomial(2*n+1, 2), n=0..46); # Zerinvary Lajos, Jan 21 2007

MATHEMATICA

A014105[n_Integer] := n*(2*n + 1); Table[A014105[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

PROG

(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

(MAGMA) [ n*(2*n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

CROSSREFS

Cf. A000217, A000384.

Second column of array A094416.

Cf. A100040, A100041, 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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Jun 14 1998

EXTENSIONS

Link added and minor errors corrected by Johannes W. Meijer, Feb 04 2010

STATUS

approved

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Last modified October 20 09:00 EDT 2014. Contains 248329 sequences.