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A002412 Hexagonal pyramidal numbers, or greengrocer's numbers.

%I A002412 M4374 N1839 #180 Feb 16 2025 08:32:25
%S A002412 0,1,7,22,50,95,161,252,372,525,715,946,1222,1547,1925,2360,2856,3417,
%T A002412 4047,4750,5530,6391,7337,8372,9500,10725,12051,13482,15022,16675,
%U A002412 18445,20336,22352,24497,26775,29190,31746,34447,37297,40300
%N A002412 Hexagonal pyramidal numbers, or greengrocer's numbers.
%C A002412 Binomial transform of (1, 6, 9, 4, 0, 0, 0, ...). - _Gary W. Adamson_, Oct 16 2007
%C A002412 a(n) is the sum of the maximum(m,n) over {(m,n): m,n in positive integers, m<=n}. - _Geoffrey Critzer_, Oct 11 2009
%C A002412 We obtain these numbers for d=2 in the identity n*(n*(d*n-d+2)/2)-sum(k*(d*k-d+2)/2, k=0..n-1) = n*(n+1)*(2*d*n-2*d+3)/6 (see Klaus Strassburger in Formula lines). - _Bruno Berselli_, Apr 21 2010, Nov 16 2010
%C A002412 q^a(n) is the Hankel transform of the q-Catalan numbers. - _Paul Barry_, Dec 15 2010
%C A002412 Row 1 of the convolution array A213835. - _Clark Kimberling_, Jul 04 2012
%C A002412 From _Ant King_, Oct 24 2012: (Start)
%C A002412 For n>0, the digital roots of this sequence A010888(A002412(n)) form the purely periodic 27-cycle {1,7,4,5,5,8,9,3,3,4,1,7,8,8,2,3,6,6,7,4,1,2,2,5,6,9,9}.
%C A002412 For n>0, the units' digits of this sequence A010879(A002412(n)) form the purely periodic 20-cycle {1,7,2,0,5,1,2,2,5,5,6,2,7,5,0,6,7,7,0,0}.
%C A002412 (End)
%C A002412 Partial sums of A000384. - _Omar E. Pol_, Jan 12 2013
%C A002412 Row sums of A094728. - _J. M. Bergot_, Jun 14 2013
%C A002412 Number of orbits of Aut(Z^7) as function of the infinity norm (n+1) of the representative integer lattice point of the orbit, when the cardinality of the orbit is  equal to 40320. - _Philippe A.J.G. Chevalier_, Dec 28 2015
%C A002412 Coefficients in the hypergeometric series identity 1 - 7*(x - 1)/(3*x + 1) + 22*(x - 1)*(x - 2)/((3*x + 1)*(3*x + 2)) - 50*(x - 1)*(x - 2)*(x - 3)/((3*x + 1)*(3*x + 2)*(3*x + 3)) + ... = 0, valid for Re(x) > 1. Cf. A000326 and A002418. Column 3 of A103450. - _Peter Bala_, Mar 14 2019
%D A002412 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A002412 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D A002412 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 A002412 T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%D A002412 I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
%D A002412 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002412 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002412 T. D. Noe and William A. Tedeschi, <a href="/A002412/b002412.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms computed by T. D. Noe)
%H A002412 Abdullah Atmaca and A. Yavuz Oruç, <a href="https://doi.org/10.1016/j.akcej.2017.11.008">On the size of two families of unlabeled bipartite graphs</a>, AKCE International Journal of Graphs and Combinatorics, Vo. 16, No. 2 (2019), pp. 222-229.
%H A002412 Bruno Berselli, A description of the transform in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H A002412 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 A002412 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A002412 Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%H A002412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalPyramidalNumber.html">Hexagonal Pyramidal Number</a>.
%H A002412 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A002412 a(n) = n(n + 1)(4n - 1)/6.
%F A002412 G.f.: x*(1+3*x)/(1-x)^4. - _Simon Plouffe_ in his 1992 dissertation.
%F A002412 a(n) = n^3 - Sum_{i=1..n-1} i^2. - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%F A002412 Partial sums of n odd-indexed triangular numbers, e.g., a(3) = t(1)+t(3)+t(5) = 1+6+15 = 22. - _Jon Perry_, Jul 23 2003
%F A002412 a(n) = Sum_{i=0..n-1} (n - i)*(n + i). - _Jon Perry_, Sep 26 2004
%F A002412 a(n) = n*A000292(n) - (n-1)*A000292(n-1) = n*binomial((n+2),3) - (n-1)*binomial((n+1),3); e.g., a(5) = 95 = 5*35 - 4*20. - _Gary W. Adamson_, Dec 28 2007
%F A002412 a(n) = Sum_{i=0..n} (2i^2 + 3i + 1), for n >= 0 (Omits the leading 0). - _William A. Tedeschi_, Aug 25 2010
%F A002412 a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4), with a(0)=0, a(1)=1, a(2)=7, a(3)=22. - _Harvey P. Dale_, Jul 16 2011
%F A002412 a(n) = sum a*b, where the summing is over all unordered partitions 2*n = a+b. - _Vladimir Shevelev_, May 11 2012
%F A002412 From _Ant King_, Oct 24 2012: (Start)
%F A002412 a(n) = a(n-1) + n*(2*n-1).
%F A002412 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 4.
%F A002412 a(n) = (n+1)*(2*A000384(n) + n)/6 = (4*n-1)*A000217(n)/3.
%F A002412 a(n) = A000292(n) + 3*A000292(n-1) = A002411(n) + A000292(n-1).
%F A002412 a(n) = binomial(n+2,3) + 3*binomial(n+1,3) = (4*n-1)*binomial(n+1,2)/3.
%F A002412 Sum_{n>=1} 1/a(n) = 6*(12*log(2)-2*Pi-1)/5 = 1.2414...
%F A002412 (End)
%F A002412 a(n) = Sum_{i=1..n} Sum_{j=1..n} max(i,j) = Sum_{i=1..n} i*(2*n-i). - _Enrique Pérez Herrero_, Jan 15 2013
%F A002412 a(n) = A005900(n+1) - A000326(n+1) = Octahedral - Pentagonal Numbers. - _Richard R. Forberg_, Aug 07 2013
%F A002412 a(n) = n*A000217(n) + Sum_{i=0..n-1} A000217(i). - _Bruno Berselli_, Dec 18 2013
%F A002412 a(n) = 2n * A000217(n) - A000330(n). - _J. M. Bergot_, Apr 05 2014
%F A002412 a(n) = A080851(4,n-1). - _R. J. Mathar_, Jul 28 2016
%F A002412 E.g.f.: x*(6 + 15*x + 4*x^2)*exp(x)/6. - _Ilya Gutkovskiy_, May 12 2017
%F A002412 Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(1 + 2*sqrt(2)*Pi - 2*(3+sqrt(2))*log(2) + 4*sqrt(2)*log(2-sqrt(2)))/5. - _Amiram Eldar_, Jan 04 2022
%e A002412 Let n=5, 2*n=10. Since 10 = 1 + 9 = 2 + 8 = 3 + 7 = 4 + 6 = 5 + 5, a(5) = 1*9 + 2*8 + 3*7 + 4*6 + 5*5 = 95. - _Vladimir Shevelev_, May 11 2012
%p A002412 seq(sum(i*(2*k-i), i=1..k), k=0..100); # _Wesley Ivan Hurt_, Sep 25 2013
%t A002412 Figurate[ ngon_, rank_, dim_] := Binomial[rank + dim - 2, dim - 1] ((rank - 1)*(ngon - 2) + dim)/dim; Table[ Figurate[6, r, 3], {r, 0, 40}] (* _Robert G. Wilson v_, Aug 22 2010 *)
%t A002412 Table[n(n+1)(4n-1)/6, {n,0,40}] (* or *) LinearRecurrence[{4,-6,4,-1}, {0,1,7,22}, 40] (* _Harvey P. Dale_, Jul 16 2011 *)
%o A002412 (PARI) v=vector(40,i,(i*(i+1))\2); s=0; print1(s","); forstep(i=1,40,2,s+=v[i]; print1(s","))
%o A002412 (Maxima) A002412(n):=n*(n+1)*(4*n-1)/6$ makelist(A002412(n),n,0,20); /* _Martin Ettl_, Dec 12 2012 */
%o A002412 (Magma) [n*(n+1)*(4*n-1)/6: n in [0..40]]; // _Vincenzo Librandi_, Nov 28 2015
%o A002412 (GAP) List([0..40],n->n*(n+1)*(4*n-1)/6); # _Muniru A Asiru_, Mar 18 2019
%o A002412 (Python) print([n*(n+1)*(4*n-1)//6 for n in range(40)]) # _Michael S. Branicky_, Mar 28 2022
%Y A002412 Bisection of A002623. Equals A000578(n) - A000330(n-1).
%Y A002412 Cf. A000292, A016061.
%Y A002412 a(n) = A093561(n+2, 3), (4, 1)-Pascal column.
%Y A002412 Cf. A220084 for a list of numbers of the form n*P(k,n)-(n-1)*P(k,n-1), where P(k,n) is the n-th k-gonal pyramidal number (see Adamson's formula).
%Y A002412 Cf. similar sequences listed in A237616.
%Y A002412 Orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008585, A005843, A001477, A000217.
%K A002412 nonn,easy,nice
%O A002412 0,3
%A A002412 _N. J. A. Sloane_