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 A005449 Second pentagonal numbers: a(n) = n*(3*n + 1)/2. 138
 0, 2, 7, 15, 26, 40, 57, 77, 100, 126, 155, 187, 222, 260, 301, 345, 392, 442, 495, 551, 610, 672, 737, 805, 876, 950, 1027, 1107, 1190, 1276, 1365, 1457, 1552, 1650, 1751, 1855, 1962, 2072, 2185, 2301, 2420, 2542, 2667, 2795, 2926, 3060, 3197, 3337, 3480 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 Also number of cards to build an n-tier house of cards. - Martin Wohlgemuth, Aug 11 2002 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 Row sums of triangle A134403. Bisection of A001318. - Omar E. Pol, Aug 22 2011 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 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 Partial sums give A006002. - Denis Borris, Jan 07 2013 A002260 is the following array A read by antidiagonals:   0,  1,  2,  3,  4,  5, ...   0,  1,  2,  3,  4,  5, ...   0,  1,  2,  3,  4,  5, ...   0,  1,  2,  3,  4,  5, ...   0,  1,  2,  3,  4,  5, ...   0,  1,  2,  3,  4,  5, ... 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 From Klaus Purath, May 13 2021: (Start) This sequence and A000326 provide all integer m such that 24*m + 1 is a square. The union of the two sequences is A001318. 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) 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 REFERENCES Henri Cohen, A Course in Computational Algebraic Number Theory, vol. 138 of Graduate Texts in Mathematics, Springer-Verlag, 2000. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..2000 A. O. L. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp., Vol. 61, No. 203 (1993), pp. 29-68. Charles H. Conley and Valentin Ovsienko, Quiddities of polygon dissections and the Conway-Coxeter frieze equation, arXiv:2107.01234 [math.CO], 2021. Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, Acta Academiae Scientiarum Imperialis Petropolitanae, Vol. 1780: I, pp. 56-75. Leonhard Euler, Observatio de summis divisorum, Novi Commentarii academiae scientiarum Petropolitanae, Vol. 5, pp. 59-74. Leonhard Euler, An observation on the sums of divisors, arXiv:math/0411587 [math.HO], 2004-2009, p. 8. Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005. D. Suprijanto and I. W. Suwarno, Observation on Sums of Powers of Integers Divisible by 3k-1, Applied Mathematical Sciences, Vol. 8, No. 45 (2014), pp. 2211-2217. Martin Wohlgemuth, Pentagon, Kartenhaus und Summenzerlegung. Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = A110449(n, 1) for n>0. 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 a(n) = A001844(n) - A000217(n+1) = A101164(n+2,2) for n>0. - Reinhard Zumkeller, Dec 03 2004 a(n) = right term of M^n * [1 0 0] where M = the 3 X 3 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, Dec 19 2004 a(n) = Sum_{j=1..n} n+j. - Zerinvary Lajos, Sep 12 2006 a(n) = A126890(n,n). - Reinhard Zumkeller, Dec 30 2006 a(n) = 2*C(3*n,4)/C(3*n,2), n>=1. - Zerinvary Lajos, Jan 02 2007 a(n) = A000217(n) + A000290(n). - Zak Seidov, Apr 06 2008 a(n) = a(n-1) + 3*n-1 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010 a(n) = A129267(n+5,n). - Philippe Deléham, Dec 21 2011 a(n) = 2*A000217(n) + A000217(n-1). - Philippe Deléham, Mar 25 2013 a(n) = A130518(3*n+1). - Philippe Deléham, Mar 26 2013 a(n) = (12/(n+2)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+2). - Vladimir Kruchinin, Jun 04 2013 a(n) = floor(n/(1-exp(-2/(3*n)))) for n>0. - Richard R. Forberg, Jun 22 2013 a(n) = Sum_{i=1..n} 2*i - 1 + (i mod 2). - Wesley Ivan Hurt, Oct 11 2013 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 Sum_{n>=1} 1/a(n) = 6 - Pi/sqrt(3) - 3*log(3) = 0.89036376976145307522... . - Vaclav Kotesovec, Apr 27 2016 a(n) = A000217(2*n) - A000217(n). - Bruno Berselli, Sep 21 2016 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/sqrt(3) + 4*log(2) - 6. - Amiram Eldar, Jan 18 2021 From Klaus Purath, May 13 2021: (Start) Partial sums of A016789 for n>0. a(n) = 3*n^2 - A000326(n). a(n) = A000326(n) + n. a(n) = A002378(n+1) + A000217(n). (End) From Klaus Purath, Jul 14 2021: (Start) 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. a(2*n) = n*(a(n+1) - a(n-1)), n > 0. a(2*n+1) = n*(a(n+1) - a(n)). (End) EXAMPLE From Omar E. Pol, Aug 22 2011: (Start) Illustration of initial terms: .                                               O .                                             O O .                                 O         O O O .                               O O       O O O O .                     O       O O O     O O O O O .                   O O     O O O O     O O O O O .           O     O O O     O O O O     O O O O O .         O O     O O O     O O O O     O O O O O .    O    O O     O O O     O O O O     O O O O O .    O    O O     O O O     O O O O     O O O O O . .    2     7        15         26           40 . (End) MAPLE A005449:=n->n*(3*n + 1)/2; seq(A005449(k), k=0..100); # Wesley Ivan Hurt, Oct 11 2013 MATHEMATICA Table[n (3 n + 1)/2, {n, 0, 100}] (* Zak Seidov, Jan 31 2012 *) PROG (PARI) {a(n) = n * (3*n + 1) / 2} /* Michael Somos, Jul 18 2003 */ (Magma) [n*(3*n + 1) / 2: n in [0..40]]; // Vincenzo Librandi, May 02 2011 CROSSREFS Cf. A000217, A000320, A000326, A001318, A033568, A049451, A101165, A101166. 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. Cf. numbers of the form n*(n*k-k+4))/2 listed in A226488 (this sequence is the case k=3). Cf. numbers of the form n*((2*k+1)*n+1)/2 listed in A022289 (this sequence is the case k=1). Sequence in context: A194140 A029888 A194112 * A293401 A323038 A333637 Adjacent sequences:  A005446 A005447 A005448 * A005450 A005451 A005452 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)