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A177065
a(n) = (8*n+3)*(8*n+5).
2
15, 143, 399, 783, 1295, 1935, 2703, 3599, 4623, 5775, 7055, 8463, 9999, 11663, 13455, 15375, 17423, 19599, 21903, 24335, 26895, 29583, 32399, 35343, 38415, 41615, 44943, 48399, 51983, 55695, 59535, 63503, 67599, 71823, 76175, 80655, 85263, 89999, 94863, 99855
OFFSET
0,1
COMMENTS
Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 64*A002061(n+1) - 49. - Bruno Berselli, Aug 24 2010
FORMULA
a(n) = 128*n + a(n-1) with n > 0, a(0)=15.
a(n) = A125169(A016754(n) - 1). - Reinhard Zumkeller, Jul 05 2010
a(0)=15, a(1)=143, a(2)=399, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 13 2013
G.f.: (15+98*x+15*x^2)/(1-x)^3. - Vincenzo Librandi, Apr 08 2013
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017101(n)*A004770(n).
Sum_{n>=0} 1/a(n) = (sqrt(2)-1)*Pi/16.
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(tan(3*Pi/16)) + sin(Pi/8) * log(cot(Pi/16)))/4.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/8)*cos(Pi/(4*sqrt(2))).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/8). (End)
E.g.f.: exp(x)*(15 + 64*x*(2 + x)). - Elmo R. Oliveira, Oct 25 2024
MAPLE
A177065:=n->(8*n+3)*(8*n+5): seq(A177065(n), n=0..100); # Wesley Ivan Hurt, Apr 24 2017
MATHEMATICA
Table[(8n+3)(8n+5), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {15, 143, 399}, 40] (* Harvey P. Dale, Mar 13 2013 *)
CoefficientList[Series[(15 + 98 x + 15 x^2)/(1-x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Apr 08 2013 *)
PROG
(Magma) [(8*n+3)*(8*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013
(PARI) a(n)=(8*n+3)*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Vincenzo Librandi, May 31 2010
EXTENSIONS
Edited by N. J. A. Sloane, Jun 22 2010
STATUS
approved