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A002413
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Heptagonal (or 7-gonal) pyramidal numbers: n*(n+1)*(5*n-2)/6.
(Formerly M4498 N1904)
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24
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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;
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history;
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OFFSET
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0,3
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COMMENTS
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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]
Contribution from Ant King, Oct 25 2012: (Start)
For n > 0, the digital roots of this sequence A01088(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)
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number.
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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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.]
Contribution from Ant King, Oct 25 2012: (Start)
a(n) = a(n - 1) + n*(5n - 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 >= 0} 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)
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MATHEMATICA
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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 *)
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PROG
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(Maxima) A002413(n):=n*(n+1)*(5*n-2)/6$ makelist(A002413(n), n, 0, 20); /* Martin Ettl, Dec 12 2012 */
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CROSSREFS
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Cf. A093562 ((5, 1) Pascal, column m = 3).
Sequence in context: A215097 A111694 A129111 * A218325 A163121 A213039
Adjacent sequences: A002410 A002411 A002412 * A002414 A002415 A002416
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Dec 23 1999
a(0)=0 prepended by Max Alekseyev, Nov 23 2011
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STATUS
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approved
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