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%I #225 Mar 02 2024 14:09:03
%S 0,2,7,15,26,40,57,77,100,126,155,187,222,260,301,345,392,442,495,551,
%T 610,672,737,805,876,950,1027,1107,1190,1276,1365,1457,1552,1650,1751,
%U 1855,1962,2072,2185,2301,2420,2542,2667,2795,2926,3060,3197,3337,3480
%N Second pentagonal numbers: a(n) = n*(3*n + 1)/2.
%C 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_, Jan 07 2002
%C Also number of cards to build an n-tier house of cards. - _Martin Wohlgemuth_, Aug 11 2002
%C The modular form Delta(q) = q*Product_{n>=1} (1-q^n)^24 = q*(1 + Sum_{n>=1} (-1)^n*(q^(n*(3*n-1)/2)+q^(n*(3*n+1)/2))^24 = q*(1 + Sum_{n>=1} A033999(n)*(q^A000326(n)+q^a(n))^24. - _Jonathan Vos Post_, Mar 15 2006
%C Row sums of triangle A134403.
%C Bisection of A001318. - _Omar E. Pol_, Aug 22 2011
%C Sequence found by reading the line from 0 in the direction 0, 7, ... and the line from 2, in the direction 2, 15, ... in the square spiral whose vertices are the generalized pentagonal numbers A001318. - _Omar E. Pol_, Sep 08 2011
%C A general formula for the n-th second k-gonal number is given by T(n, k) = n*((k-2)*n+k-4)/2, n>=0, k>=5. - _Omar E. Pol_, Aug 04 2012
%C Partial sums give A006002. - _Denis Borris_, Jan 07 2013
%C A002260 is the following array A read by antidiagonals:
%C 0, 1, 2, 3, 4, 5, ...
%C 0, 1, 2, 3, 4, 5, ...
%C 0, 1, 2, 3, 4, 5, ...
%C 0, 1, 2, 3, 4, 5, ...
%C 0, 1, 2, 3, 4, 5, ...
%C 0, 1, 2, 3, 4, 5, ...
%C and a(n) is the hook sum: Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). - _R. J. Mathar_, Jun 30 2013
%C From _Klaus Purath_, May 13 2021: (Start)
%C This sequence and A000326 provide all integer m such that 24*m + 1 is a square. The union of the two sequences is A001318.
%C If A is a sequence satisfying the recurrence t(n) = 3*t(n-1) - 2*t(n-2) with the initial values either A(0) = 1, A(1) = n + 2 or A(0) = -1, A(1) = n-1, then a(n) = (A(i)^2 - A(i-1)*A(i+1))/2^i + n^2 for i>0. (End)
%C a(n+1) is the number of Dyck paths of size (3,3n+2), i.e., the number of NE lattice paths from (0,0) to (3,3n+2) which stay above the line connecting these points. - _Harry Richman_, Jul 13 2021
%C Binomial transform of [0, 2, 3, 0, 0, 0, ...], being a(n) = 2*binomial(n,1) + 3*binomial(n,2). a(3) = 15 = [0, 2, 3, 0] dot [1, 3, 3, 1] = [0 + 6 + 9 + 0]. - _Gary W. Adamson_, Dec 17 2022
%C a(n) is the sum of longest side length of all nondegenerate integer-sided triangles with shortest side length n and middle side length (n + 1), n > 0. - _Torlach Rush_, Feb 04 2024
%D Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000.
%H Vincenzo Librandi, <a href="/A005449/b005449.txt">Table of n, a(n) for n = 0..2000</a>
%H A. O. L. Atkin and F. Morain, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1199989-X">Elliptic Curves and Primality Proving</a>, Math. Comp., Vol. 61, No. 203 (1993), pp. 29-68.
%H Charles H. Conley and Valentin Ovsienko, <a href="https://arxiv.org/abs/2107.01234">Quiddities of polygon dissections and the Conway-Coxeter frieze equation</a>, arXiv:2107.01234 [math.CO], 2021.
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/542/">De mirabilibus proprietatibus numerorum pentagonalium</a>, Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 1780: I, pp. 56-75.
%H Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/243/">Observatio de summis divisorum</a>, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 5, pp. 59-74.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009, p. 8.
%H Leonhard Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.
%H Alfred Hoehn, <a href="/A000326/a000326.pdf">Illustration of initial terms of A000326, A005449, A045943, A115067</a>.
%H D. Suprijanto and I. W. Suwarno, <a href="http://dx.doi.org/10.12988/ams.2014.4139">Observation on Sums of Powers of Integers Divisible by 3k-1</a>, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2211-2217.
%H Martin Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">Pentagon, Kartenhaus und Summenzerlegung</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = A110449(n, 1) for n>0.
%F G.f.: x*(2+x)/(1-x)^3. E.g.f.: exp(x)*(2*x + 3*x^2/2). a(n) = n*(3*n + 1)/2. a(-n) = A000326(n). - _Michael Somos_, Jul 18 2003
%F a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - _Reinhard Zumkeller_, Dec 03 2004
%F a(n) = Sum_{j=1..n} n+j. - _Zerinvary Lajos_, Sep 12 2006
%F a(n) = A126890(n,n). - _Reinhard Zumkeller_, Dec 30 2006
%F a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - _Zerinvary Lajos_, Jan 02 2007
%F a(n) = A000217(n) + A000290(n). - _Zak Seidov_, Apr 06 2008
%F a(n) = a(n-1) + 3*n-1 for n>0, a(0)=0. - _Vincenzo Librandi_, Nov 18 2010
%F a(n) = A129267(n+5,n). - _Philippe Deléham_, Dec 21 2011
%F a(n) = 2*A000217(n) + A000217(n-1). - _Philippe Deléham_, Mar 25 2013
%F a(n) = A130518(3*n+1). - _Philippe Deléham_, Mar 26 2013
%F a(n) = (12/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - _Vladimir Kruchinin_, Jun 04 2013
%F a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - _Richard R. Forberg_, Jun 22 2013
%F a(n) = Sum_{i=1..n} 2*i - 1 + (i mod 2). - _Wesley Ivan Hurt_, Oct 11 2013
%F a(n) = (A000292(6*n+k+1)-A000292(k))/(6*n+1)-A000217(3*n+k+1), for any k >= 0. - _Manfred Arens_, Apr 26 2015
%F Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - _Vaclav Kotesovec_, Apr 27 2016
%F a(n) = A000217(2*n) - A000217(n). - _Bruno Berselli_, Sep 21 2016
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - _Amiram Eldar_, Jan 18 2021
%F From _Klaus Purath_, May 13 2021: (Start)
%F Partial sums of A016789 for n > 0.
%F a(n) = 3*n^2 - A000326(n).
%F a(n) = A000326(n) + n.
%F a(n) = A002378(n+1) + A000217(n). (End)
%F From _Klaus Purath_, Jul 14 2021: (Start)
%F In addition to the May 13 2021 comment: The same square b^2 = 24*a(n) + 1 we get by b^2 = (a(n+1) - a(n-1))^2 = (a(2*n)/n)^2.
%F a(2*n) = n*(a(n+1) - a(n-1)), n > 0.
%F a(2*n+1) = n*(a(n+1) - a(n)). (End)
%e From _Omar E. Pol_, Aug 22 2011: (Start)
%e Illustration of initial terms:
%e . O
%e . O O
%e . O O O O
%e . O O O O O O
%e . O O O O O O O O O
%e . O O O O O O O O O O O
%e . O O O O O O O O O O O O O
%e . O O O O O O O O O O O O O O
%e . O O O O O O O O O O O O O O O
%e . O O O O O O O O O O O O O O O
%e .
%e . 2 7 15 26 40
%e .
%e (End)
%p A005449:=n->n*(3*n + 1)/2; seq(A005449(k), k=0..100); # _Wesley Ivan Hurt_, Oct 11 2013
%t Table[n (3 n + 1)/2, {n, 0, 100}] (* _Zak Seidov_, Jan 31 2012 *)
%o (PARI) {a(n) = n * (3*n + 1) / 2} /* _Michael Somos_, Jul 18 2003 */
%o (Magma) [n*(3*n + 1) / 2: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011
%Y Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166.
%Y The generalized pentagonal numbers b*n+3*n*(n-1)/2, for b = 1 through 12, form sequences A000326, this sequence, A045943, A115067, A140090, A140091, A059845, A140672-A140675, A151542.
%Y Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488 (this sequence is the case k=3).
%Y Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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