A069190 - OEIS

%I #64 Aug 01 2024 12:22:12

%S 1,25,73,145,241,361,505,673,865,1081,1321,1585,1873,2185,2521,2881,

%T 3265,3673,4105,4561,5041,5545,6073,6625,7201,7801,8425,9073,9745,

%U 10441,11161,11905,12673,13465,14281,15121,15985,16873,17785,18721,19681,20665,21673

%N Centered 24-gonal numbers.

%C Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - _Omar E. Pol_, Sep 16 2011

%H Ivan Panchenko, <a href="/A069190/b069190.txt">Table of n, a(n) for n = 1..1000</a>

%H John Elias, <a href="/A069190/a069190.png">Illustration: Odd Ordered Star Perimeters</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.

%H R. Yin, J. Mu, and T. Komatsu, <a href="https://doi.org/10.20944/preprints202407.2280.v1">The p-Frobenius Number for the Triple of the Generalized Star Numbers</a>, Preprints 2024, 2024072280. See p. 2.

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 12*n^2 - 12*n + 1.

%F a(n) = 24*n + a(n-1) - 24 with a(1)=1. - _Vincenzo Librandi_, Aug 08 2010

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - _Harvey P. Dale_, Jul 17 2011

%F G.f.: x*(1+22*x+x^2)/(1-x)^3. - _Harvey P. Dale_, Jul 17 2011

%F Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 26 2011

%F From _Amiram Eldar_, Jun 21 2020: (Start)

%F Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).

%F Sum_{n>=1} a(n)/n! = 13*e - 1.

%F Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)

%F E.g.f.: exp(x)*(1 + 12*x^2) - 1. - _Stefano Spezia_, May 31 2022

%e a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

%t FoldList[#1 + #2 &, 1, 24 Range@ 45] (* _Robert G. Wilson v_ *)

%t Table[12n^2-12n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,25,73},50] (* _Harvey P. Dale_, Jul 17 2011 *)

%o (PARI) a(n)=12*n^2-12*n+1 \\ _Charles R Greathouse IV_, Sep 24 2015

%Y Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

%K nonn,easy

%O 1,2

%A _Terrel Trotter, Jr._, Apr 10 2002

%E More terms from _Harvey P. Dale_, Jul 17 2011