OFFSET
1,2
COMMENTS
The Ca4 triangle sums of A139600 are given by the terms of this sequence. For the definitions of the Ca4 and other triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Polygonal number.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = A062026(n)/2.
G.f.: -x*(1-2*x+4*x^2)/(x-1)^5. - Simon Plouffe in his 1992 dissertation.
a(n) = Sum_{i=0..n} (i(i-2)^2 + i^2)/2.
a(n) = binomial(n+3,4) - 2*binomial(n+2,4) + 4*binomial(n+1,4).
From Elmo R. Oliveira, Sep 09 2025: (Start)
E.g.f.: exp(x)*x*(x^3 + 4*x^2 + 4*x + 8)/8.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 32*log(2)/15 - 11/15 + 4*Pi*sech(sqrt(15)*Pi/2)/15. - Amiram Eldar, Nov 01 2025
MATHEMATICA
Table[Sum[(i*(i - 2)^2 + i^2)/2, {i, 0, n}], {n, 38}]
Accumulate[Table[(n (n-2)^2+n^2)/2, {n, 50}]] (* Harvey P. Dale, Aug 05 2011 *)
Table[n(n+1)(n^2-3n+6)/8, {n, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {1, 3, 9, 25, 60}, 40] (* Harvey P. Dale, May 27 2023 *)
PROG
(Magma) [n*(n+1)*(n^2-3*n+6)/8: n in [1..50]]; // Vincenzo Librandi, Jun 07 2013
(PARI) a(n)=n*(n+1)*(n^2-3*n+6)/8 \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dennis S. Kluk (mathemagician(AT)ameritech.net)
STATUS
approved
