%I M4116 N1709 #312 Feb 16 2025 08:32:25

%S 0,1,6,18,40,75,126,196,288,405,550,726,936,1183,1470,1800,2176,2601,

%T 3078,3610,4200,4851,5566,6348,7200,8125,9126,10206,11368,12615,13950,

%U 15376,16896,18513,20230,22050,23976,26011,28158,30420,32800,35301,37926,40678

%N Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.

%C a(n) = n^2(n+1)/2 is half the number of colorings of three points on a line with n+1 colors. - _R. H. Hardin_, Feb 23 2002

%C Sum of n smallest multiples of n. - _Amarnath Murthy_, Sep 20 2002

%C a(n) = number of (n+6)-bit binary sequences with exactly 7 1's none of which is isolated. A 1 is isolated if its immediate neighbor(s) are 0. - _David Callan_, Jul 15 2004

%C Also as a(n) = (1/6)*(3*n^3+3*n^2), n > 0: structured trigonal prism numbers (cf. A100177 - structured prisms; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

%C Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Nov 18 2005

%C If Y is a 3-subset of an n-set X then, for n >= 5, a(n-4) is the number of 5-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Nov 23 2007

%C a(n-1), n >= 2, is the number of ways to have n identical objects in m=2 of altogether n distinguishable boxes (n-2 boxes stay empty). - _Wolfdieter Lang_, Nov 13 2007

%C a(n+1) is the convolution of (n+1) and (3n+1). - _Paul Barry_, Sep 18 2008

%C The number of 3-character strings from an alphabet of n symbols, if a string and its reversal are considered to be the same.

%C Partial sums give A001296. - _Jonathan Vos Post_, Mar 26 2011

%C a(n-1):=N_1(n), n >= 1, is the number of edges of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p.506. - _Wolfdieter Lang_, May 27 2011

%C Partial sums of pentagonal numbers A000326. - _Reinhard Zumkeller_, Jul 07 2012

%C From _Ant King_, Oct 23 2012: (Start)

%C For n > 0, the digital roots of this sequence A010888(A002411(n)) form the purely periodic 9-cycle {1,6,9,4,3,9,7,9,9}.

%C For n > 0, the units' digits of this sequence A010879(A002411(n)) form the purely periodic 20-cycle {1,6,8,0,5,6,6,8,5,0,6,6,3,0,0,6,1,8,0,0}.

%C (End)

%C a(n) is the number of inequivalent ways to color a path graph having 3 nodes using at most n colors. Note, here there is no restriction on the color of adjacent nodes as in the above comment by _R. H. Hardin_ (Feb 23 2002). Also, here the structures are counted up to graph isomorphism, where as in the above comment the "three points on a line" are considered to be embedded in the plane. - _Geoffrey Critzer_, Mar 20 2013

%C After 0, row sums of the triangle in A101468. - _Bruno Berselli_, Feb 10 2014

%C Latin Square Towers: Take a Latin square of order n, with symbols from 1 to n, and replace each symbol x with a tower of height x. Then the total number of unit cubes used is a(n). - _Arun Giridhar_, Mar 29 2015

%C This is the case k = n+4 of b(n,k) = n*((k-2)*n-(k-4))/2, which is the n-th k-gonal number. Therefore, this is the 3rd upper diagonal of the array in A139600. - _Luciano Ancora_, Apr 11 2015

%C For n > 0, a(n) is the number of compositions of n+7 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016

%C Also the Wiener index of the n-antiprism graph. - _Eric W. Weisstein_, Sep 07 2017

%C For n > 0, a(2n+1) is the number of non-isomorphic 5C_m-snakes, where m = 2n+1 or m = 2n (for n >= 2). A kC_n-snake is a connected graph in which the k >= 2 blocks are isomorphic to the cycle C_n and the block-cutpoint graph is a path. - _Christian Barrientos_, May 15 2019

%C For n >= 1, a(n-1) is the number of 0°- and 45°-tilted squares that can be drawn by joining points in an n X n lattice. - _Paolo Xausa_, Apr 13 2021

%C a(n) is the number of all possible products of n rolls of a six-sided die. This can be easily seen by the recursive formula a(n) = a(n - 1) + 2 * binomial(n, 2) + binomial(n + 1, 2). - _Rafal Walczak_, Jun 15 2024

%D V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_1.

%D Christian Barrientos, Graceful labelings of cyclic snakes, Ars Combin., Vol. 60 (2001), pp. 85-96.

%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.

%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 166, Table 10.4/I/5).

%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.

%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see Vol. 2, p. 2.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002411/b002411.txt">Table of n, a(n) for n = 0..1000</a>

%H Somaya Barati, Beáta Bényi, Abbas Jafarzadeh and Daniel Yaqubi, <a href="https://arxiv.org/abs/1812.02955">Mixed restricted Stirling numbers</a>, arXiv:1812.02955 [math.CO], 2018.

%H Phyllis Chinn and Silvia Heubach, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL6/Heubach/heubach5.html">Integer Sequences Related to Compositions without 2's</a>, J. Integer Seq., Vol. 6 (2003), Article 03.2.3.

%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 17.

%H Lancelot Hogben, <a href="https://archive.org/details/chanceandchoiceb029729mbp/page/n39">Choice and Chance by Cardpack and Chessboard</a>, Vol. 1, Max Parrish and Co, London, 1950, p. 36.

%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, Vol 19 (2016), Article 16.7.3.

%H C. Krishnamachaki, <a href="/A001296/a001296.pdf">The operator (xD)^n</a>, J. Indian Math. Soc., Vol. 15 (1923), pp. 3-4. [Annotated scanned copy]

%H S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalPyramidalNumber.html">Pentagonal Pyramidal Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>.

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>.

%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F Average of n^2 and n^3.

%F G.f.: x*(1+2*x)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation

%F a(n) = n*Sum_{k=0..n} (n-k) = n*Sum_{k=0..n} k. - _Paul Barry_, Jul 21 2003

%F a(n) = n*A000217(n). - Xavier Acloque, Oct 27 2003

%F a(n) = (1/2)*Sum_{j=1..n} Sum_{i=1..n} (i+j) = (1/2)*(n^2+n^3) = (1/2)*A011379(n). - _Alexander Adamchuk_, Apr 13 2006

%F Row sums of triangle A127739, triangle A132118; and binomial transform of [1, 5, 7, 3, 0, 0, 0, ...] = (1, 6, 18, 40, 75, ...). - _Gary W. Adamson_, Aug 10 2007

%F G.f.: x*F(2,3;1;x). - _Paul Barry_, Sep 18 2008

%F Sum_{j>=1} 1/a(j) = hypergeom([1, 1, 1], [2, 3], 1) = -2 + 2*zeta(2) = A195055 - 2. - _Stephen Crowley_, Jun 28 2009

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=6, a(3)=18. - _Harvey P. Dale_, Oct 20 2011

%F From _Ant King_, Oct 23 2012: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 3.

%F a(n) = (n+1)*(2*A000326(n)+n)/6 = A000292(n) + 2*A000292(n-1).

%F a(n) = A000330(n)+A000292(n-1) = A000217(n) + 3*A000292(n-1).

%F a(n) = binomial(n+2,3) + 2*binomial(n+1,3).

%F (End)

%F a(n) = (A000330(n) + A002412(n))/2 = (A000292(n) + A002413(n))/2. - _Omar E. Pol_, Jan 11 2013

%F a(n) = (24/(n+3)!)*Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*j^(n+3). - _Vladimir Kruchinin_, Jun 04 2013

%F Sum_{n>=1} a(n)/n! = (7/2)*exp(1). - _Richard R. Forberg_, Jul 15 2013

%F E.g.f.: x*(2 + 4*x + x^2)*exp(x)/2. - _Ilya Gutkovskiy_, May 31 2016

%F From _R. J. Mathar_, Jul 28 2016: (Start)

%F a(n) = A057145(n+4,n).

%F a(n) = A080851(3,n-1). (End)

%F For n >= 1, a(n) = (Sum_{i=1..n} i^2) + Sum_{i=0..n-1} i^2*((i+n) mod 2). - _Paolo Xausa_, Apr 13 2021

%F a(n) = Sum_{k=1..n} GCD(k,n) * LCM(k,n). - _Vaclav Kotesovec_, May 22 2021

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2 + Pi^2/6 - 4*log(2). - _Amiram Eldar_, Jan 03 2022

%e a(3)=18 because 4 identical balls can be put into m=2 of n=4 distinguishable boxes in binomial(4,2)*(2!/(1!*1!) + 2!/2!) = 6*(2+1) = 18 ways. The m=2 part partitions of 4, namely (1,3) and (2,2), specify the filling of each of the 6 possible two-box choices. - _Wolfdieter Lang_, Nov 13 2007

%p seq(n^2*(n+1)/2, n=0..40);

%t Table[n^2 (n + 1)/2, {n, 0, 40}]

%t LinearRecurrence[{4, -6, 4, -1}, {0, 1, 6, 18}, 50] (* _Harvey P. Dale_, Oct 20 2011 *)

%t Nest[Accumulate, Range[1, 140, 3], 2]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2012 *)

%t CoefficientList[Series[x (1 + 2 x) / (1 - x)^4, {x, 0, 45}], x] (* _Vincenzo Librandi_, Jan 08 2016 *)

%o (PARI) a(n)=n^2*(n+1)/2

%o (Haskell)

%o a002411 n = n * a000217 n -- _Reinhard Zumkeller_, Jul 07 2012

%o (Magma) [n^2*(n+1)/2: n in [0..40]]; // _Wesley Ivan Hurt_, May 25 2014

%o (PARI) concat(0, Vec(x*(1+2*x)/(1-x)^4 + O(x^100))) \\ _Altug Alkan_, Jan 07 2016

%o (GAP) List([0..45], n->n^2*(n+1)/2); # _Muniru A Asiru_, Feb 19 2018

%Y Cf. A001296, A011379, A015223, A015224, A014799, A014800, A127739, A132118, A139600.

%Y A006002(n) = -a(-1-n).

%Y a(n) = A093560(n+2, 3), (3, 1)-Pascal column.

%Y A row or column of A132191.

%Y Second column of triangle A103371.

%Y Cf. similar sequences listed in A237616.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_