A081585 Third row of Pascal-(1,3,1) array A081578.

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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
(Sage) [8*n^2 +1 for n in (0..40)] # G. C. Greubel, May 26 2021

Crossrefs

Cf. A016813, A081578, A081586, A157912.

Sequence in context: A092562 A103602 A205796 * A227221 A273316 A101990

Adjacent sequences: A081582 A081583 A081584 * A081586 A081587 A081588

Keyword

easy,nonn

Author

Paul Barry, Mar 23 2003

Status

approved