OFFSET
0,2
COMMENTS
4-dimensional pyramidal number, composed of consecutive 3-dimensional slices; each of which is a 3-dimensional 12-gonal (or dodecagonal) pyramidal number; which in turn is composed of consecutive 2-dimensional slices 12-gonal numbers. - Jonathan Vos Post, Mar 17 2006
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 211.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
Murray R. Spiegel, Calculus of Finite Differences and Difference Equations, "Schaum's Outline Series", McGraw-Hill, 1971, pp. 10-20, 79-94.
LINKS
Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = C(n+3, 3)*(5*n+2)/2 = (n+1)*(n+2)*(n+3)*(5*n+2)/12.
G.f.: (1+9*x)/(1-x)^5.
From Amiram Eldar, Feb 11 2022: (Start)
Sum_{n>=0} 1/a(n) = (125*log(5) + 10*sqrt(5*(5-2*sqrt(5)))*Pi - 50*sqrt(5)*log(phi) - 84)/104, where phi is the golden ratio (A001622).
Sum_{n>=0} (-1)^n/a(n) = (50*sqrt(5)*log(phi) + 5*sqrt(50-10*sqrt(5))*Pi - 256*log(2) + 90)/52. (End)
MATHEMATICA
Accumulate[Table[n(n+1)(10n-7)/6, {n, 0, 50}]] (* Harvey P. Dale, Nov 13 2013 *)
PROG
(Magma) /* A000027 convolved with A051624 (excluding 0): */ A051624:=func<n | n*(5*n-4)>; [&+[(n-i+1)*A051624(i): i in [1..n]]: n in [1..35]]; // Bruno Berselli, Dec 07 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 11 1999
STATUS
approved
