Search: id:A014105
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%I A014105
%S A014105 0,3,10,21,36,55,78,105,136,171,210,253,300,351,406,465,528,595,666,
%T A014105 741,820,903,990,1081,1176,1275,1378,1485,1596,1711,1830,1953,2080,
%U A014105 2211,2346,2485,2628,2775,2926,3081,3240,3403,3570,3741,3916,4095,4278
%N A014105 Second hexagonal numbers: n(2n+1).
%C A014105 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 (se16(AT)btinternet.com), Jan 22 2001
%C A014105 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
%C A014105 a(n) = A084849(n) - 1; A100035(a(n)+1) = 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 31 2004
%C A014105 Also a(n)=3*Sum(tan^2(k*pi/(2(n+1))), k, 1, n); - Ignacio Larrosa (ignacio.larrosa(AT)eresmas.net), 
               Apr 17 2001
%C A014105 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 R. Janjic (agnus(AT)blic.net), 
               Oct 28 2007
%C A014105 a(2*n) = A033585(n); a(3*n) = A144314(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 17 2008]
%H A014105 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A014105 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A014105 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative 
               Functions</a>
%F A014105 a(n)^2 = n*(a(n)+1 + a(n)+2 + ... + a(n)+2n), e.g. 10^2 = 2*(11 + 12 
               + 13 +14) - Charlie Marion (charliem(AT)bestweb.net), Jun 15 2003
%F A014105 G.f.: x(3+x)/(1-x)^3. E.g.f.: exp(x)(3x+2x^2). a(n)=A000217(2n)=A000384(-n).
%F A014105 Partial sums of odd numbers 3 mod 4, i.e. 3, 3+7, 3+7+11, ... Cf. A001107. 
               - Jon Perry (perry(AT)globalnet.co.uk), Dec 18 2004
%F A014105 a(n) = A126890(n,k) + A126890(n,n-k), 0<=k<=n. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 30 2006
%F A014105 a(n)=4*n+a(n-1)-5 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A014105 For n=2, a(2)=4*2+0-5=3; n=3, a(3)=4*3+3-5=10; n=4, a(4)=4*4+10-5=21 
               [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A014105 seq(binomial(2*n+1,2), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 21 2007
%p A014105 a:=n->sum(j, j=0..n): seq(a(2*n), n=0..46); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 30 2007
%p A014105 with(finance):seq(add(cashflows([n,k,k], 0 ),k=1..n),n=0..51); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 30 2008
%p A014105 a:=n->sum(1+sum(2, k=1..n), k=1..n):seq(a(n), n=0...43); [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Aug 24 2008]
%t A014105 s=0;lst={s};Do[s+=n++ +3;AppendTo[lst, s], {n, 0, 6!, 4}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 16 2008]
%o A014105 (PARI) a(n)=n*(2*n+1)
%Y A014105 Cf. A000217, A000384.
%Y A014105 Second column of array A094416.
%Y A014105 Cf. A100040, A100041.
%Y A014105 A081266, A144312. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Sep 17 2008]
%Y A014105 Sequence in context: A073604 A004194 A097590 this_sequence A146012 A027917 
               A038347
%Y A014105 Adjacent sequences: A014102 A014103 A014104 this_sequence A014106 A014107 
               A014108
%K A014105 nonn,easy,new
%O A014105 0,2
%A A014105 N. J. A. Sloane (njas(AT)research.att.com).

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