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%I #144 Aug 01 2024 11:58:47
%S 1,11,31,61,101,151,211,281,361,451,551,661,781,911,1051,1201,1361,
%T 1531,1711,1901,2101,2311,2531,2761,3001,3251,3511,3781,4061,4351,
%U 4651,4961,5281,5611,5951,6301,6661,7031,7411,7801,8201,8611,9031,9461,9901,10351,10811
%N Centered 10-gonal numbers.
%C Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - _Amarnath Murthy_, Dec 11 2003
%C All divisors of a(n) are congruent to 1 or -1, modulo 10; that is, they end in the decimal digit 1 or 9. Proof: If p is an odd prime different from 5 then 5n^2 - 5n + 1 == 0 (mod p) implies 25(2n - 1)^2 == 5 (mod p), whence p == 1 or -1 (mod 10). - Nick Hobson, Nov 13 2006
%C Centered decagonal numbers. - _Omar E. Pol_, Oct 03 2011
%C The partial sums of this sequence give A004466. - _Leo Tavares_, Oct 04 2021
%C The continued fraction expansion of sqrt(5*a(n)) is [5n-3; {2, 2n-2, 2, 10n-6}]. For n=1, this collapses to [2; {4}]. - _Magus K. Chu_, Sep 12 2022
%C Numbers m such that 20*m + 5 is a square. Also values of the Fibonacci polynomial y^2 - x*y - x^2 for x = n and y = 3*n - 1. This is a subsequence of A089270. - _Klaus Purath_, Oct 30 2022
%C All terms can be written as a difference of two consecutive squares a(n) = A005891(n-1)^2 - A028895(n-1)^2, and they can be represented by the forms (x^2 + 2mxy + (m^2-1)y^2) and (3x^2 + (6m-2)xy + (3m^2-2m)y^2), both of discriminant 4. - _Klaus Purath_, Oct 17 2023
%H T. D. Noe, <a href="/A062786/b062786.txt">Table of n, a(n) for n = 1..1000</a>
%H Leo Tavares, <a href="/A062786/a062786.jpg">Illustration: Pentagonal Stars</a>.
%H Leo Tavares, <a href="/A062786/a062786_1.jpg">Illustration: Mid-section Stars</a>.
%H Leo Tavares, <a href="/A062786/a062786_2.jpg">Illustration: Mid-section Star Pillars</a>.
%H Leo Tavares, <a href="/A062786/a062786_3.jpg">Illustration: Trapezoidal Rays</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) = 5*n*(n-1) + 1.
%F From _Gary W. Adamson_, Dec 29 2007: (Start)
%F Binomial transform of [1, 10, 10, 0, 0, 0, ...];
%F Narayana transform (A001263) of [1, 10, 0, 0, 0, ...]. (End)
%F a(n) = 10*n + a(n-1) - 10; a(1)=1. - _Vincenzo Librandi_, Aug 07 2010
%F G.f.: x*(1+8*x+x^2) / (1-x)^3. - _R. J. Mathar_, Feb 04 2011
%F a(n) = A124080(n-1) + 1. - _Omar E. Pol_, Oct 03 2011
%F a(n) = A101321(10,n-1). - _R. J. Mathar_, Jul 28 2016
%F a(n) = A028387(A016861(n-1))/5 for n > 0. - _Art Baker_, Mar 28 2019
%F E.g.f.: (1+5*x^2)*exp(x) - 1. - _G. C. Greubel_, Mar 30 2019
%F Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(5))) / sqrt(5). - _Vaclav Kotesovec_, Jul 23 2019
%F From _Amiram Eldar_, Jun 20 2020: (Start)
%F Sum_{n>=1} a(n)/n! = 6*e - 1.
%F Sum_{n>=1} (-1)^n * a(n)/n! = 6/e - 1. (End)
%F a(n) = A005891(n-1) + 5*A000217(n-1). - _Leo Tavares_, Jul 14 2021
%F a(n) = A003154(n) - 2*A000217(n-1). See Mid-section Stars illustration. - _Leo Tavares_, Sep 06 2021
%F From _Leo Tavares_, Oct 06 2021: (Start)
%F a(n) = A144390(n-1) + 2*A028387(n-1). See Mid-section Star Pillars illustration.
%F a(n) = A000326(n) + A000217(n) + 3*A000217(n-1). See Trapezoidal Rays illustration.
%F a(n) = A060544(n) + A000217(n-1). (End)
%F From _Leo Tavares_, Oct 31 2021: (Start)
%F a(n) = A016754(n-1) + 2*A000217(n-1).
%F a(n) = A016754(n-1) + A002378(n-1).
%F a(n) = A069099(n) + 3*A000217(n-1).
%F a(n) = A069099(n) + A045943(n-1).
%F a(n) = A003215(n-1) + 4*A000217(n-1).
%F a(n) = A003215(n-1) + A046092(n-1).
%F a(n) = A001844(n-1) + 6*A000217(n-1).
%F a(n) = A001844(n-1) + A028896(n-1).
%F a(n) = A005448(n) + 7*A000217(n).
%F a(n) = A005448(n) + A024966(n). (End)
%F From _Klaus Purath_, Oct 30 2022: (Start)
%F a(n) = a(n-2) + 10*(2*n-3).
%F a(n) = 2*a(n-1) - a(n-2) + 10.
%F a(n) = A135705(n-1) + n.
%F a(n) = A190816(n) - n.
%F a(n) = 2*A005891(n-1) - 1. (End)
%t FoldList[#1+#2 &, 1, 10Range@ 45] (* _Robert G. Wilson v_, Feb 02 2011 *)
%t 1+5*Pochhammer[Range[50]-1, 2] (* _G. C. Greubel_, Mar 30 2019 *)
%o (PARI) j=[]; for(n=1,75,j=concat(j,(5*n*(n-1)+1))); j
%o (PARI) for (n=1, 1000, write("b062786.txt", n, " ", 5*n*(n - 1) + 1) ) \\ _Harry J. Smith_, Aug 11 2009
%o (Magma) [1+5*n*(n-1): n in [1..50]]; // _G. C. Greubel_, Mar 30 2019
%o (Sage) [1+5*rising_factorial(n-1, 2) for n in (1..50)] # _G. C. Greubel_, Mar 30 2019
%o (GAP) List([1..50], n-> 1+5*n*(n-1)) # _G. C. Greubel_, Mar 30 2019
%o (Python) def a(n): return(5*n**2-5*n+1) # _Torlach Rush_, May 10 2024
%Y Cf. A001263, A124080, A101321, A028387, A016861, A003154, A005891, A000217, A004466, A144390, A000326, A060544.
%Y Cf. also A016754, A002378, A069099, A045943, A003215, A046092, A001844, A028896, A005448, A024966, A082970.
%K easy,nonn
%O 1,2
%A _Jason Earls_, Jul 19 2001
%E Better description from _Terrel Trotter, Jr._, Apr 06 2002