cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A038163 G.f.: 1/((1-x)*(1-x^2))^3.

Original entry on oeis.org

1, 3, 9, 19, 39, 69, 119, 189, 294, 434, 630, 882, 1218, 1638, 2178, 2838, 3663, 4653, 5863, 7293, 9009, 11011, 13377, 16107, 19292, 22932, 27132, 31892, 37332, 43452, 50388, 58140, 66861, 76551, 87381, 99351, 112651, 127281
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of symmetric nonnegative integer 6 X 6 matrices with sum of elements equal to 4*n, under action of dihedral group D_4. - Vladeta Jovovic, May 14 2000
Equals the triangular sequence convolved with the aerated triangular sequence, [1, 0, 3, 0, 6, 0, 10, ...]. - Gary W. Adamson, Jun 11 2009
Number of partitions of n (n>=1) into 1s and 2s if there are three kinds of 1s and three kinds of 2s. Example: a(2)=9 because we have 11, 11', 11", 1'1', 1'1", 1"1", 2, 2', and 2". - Emeric Deutsch, Jun 26 2009
Equals the tetrahedral numbers with repeats convolved with the natural numbers: (1 + x + 4x^2 + 4x^3 + ...) * (1 + 2x + 3x^2 + 4x^3 + ...) = (1 + 3x + 9x^2 + 19x^3 + ...). - Gary W. Adamson, Dec 22 2010

Links

Crossrefs

Cf. A008619, A006918, A001753.
Cf. A096338.
Column k=3 of A210391. - Alois P. Heinz, Mar 22 2012
Cf. A000217.

Programs

  • Haskell
    import Data.List (inits, intersperse)
a038163 n = a038163_list !! n
a038163_list = map
    (sum . zipWith (*) (intersperse 0 $ tail a000217_list) . reverse) $
    tail $ inits $ tail a000217_list where
-- Reinhard Zumkeller, Feb 27 2015
  • Maple
    G := 1/((1-x)^3*(1-x^2)^3): Gser := series(G, x = 0, 42): seq(coeff(Gser, x, n), n = 0 .. 37); # Emeric Deutsch, Jun 26 2009
# alternative
A038163 := proc(n)
    (4*n^5+90*n^4+760*n^3+2970*n^2+5266*n+3285+(-1)^n*(30*n^2+270*n+555))/3840 ;
end proc:
seq(A038163(n),n=0..30) ; # R. J. Mathar, Feb 22 2021
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-x^2))^3, {x, 0, 40}], x] (* Jean-François Alcover, Mar 11 2014 *)
LinearRecurrence[{3,0,-8,6,6,-8,0,3,-1},{1,3,9,19,39,69,119,189,294},50] (* Harvey P. Dale, Nov 24 2022 *)

Formula

a(2*k) = (4*k + 5)*binomial(k + 4, 4)/5 = A034263(k); a(2*k + 1) = binomial(k + 4, 4)*(15 + 4*k)/5 = A059599(k), k >= 0.
a(n) = (1/3840)*(4*n^5 + 90*n^4 + 760*n^3 + 2970*n^2 + 5266*n + 3285 + (-1)^n*(30*n^2 + 270*n + 555)). Recurrence: a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9). - Vladeta Jovovic, Apr 24 2002
a(n+1) - a(n) = A096338(n+2). - R. J. Mathar, Nov 04 2008

A034263 a(n) = binomial(n+4,4)*(4*n+5)/5.

Original entry on oeis.org

1, 9, 39, 119, 294, 630, 1218, 2178, 3663, 5863, 9009, 13377, 19292, 27132, 37332, 50388, 66861, 87381, 112651, 143451, 180642, 225170, 278070, 340470, 413595, 498771, 597429, 711109, 841464, 990264, 1159400, 1350888, 1566873, 1809633, 2081583, 2385279
Offset: 0

Views

Author

Clark Kimberling, Barry E. Williams, Dec 13 1999

Keywords

Comments

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
5-dimensional form of hexagonal-based pyramid numbers. - Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005
Convolution of triangular numbers (A000217) and hexagonal numbers (A000384). - Bruno Berselli, Jun 27 2013

Examples

			By the third comment: A000217(1..6) and A000384(1..6) give the term a(5) = 1*21+5*15+12*10+22*6+35*3+51*1 = 630. - _Bruno Berselli_, Jun 27 2013

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
  • Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
  • S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/4).

Links

Crossrefs

Partial sums of A002417.
Cf. A000217, A000384, A000292, A034261, A059599, A093561.
Cf. similar sequences listed in A254142.

Programs

  • GAP
    List([0..35], n-> (4*n+5)*Binomial(n+4,4)/5); # G. C. Greubel, Aug 28 2019
  • Magma
    [(4*n+5)*Binomial(n+4,4)/5: n in [0..35]]; // G. C. Greubel, Aug 28 2019
  • Maple
    a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120: seq(a(n),n=0..35); # Emeric Deutsch, Nov 18 2005
  • Mathematica
    Table[Binomial[n+4, 4]*(4*n+5)/5, {n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
a[n_] := (1+n)(2+n)(3+n)(4+n)(4n+5)/120; Array[a, 36, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 39, 119, 294, 630}, 36] (* or *)
CoefficientList[ Series[(1+3*x)/(1-x)^6, {x, 0, 35}], x] (* Robert G. Wilson v, Feb 26 2015 *)
Table[Sum[-x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 1, 30}]/4 (* Horst H. Manninger, Jun 19 2025 *)
  • PARI
    a(n)=(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120 \\ Charles R Greathouse IV, Sep 24 2015, corrected by Altug Alkan, Aug 15 2017
  • Sage
    [(4*n+5)*binomial(n+4,4)/5 for n in (0..35)] # G. C. Greubel, Aug 28 2019

Formula

a(n) = A093561(n+5, 5).
a(n) = A034261(n+1, 3).
G.f.: (1+3*x)/(1-x)^6.
a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120. - Emeric Deutsch and Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005, corrected by Eric Rowland, Aug 15 2017
a(-n-4) = -A059599(n). - Bruno Berselli, Aug 23 2011
a(n) = Sum_{i=1..n+1} i*A000292(i). - Bruno Berselli, Jan 23 2015
Sum_{n>=0} 1/a(n) = 28300/231 - 1280*Pi/77 - 7680*log(2)/77. - Amiram Eldar, Feb 15 2022

Extensions

Corrected and extended by N. J. A. Sloane, Apr 21 2000
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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