OFFSET
0,2
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Irving Adler, Three Diophantine equations - Part II, Fib. Quart., Vol. 7, No. 2 (1969), pp. 181-193.
Edgar I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., Vol. 7, No. 3 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
G.f.: (1+5*x)/(1-x)^9.
From G. C. Greubel, Jan 19 2020: (Start)
a(n) = 6*binomial(n+8, 8) - 5*binomial(n+7, 7).
E.g.f.: (20160 +262080*x +635040*x^2 +540960*x^3 +205800*x^4 +38808*x^5 +3724*x^6 +172*x^7 +3*x^8)*exp(x)/20160. (End)
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Wesley Ivan Hurt, Jun 07 2021
From Amiram Eldar, Nov 07 2025: (Start)
Sum_{n>=0} 1/a(n) = 45992/255 - 2187*sqrt(3)*Pi/187 - 19683*log(3)/187.
Sum_{n>=0} (-1)^n/a(n) = 4374*sqrt(3)*Pi/187 - 35840*log(2)/187 + 18256/2805. (End)
MAPLE
seq( (3*n+4)*binomial(n+7, 7)/4, n=0..40); # G. C. Greubel, Jan 19 2020
MATHEMATICA
CoefficientList[Series[(1+5x)/(1-x)^9, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)
Table[6*Binomial[n+8, 8] -5*Binomial[n+7, 7], {n, 0, 40}] (* G. C. Greubel, Jan 19 2020 *)
LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 14, 90, 390, 1320, 3762, 9438, 21450, 45045}, 30] (* Harvey P. Dale, Jul 19 2022 *)
PROG
(Magma) [((3*n+4)*Binomial(n+7, 7))/4: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014
(PARI) a(n) = (3*n+4)*binomial(n+7, 7)/4; \\ Michel Marcus, Sep 07 2017
(SageMath) [(3*n+4)*binomial(n+7, 7)/4 for n in (0..40)] # G. C. Greubel, Jan 19 2020
(GAP) List([0..40], n-> (3*n+4)*Binomial(n+7, 7)/4 ); # G. C. Greubel, Jan 19 2020
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, May 06 2000
EXTENSIONS
Corrected and extended by James Sellers, May 10 2000
STATUS
approved