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 A002413 Heptagonal (or 7-gonal) pyramidal numbers: a(n) = n*(n+1)*(5*n-2)/6. (Formerly M4498 N1904) 32
 0, 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, 1166, 1508, 1911, 2380, 2920, 3536, 4233, 5016, 5890, 6860, 7931, 9108, 10396, 11800, 13325, 14976, 16758, 18676, 20735, 22940, 25296, 27808, 30481, 33320, 36330, 39516, 42883, 46436, 50180, 54120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The partial sums of A000566. - R. J. Mathar, Mar 19 2008 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 From Ant King, Oct 25 2012: (Start) 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}. 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}. (End) REFERENCES A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194. E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. 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. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n = 0..1000 Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number. Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1). FORMULA a(n) = n*(n + 1)*(5*n - 2)/6. G.f.: x*(1 + 4*x)/(1 - x)^4. [Suggested by Simon Plouffe in his 1992 dissertation.] From Ant King, Oct 25 2012: (Start) a(n) = a(n - 1) + n*(5*n - 3)/2. a(n) = 3*a(n - 1) - 3*a(n - 2) + a(n - 3) + 5. a(n) = 4*a(n - 1) - 6*a(n - 2) + 4*a(n - 3) - a(n - 4) a(n) = (n + 1)*(2*A000566(n) + n)/6 = (5*n - 2)*A000217(n)/3. a(n) = A000292(n) + 4*A000292(n - 1) a(n) = A002412(n) + A000292(n - 1) a(n) = A000217(n) + 5*A000292(n - 1) a(n) = binomial(n + 2, 3) + 4*binomial(n + 1, 3) = (5*n - 2) * binomial(n + 1, 2)/3. 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... (End) a(n) = Sum_{i=0..n-1} (n-i)*(5*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014 a(n) = A080851(5,n-1). - R. J. Mathar, Jul 28 2016 E.g.f.: x*(6 + 18*x + 5*x^2)*exp(x)/6. - Ilya Gutkovskiy, May 12 2017 EXAMPLE 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 MAPLE A002413:=n->n*(n+1)*(5*n-2)/6: seq(A002413(n), n=0..60); # Wesley Ivan Hurt, Apr 14 2017 MATHEMATICA LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 60}, 40] (* Ant King, Oct 25 2012 *) Table[(5n^3 + 3n^2 - 2n)/6, {n, 0, 39}] (* Alonso del Arte, Oct 25 2012 *) PROG (Maxima) A002413(n):=n*(n+1)*(5*n-2)/6\$ makelist(A002413(n), n, 0, 20); /* Martin Ettl, Dec 12 2012 */ (PARI) a(n)=n*(n+1)*(5*n-2)/6 \\ Charles R Greathouse IV, Sep 24 2015 (MAGMA) [n*(n + 1)*(5*n - 2)/6: n in [0..50]]; // G. C. Greubel, Nov 04 2017 CROSSREFS Cf. A093562 ((5, 1) Pascal, column m = 3). Cf. similar sequences listed in A237616. Sequence in context: A215097 A111694 A129111 * A218325 A252870 A163121 Adjacent sequences:  A002410 A002411 A002412 * A002414 A002415 A002416 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS More terms from James A. Sellers, Dec 23 1999 a(0)=0 prepended by Max Alekseyev, Nov 23 2011 STATUS approved

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Last modified October 14 12:02 EDT 2019. Contains 328004 sequences. (Running on oeis4.)