Search: id:A005449
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%I A005449
%S A005449 0,2,7,15,26,40,57,77,100,126,155,187,222,260,301,345,392,442,495,551,
%T A005449 610,672,737,805,876,950,1027,1107,1190,1276,1365,1457,1552,1650,1751,
%U A005449 1855,1962,2072,2185,2301,2420,2542,2667,2795,2926,3060,3197,3337,3480
%N A005449 Second pentagonal numbers: n*(3n+1)/2.
%C A005449 Number of edges in the join of the complete graph and the cycle graph, 
               both of order n, K_n * C_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), 
               Jan 07 2002
%C A005449 Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth 
               (mail(AT)matroid.com), Aug 11 2002
%C A005449 a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard 
               Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 03 2004
%C A005449 Also sum of next n consecutive numbers greater than n: a(n)=A014105(n)-A000217(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 13 2005
%C A005449 The modular form Delta(q) = q*(1 + SUM[1..infinity]((-1)^n)*((q^(n*(3*n-1)/
               2))+((q^(n*(3*n+1)/2))))) = q*Prod[1..infinity] (1-q^n)^24. Thus 
               Delta(q) = q*(1 + SUM[1..infinity](A033999(n)*(q^A000326(n))+(q^A005449(n)))). 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 15 2006
%C A005449 a(n) = A126890(n,n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Dec 30 2006
%C A005449 a(n) = A000217(n) + A000290(n) - Zak Seidov (zakseidov(AT)yahoo.com), 
               Apr 06 2008
%C A005449 Row sums of triangle A134403.
%D A005449 A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. 
               Comp. 61:29-68, 1993
%D A005449 H. Cohen, A Course in Computational Algebraic Number Theory, vol. 138 
               of Graduate Texts in Mathematics, Springer-Verlag, 2000.
%H A005449 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
               a>
%H A005449 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A005449 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">
               De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 1
%H A005449 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
               Observatio de summis divisorum</a> p. 8.
%H A005449 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation 
               on the sums of divisors</a> p. 8.
%H A005449 L. Euler, <a href="http://arXiv.org/abs/math.HO/0505373">On the remarkable 
               properties of the pentagonal numbers</a>
%H A005449 M. Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">
               Pentagon, Kartenhaus und Summenzerlegung</a>
%H A005449 Alfred Hoehn, <a href="a000326.jpg">Illustration of initial terms of 
               A000326, A005449, A045943, A115067</a>
%F A005449 G.f.: x(2+x)/(1-x)^3. E.g.f.: exp(x)(2x+3x^2/2). a(n)=n(3n+1)/2. a(-n)=A000326(n).
%F A005449 a(n) = right term of M^n * [1 0 0] where M = the 3X3 matrix [1 0 0 / 
               1 1 0 / 2 3 1]. M^n * [1 0 0] = [1 n a(n)]. E.g. a(4) = 26 since 
               M^4 * [1 0 0] = [1 4 26] = [1 n a(n)]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Dec 19 2004
%F A005449 a(n) = n*(3n+1)/2. G.f.: x(2+x)/(1-x)^3. E.g.f.: exp(x)(2x+3x^2/2). a(-n)=A000326(n). 
               - Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 15 2006
%F A005449 sum (n+j,j=1..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Sep 
               12 2006
%F A005449 a(n)=2*C(3*n,4)/C(3*n,2),n>=1 . - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 02 2007
%F A005449 a(n)=3*n+a(n-1)-4 (with a(1)=0) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 08 2009]
%e A005449 For n=2, a(2)=3*2+0-4=2; n=3, a(3)=3*3+2-4=7; n=4, a(4)=3*4+7-4=15 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 08 2009]
%p A005449 [seq(2*binomial(3*n,4)/binomial(3*n,2),n=1..49)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 02 2007
%p A005449 a:=n->sum(n/3, j=0..n): seq(a(3*n)/2, n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Apr 30 2007
%p A005449 a:=n->sum(k+sum(1, k=1..n),k=1..n):seq(a(n), n=0...48); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jun 17 2008
%p A005449 seq(sum(binomial(n,m), m=1..2)+n^2,n=0..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 17 2008
%t A005449 lst={};Do[AppendTo[lst, n*(3*n+1)/2], {n, 0, 6!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Nov 06 2008]
%o A005449 (PARI) a(n)=n*(3*n+1)/2
%Y A005449 a(n) = A110449(n, 1) for n>0.
%Y A005449 Cf. A001318, A000326, A049451, A033568, A101165, A101166, A000320.
%Y A005449 The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 
               12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, 
               A059845, A140672, A140673, A140674, A140675, A151542.
%Y A005449 Sequence in context: A132746 A167543 A029888 this_sequence A113422 A061802 
               A003452
%Y A005449 Adjacent sequences: A005446 A005447 A005448 this_sequence A005450 A005451 
               A005452
%K A005449 nonn,new
%O A005449 0,2
%A A005449 N. J. A. Sloane (njas(AT)research.att.com).

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