OFFSET
0,2
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
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
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).
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 215.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
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
E.g.f.: exp(x)*(120 + 960*x + 1320*x^2 + 560*x^3 + 85*x^4 + 4*x^5)/120. - Stefano Spezia, Oct 29 2025
EXAMPLE
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
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 *)
PROG
(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
(Magma) [(4*n+5)*Binomial(n+4, 4)/5: n in [0..35]]; // G. C. Greubel, Aug 28 2019
(SageMath) [(4*n+5)*binomial(n+4, 4)/5 for n in (0..35)] # G. C. Greubel, Aug 28 2019
(GAP) List([0..35], n-> (4*n+5)*Binomial(n+4, 4)/5); # G. C. Greubel, Aug 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Barry E. Williams, Dec 13 1999
EXTENSIONS
Corrected and extended by N. J. A. Sloane, Apr 21 2000
STATUS
approved