OFFSET
0,1
COMMENTS
Al-Saedi has discovered that p(10*n+6,4) + p(10*n+7,4) + p(10*n+8,4) == 0 (mod 5), where p(m,k) denote the number of partitions of m into at most k parts [see Theorem 3.6, formula 23, in Links and References sections].
Hirschhorn showed that p(10*n+6,4) + p(10*n+7,4) + p(10*n+8,4) = (5*n+5)*(5*n+6)*(5*n+7)/6 [see References section: paragraph 3, "Proofs of (1.5)-(1.8)"].
REFERENCES
Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, Journal of Number Theory, Vol. 178, 2017, pages 158-178.
Michael D. Hirschhorn, Congruences modulo 5 for partitions into at most four parts, The Fibonacci Quarterly, Vol. 56, Number 1, 2018, pages 34-37.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Ali H. Al-Saedi, Using Periodicity to Obtain Partition Congruences, arXiv:1609.03633 [math.NT], 2017, pages 12-13.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
O.g.f.: 5*(7 + 16*x + 2*x^2)/(1 - x)^4 [formula 4.2 in Hirschhorn's paper].
E.g.f.: 5*(42 + 222*x + 165*x^2 + 25*x^3)*exp(x)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(-n) = -A300522(n-2).
MATHEMATICA
Table[(5 n + 5) (5 n + 6) (5 n + 7)/6, {n, 0, 40}]
Table[Times@@(5n+{5, 6, 7})/6, {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {35, 220, 680, 1540}, 40] (* Harvey P. Dale, Oct 22 2019 *)
PROG
(PARI) vector(40, n, n--; (5*n+5)*(5*n+6)*(5*n+7)/6)
(Sage) [(5*n+5)*(5*n+6)*(5*n+7)/6 for n in (0..40)]
(Maxima) makelist((5*n+5)*(5*n+6)*(5*n+7)/6, n, 0, 40);
(GAP) List([0..40], n -> (5*n+5)*(5*n+6)*(5*n+7)/6);
(Magma) [(5*n+5)*(5*n+6)*(5*n+7)/6: n in [0..40]];
(Python) [(5*n+5)*(5*n+6)*(5*n+7)/6 for n in range(40)]
(Julia) [div((5*n+5)*(5*n+6)*(5*n+7), 6) for n in 0:40] |> println
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Mar 08 2018
STATUS
approved