cp's OEIS Frontend

A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.

%I A002413 M4498 N1904 #99 Jul 02 2025 16:01:54
%S A002413 0,1,8,26,60,115,196,308,456,645,880,1166,1508,1911,2380,2920,3536,
%T A002413 4233,5016,5890,6860,7931,9108,10396,11800,13325,14976,16758,18676,
%U A002413 20735,22940,25296,27808,30481,33320,36330,39516,42883,46436,50180,54120
%N A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6.
%C A002413 The partial sums of A000566. - _R. J. Mathar_, Mar 19 2008
%C A002413 A002413(n + 1) is the number of 4-tuples (w, x, y, z) having all terms in {0, ..., n} and w = floor((x + y + z)/2).  - _Clark Kimberling_, May 28 2012
%C A002413 From _Ant King_, Oct 25 2012: (Start)
%C A002413 For n > 0, the digital roots of this sequence A010888(A002413(n)) form the purely periodic 27-cycle {1, 8, 8, 6, 7, 7, 2, 6, 6, 7, 5, 5, 3, 4, 4, 8, 3, 3, 4, 2, 2, 9, 1, 1, 5, 9, 9}.
%C A002413 For n > 0, the units' digits of this sequence A010879(A002413(n)) form the purely periodic 20-cycle {1, 8, 6, 0, 5, 6, 8, 6, 5, 0, 6, 8, 1, 0, 0, 6, 3, 6, 0, 0}.
%C A002413 (End)
%D A002413 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D A002413 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D A002413 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 A002413 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002413 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002413 T. D. Noe, <a href="/A002413/b002413.txt">Table of n, a(n) for n = 0..1000</a>
%H A002413 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 A002413 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002413 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalPyramidalNumber.html">Heptagonal Pyramidal Number.</a>
%H A002413 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H A002413 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A002413 a(n) = n*(n + 1)*(5*n - 2)/6.
%F A002413 G.f.: x*(1 + 4*x)/(1 - x)^4. [Suggested by _Simon Plouffe_ in his 1992 dissertation.]
%F A002413 From _Ant King_, Oct 25 2012: (Start)
%F A002413 a(n) = a(n - 1) + n*(5*n - 3)/2.
%F A002413 a(n) = 3*a(n - 1) - 3*a(n - 2) + a(n - 3) + 5.
%F A002413 a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4)
%F A002413 a(n) = (n + 1)*(2*A000566(n) + n)/6 = (5*n - 2)*A000217(n)/3.
%F A002413 a(n) = A000292(n) + 4*A000292(n - 1)
%F A002413 a(n) = A002412(n) + A000292(n - 1)
%F A002413 a(n) = A000217(n) + 5*A000292(n - 1)
%F A002413 a(n) = binomial(n + 2, 3) + 4*binomial(n + 1, 3) = (5*n - 2) * binomial(n + 1, 2)/3.
%F A002413 Sum_{n >= 1} 1/a(n) = 15*(log(3125) + sqrt(5)*log((3 - sqrt(5))/2) - 2*Pi*sqrt(5*(5 - 2*sqrt(5)))/5 - 8/5)/28 = 1.207293...
%F A002413 (End)
%F A002413 a(n) = Sum_{i=0..n-1} (n-i)*(5*i+1). - _Bruno Berselli_, Feb 10 2014
%F A002413 a(n) = A080851(5,n-1). - _R. J. Mathar_, Jul 28 2016
%F A002413 E.g.f.: x*(6 + 18*x + 5*x^2)*exp(x)/6. - _Ilya Gutkovskiy_, May 12 2017
%F A002413 a(n) = Sum_{i=0..n-1} (n+2*i)*(n-i). - _Leonid Bedratyuk_, Jul 09 2024
%e A002413 For n=7, a(7) = 7*1 + 6*6 + 5*11 + 4*16 + 3*21 + 2*26 + 1*31 = 308. - _Bruno Berselli_, Feb 10 2014
%p A002413 A002413:=n->n*(n+1)*(5*n-2)/6: seq(A002413(n), n=0..60); # _Wesley Ivan Hurt_, Apr 14 2017
%t A002413 LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, 40] (* _Ant King_, Oct 25 2012 *)
%t A002413 Table[(5n^3 + 3n^2 - 2n)/6, {n, 0, 39}] (* _Alonso del Arte_, Oct 25 2012 *)
%o A002413 (Maxima) A002413(n):=n*(n+1)*(5*n-2)/6$ makelist(A002413(n),n,0,20); /* _Martin Ettl_, Dec 12 2012 */
%o A002413 (PARI) a(n)=n*(n+1)*(5*n-2)/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A002413 (Magma) [n*(n + 1)*(5*n - 2)/6: n in [0..50]]; // _G. C. Greubel_, Nov 04 2017
%Y A002413 Cf. A093562 ((5, 1) Pascal, column m = 3).
%Y A002413 Cf. similar sequences listed in A237616.
%K A002413 nonn,easy,nice
%O A002413 0,3
%A A002413 _N. J. A. Sloane_
%E A002413 More terms from _James Sellers_, Dec 23 1999
%E A002413 a(0)=0 prepended by _Max Alekseyev_, Nov 23 2011