This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069133 #75 Feb 16 2025 08:32:45
%S A069133 1,21,61,121,201,301,421,561,721,901,1101,1321,1561,1821,2101,2401,
%T A069133 2721,3061,3421,3801,4201,4621,5061,5521,6001,6501,7021,7561,8121,
%U A069133 8701,9301,9921,10561,11221,11901,12601,13321,14061,14821,15601,16401,17221,18061,18921,19801
%N A069133 Centered 20-gonal (or icosagonal) numbers.
%C A069133 Equals binomial transform of [1, 20, 20, 0, 0, 0, ...]. - _Gary W. Adamson_, Jun 13 2008
%C A069133 Equals Narayana transform (A001263) of [1, 20, 0, 0, 0, ...]. - _Gary W. Adamson_, Jul 28 2011
%C A069133 Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Semi-axis opposite to A033583 in the same spiral. - _Omar E. Pol_, Sep 16 2011
%H A069133 Ivan Panchenko, <a href="/A069133/b069133.txt">Table of n, a(n) for n = 1..1000</a>
%H A069133 John Elias, <a href="/A069133/a069133.png">Illustration of initial terms</a>.
%H A069133 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredPolygonalNumber.html">Centered Polygonal Numbers</a>.
%H A069133 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 A069133 <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>.
%H A069133 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A069133 a(n) = 10n^2 - 10n + 1.
%F A069133 a(n) = 20*n + a(n-1) - 20 with a(1)=1. - _Vincenzo Librandi_, Aug 08 2010
%F A069133 G.f.: x*(1 + 18*x + x^2)/(1-x)^3. - _R. J. Mathar_, Feb 04 2011
%F A069133 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=21, a(2)=61. - _Harvey P. Dale_, Apr 29 2011
%F A069133 From _Amiram Eldar_, Jun 21 2020: (Start)
%F A069133 Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3/5)*Pi/2)/(2*sqrt(15)).
%F A069133 Sum_{n>=1} a(n)/n! = 11*e - 1.
%F A069133 Sum_{n>=1} (-1)^n * a(n)/n! = 11/e - 1. (End)
%F A069133 a(n) = 12*A000217(n-1) + A016754(n-1). - _John Elias_, Oct 23 2020
%F A069133 E.g.f.: exp(x)*(1 + 10*x^2) - 1. - _Nikolaos Pantelidis_, Feb 06 2023
%e A069133 a(5)=201 because 201 = 10*5^2 - 10*5 + 1 = 250 - 50 + 1.
%t A069133 FoldList[#1 + #2 &, 1, 20 Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t A069133 Table[10n^2-10n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,61}, 50] (* _Harvey P. Dale_, Apr 29 2011 *)
%o A069133 (PARI) a(n)=10*n*(n-1)+1 \\ _Charles R Greathouse IV_, Jul 29 2011
%o A069133 (Magma) [10*n^2 - 10*n + 1 : n in [1..60]]; // _Wesley Ivan Hurt_, Oct 10 2021
%Y A069133 Cf. centered polygonal numbers listed in A069190.
%K A069133 easy,nice,nonn
%O A069133 1,2
%A A069133 _Terrel Trotter, Jr._, Apr 07 2002