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A002414
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Octagonal pyramidal numbers: n*(n+1)*(2*n-1)/2.
(Formerly M4609 N1966)
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31
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1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, 1386, 1794, 2275, 2835, 3480, 4216, 5049, 5985, 7030, 8190, 9471, 10879, 12420, 14100, 15925, 17901, 20034, 22330, 24795, 27435, 30256, 33264, 36465, 39865, 43470, 47286, 51319, 55575, 60060, 64780
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OFFSET
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1,2
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COMMENTS
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Number of ways of covering 2n x 2n lattice with 2n^2 dominoes with exactly 2 horizontal dominoes.
Equals binomial transform of [0, 1, 7, 6, 0, 0, 0,...]. - Gary W. Adamson, Jun 14 2008, corrected Oct 25 2012
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF3 denominators of A156927. See A157704 for background information. - Johannes W. Meijer, Mar 07 2009
This sequence is related to A000326 by a(n) = n*A000326(n)-sum(A000326(i), i=0..n-1) and this is the case d=3 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. - Bruno Berselli, Apr 21 2010
2*a(n) gives the principal diagonal of the convolution array A213819. - Clark Kimberling, Jul 04 2012
Contribution from Ant King, Oct 26 2012: (Start)
Partial sums of the figurate octagonal numbers A000567. For each sequence with a linear recurrence with constant coefficients, the values reduced modulo some constant m generate a periodic sequence. For this sequence, these Pisano periods have length 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17,.. for m>=1.
(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.
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The statistics of dimers on a lattice, Physica, 27(1961), 1209-1225.
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=1..1000
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.
Index entries for sequences related to dominoes
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) = odd numbers * triangular numbers = (2*n-1)* binomial(n+1,2). - Xavier Acloque Oct 27 2003
G.f.: x*(1+5*x)/(1-x)^4. [Conjectured by Simon Plouffe in his 1992 dissertation.]
a(n) = A000578(n) + A000217(n-1) - Kieren MacMillan, Mar 19 2007
a(-n) = -A160378(n). - Michael Somos Mar 17 2011
Contribution from Ant King, Oct 26 2012: (Start)
a(n) = a(n-1) + n*(3*n-2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = n*A000326(n) - A002411(n-1).
a(n) = (n+1)*(2*A000567(n)+n)/6.
a(n) = A000292(n) + 5*A000292(n-1).= binomial(n+2,3)+5*binomial(n+1,3).
a(n) = A002413(n) + A000292(n-1).
a(n) = A000217(n) + 6*A000292(n-1).
Sum_{n>=0} 1/a(n) = 2*(4*log(2)-1)/3 = 1.1817258148265...
(End)
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EXAMPLE
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a(2) = 9 since there are 9 ways to cover a 4 X 4 lattice with 8 dominoes, 2 of which is horizontal and the other 6 are vertical.
x + 9*x^2 + 30*x^3 + 70*x^4 + 135*x^5 + 231*x^6 + 364*x^7 + 540*x^8 + 765*x^9 + ...
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MAPLE
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A002414 := n-> 1/2*n*(n+1)*(2*n-1);
nmax:=38; for n from 0 to nmax do fz(n):=product((1-(k+1)*z)^(1+3*k), k=0..n); c(n):= abs(coeff(fz(n), z, 1)); end do: a:=n-> c(n): seq(a(n), n=0..nmax); # Johannes W. Meijer, Mar 07 2009
a:=n-> add (j*(n+1)+n*(j-1), j=0..n): seq(a(n), n=1..40); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 18 2009]
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MATHEMATICA
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f[n_]:=6*n+1; s1=s2=0; lst={}; Do[a=f[n]; s1+=a; s2+=s1; AppendTo[lst, s2], {n, 0, 6!}]; lst [Vladimir Joseph Stephan Orlovsky, Jun 25 2009]
Table[Sum[(n^2 - i), {i, 0, n}], {n, 1, 40}] [Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 11 2009]
LinearRecurrence[{4, -6, 4, -1}, {1, 9, 30, 70}, 40] (* Harvey P. Dale, Apr 12 2013 *)
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PROG
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(PARI) {a(n) = (2*n - 1) * n * (n + 1) / 2} /* Michael Somos Mar 17 2011 */
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CROSSREFS
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Cf. A000217, A000578, A002411, A004003, A160378.
Cf. A093563 (( 6, 1) Pascal, column m=3). A000567 (differences).
Cf. A156927, A157704. - Johannes W. Meijer, Mar 07 2009
Cf. A000326.
Sequence in context: A005919 A084370 A000439 * A212517 A000440 A161684
Adjacent sequences: A002411 A002412 A002413 * A002415 A002416 A002417
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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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Additional comments from Yong Kong (ykong(AT)curagen.com), May 06 2000
More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
Incorrect formula deleted by Ant King, Oct 04 2012
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STATUS
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approved
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