%I #81 Aug 01 2024 11:57:28
%S 1,19,55,109,181,271,379,505,649,811,991,1189,1405,1639,1891,2161,
%T 2449,2755,3079,3421,3781,4159,4555,4969,5401,5851,6319,6805,7309,
%U 7831,8371,8929,9505,10099,10711,11341,11989,12655,13339,14041,14761,15499,16255,17029,17821
%N Centered 18-gonal numbers.
%C Equals binomial transform of [1, 18, 18, 0, 0, 0, ...]. Example: a(3) = 55 = (1, 2, 1) dot (1, 18, 18) = (1 + 36 + 18). - _Gary W. Adamson_, Aug 24 2010
%C Narayana transform (A001263) of [1, 18, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%C From _Lamine Ngom_, Aug 19 2021: (Start)
%C Sequence is a spoke of the hexagonal spiral built from the terms of A016777 (see illustration in links section).
%C a(n) is a bisection of A195042.
%C a(n) is a trisection of A028387.
%C a(n) + 1 is promic (A002378).
%C a(n) + 2 is a trisection of A002061.
%C a(n) + 9 is the arithmetic mean of its neighbors.
%C 4*a(n) + 5 is a square: A016945(n)^2. (End)
%H Ivan Panchenko, <a href="/A069131/b069131.txt">Table of n, a(n) for n = 1..1000</a>
%H John Elias, <a href="/A069131/a069131.png">Illustration of Initial Terms: Triangular & Hexagonal Configurations</a>.
%H Lamine Ngom, <a href="/A069131/a069131.jpg">An origin of A069131 (illustration)</a>.
%H Leo Tavares, <a href="/A069131/a069131_1.jpg">Illustration: Tri-Hexagons</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) = 9*n^2 - 9*n + 1.
%F a(n) = 18*n + a(n-1) - 18 (with a(1)=1). - _Vincenzo Librandi_, Aug 08 2010
%F G.f.: ( x*(1+16*x+x^2) ) / ( (1-x)^3 ). - _R. J. Mathar_, Feb 04 2011
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=19, a(3)=55. - _Harvey P. Dale_, Jan 20 2014
%F From _Amiram Eldar_, Jun 21 2020: (Start)
%F Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(5)*Pi/6)/(3*sqrt(5)).
%F Sum_{n>=1} a(n)/n! = 10*e - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 10/e - 1. (End)
%F From _Lamine Ngom_, Aug 19 2021: (Start)
%F a(n) = 18*A000217(n) + 1 = 9*A002378(n) + 1.
%F a(n) = 3*A003215(n) - 2.
%F a(n) = A247792(n) - 9*n.
%F a(n) = A082040(n) + A304163(n) - a(n-1) = A016778(n) + A016790(n) - a(n-1), n > 0.
%F a(n) + a(n+1) = 2*A247792(n) = A010008(n), n > 0.
%F a(n+1) - a(n) = 18*n = A008600(n). (End)
%F From _Leo Tavares_, Oct 31 2021: (Start)
%F a(n)= A000290(n) + A139278(n-1)
%F a(n) = A069129(n) + A002378(n-1)
%F a(n) = A062786(n) + 8*A000217(n-1)
%F a(n) = A062786(n) + A033996(n-1)
%F a(n) = A060544(n) + 9*A000217(n-1)
%F a(n) = A060544(n) + A027468(n-1)
%F a(n) = A016754(n-1) + 10*A000217(n-1)
%F a(n) = A016754(n-1) + A124080
%F a(n) = A069099(n) + 11*A000217(n-1)
%F a(n) = A069099(n) + A152740(n-1)
%F a(n) = A003215(n-1) + 12*A000217(n-1)
%F a(n) = A003215(n-1) + A049598(n-1)
%F a(n) = A005891(n-1) + 13*A000217(n-1)
%F a(n) = A005891(n-1) + A152741(n-1)
%F a(n) = A001844(n) + 14*A000217(n-1)
%F a(n) = A001844(n) + A163756(n-1)
%F a(n) = A005448(n) + 15*A000217(n-1)
%F a(n) = A005448(n) + A194715(n-1). (End)
%F E.g.f.: exp(x)*(1 + 9*x^2) - 1. - _Nikolaos Pantelidis_, Feb 06 2023
%e a(5) = 181 because 9*5^2 - 9*5 + 1 = 225 - 45 + 1 = 181.
%t FoldList[#1 + #2 &, 1, 18 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t LinearRecurrence[{3,-3,1},{1,19,55},50] (* _Harvey P. Dale_, Jan 20 2014 *)
%o (PARI) a(n)=9*n^2-9*n+1 \\ _Charles R Greathouse IV_, Oct 07 2015
%o (Magma) [9*n^2 - 9*n + 1 : n in [1..50]]; // _Wesley Ivan Hurt_, May 05 2021
%Y Cf. centered polygonal numbers listed in A069190.
%Y Cf. A000217, A028387, A195042, A016945, A002378, A082040, A304163, A003215, A247792, A016777,A016778, A016790, A010008, A008600, A002061.
%Y Cf. A000290, A139278, A069129, A062786, A033996, A060544, A027468, A016754, A124080, A069099, A152740, A049598, A005891, A152741, A001844, A163756, A005448, A194715.
%K easy,nice,nonn
%O 1,2
%A _Terrel Trotter, Jr._, Apr 07 2002