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A002412
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Hexagonal pyramidal numbers, or greengrocer's numbers.
(Formerly M4374 N1839)
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51
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0, 1, 7, 22, 50, 95, 161, 252, 372, 525, 715, 946, 1222, 1547, 1925, 2360, 2856, 3417, 4047, 4750, 5530, 6391, 7337, 8372, 9500, 10725, 12051, 13482, 15022, 16675, 18445, 20336, 22352, 24497, 26775, 29190, 31746, 34447, 37297, 40300
(list;
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OFFSET
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0,3
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COMMENTS
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Binomial transform of (1, 6, 9, 4, 0, 0, 0,...). - Gary W. Adamson, Oct 16 2007
a(n) is the sum of the maximum(m,n) over {(m,n):m,n in positive integers,m<=n} [From Geoffrey Critzer, Oct 11 2009]
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
q^a(n) is the Hankel transform of the q-Catalan numbers. [From Paul Barry, Dec 15 2010]
Row 1 of the convolution array A213835. [Clark Kimberling, Jul 04 2012]
Contribution from Ant King, Oct 24 2012: (Start)
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}.
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}.
(End)
Partial sums of A000384. - Omar E. Pol, Jan 12 2013
Row sums of A094728. - J. M. Bergot, Jun 14 2013
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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.
T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
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.
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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William A. Tedeschi, Table of n, a(n) for n = 0..10000 [This replaces an earlier b-file computed by T. D. Noe]
B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).
_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, Hexagonal Pyramidal Number
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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a(n) = n*(n+1)*(4*n-1)/6.
G.f.: x*(1+3*x)/(1-x)^4. [Simon Plouffe in his 1992 dissertation.]
a(n) = n^3-sum(i^2, i=1..(n-1)). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de).
Partial sums of n odd triangular numbers, e.g. a(3)=t(1)+t(3)+t(5)=1+6+15=22 - Jon Perry, Jul 23 2003
a(n)=sum(i=0, n-1, (n-i)*(n+i)). - Jon Perry, Sep 26 2004
a(n) = n*A000292(n) - (n-1)*A000292(n-1) = n*C((n+2),3) - (n-1)*C((n+1),3); e.g. a(5) = 95 = 5*35 - 4*20. - Gary W. Adamson, Dec 28 2007
a(n) = sum(i=0..n, 2*i^2 + 3*i + 1) for n >= 0. Omits the leading 0. [William A. Tedeschi, Aug 25 2010]
a(0)=0, a(1)=1, a(2)=7, a(3)=22, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) [Harvey P. Dale, Jul 16 2011]
a(n)=sum a*b, where the summing is over all unordered partitions 2*n=a+b. - Vladimir Shevelev, May 11 2012
Contribution from Ant King, Oct 24 2012: (Start)
a(n) = a(n-1) +n*(2n-1).
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +4.
a(n) = (n+1)*(2*A000384(n)+n)/6 = (4*n-1)*A000217(n)/3.
a(n) = A000292(n) +3*A000292(n-1) = A002411(n) +A000292(n-1).
a(n) = binomial(n+2,3)+3*binomial(n+1,3) = (4*n-1)*binomial(n+1,2)/3.
Sum_{n>=0} 1/a(n) = 6*(12*log(2)-2*pi-1)/5 = 1.2414...
(End)
a(n) = sum(i=1..n, sum(j=1..n, max(i,j))) = sum(i<=n, i*(2n-i)). - Enrique Pérez Herrero, Jan 15 2013
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EXAMPLE
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Let n=5, 2*n=10. Since 10=1+9=2+8=3+7=4+6=5+5, then a(5)=1*9+2*8+3*7+4*6+5*5=95. - Vladimir Shevelev, May 11 2012
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MATHEMATICA
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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 *)
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 *)
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PROG
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(PARI) v=vector(40, i, t(i)); s=0; forstep(i=1, 40, 2, s+=v[i]; print1(s", "))
(Maxima) A002412(n):=n*(n+1)*(4*n-1)/6$ makelist(A002412(n), n, 0, 20); /* Martin Ettl, Dec 12 2012 */
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CROSSREFS
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Bisection of A002623. Equals A000578(n)-A000330(n-1).
Cf. A000292, A016061.
a(n)= A093561(n+2, 3), (4, 1)-Pascal column.
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).
Sequence in context: A014073 A129109 A224141 * A211652 A211650 A211792
Adjacent sequences: A002409 A002410 A002411 * A002413 A002414 A002415
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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