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URL: https://oeis.org/A081585

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A081585
Third row of Pascal-(1,3,1) array A081578.
15
1, 9, 33, 73, 129, 201, 289, 393, 513, 649, 801, 969, 1153, 1353, 1569, 1801, 2049, 2313, 2593, 2889, 3201, 3529, 3873, 4233, 4609, 5001, 5409, 5833, 6273, 6729, 7201, 7689, 8193, 8713, 9249, 9801, 10369, 10953, 11553, 12169, 12801, 13449, 14113
OFFSET
0,2
COMMENTS
The identity (8*n^2 +1)^2 - (64*n^2 +16)*n^2 = 1 can be written as a(n)^2 -A157912(n)*n^2 = 1 for n>0. - Vincenzo Librandi, Feb 09 2012
FORMULA
a(n) = 8*n^2 + 1.
G.f.: (1+3*x)^2/(1-x)^3.
a(n) = a(n-1) + 16*n - 8 with a(0)=1. - Vincenzo Librandi, Aug 08 2010
a(n) = sqrt(8*(A000217(2*n-1)^2 +A000217(2*n)^2) +1). - J. M. Bergot, Sep 04 2015
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(8))*coth(Pi/sqrt(8)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(8))*csch(Pi/sqrt(8)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(8))*sinh(Pi/2).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(8))*csch(Pi/sqrt(8)). (End)
E.g.f.: (1 +8*x +8*x^2)*exp(x). - G. C. Greubel, May 26 2021
MAPLE
seq(1+8*n^2, n=0..100); # Robert Israel, Sep 04 2015
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {1, 9, 33}, 40] (* Vincenzo Librandi, Feb 09 2012 *)
PROG
(Magma) I:=[1, 9, 33]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 09 2012
(PARI) for(n=0, 50, print1(8*n^2+1", ")); \\ Vincenzo Librandi, Feb 09 2012
(SageMath) [8*n^2 +1 for n in (0..40)] # G. C. Greubel, May 26 2021
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 23 2003
STATUS
approved