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A081585 Third row of Pascal-(1,3,1) array A081578. 9
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 30 19:15 EDT 2021. Contains 346359 sequences. (Running on oeis4.)