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 A062786 Centered 10-gonal numbers. 38
 1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - Amarnath Murthy, Dec 11 2003 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 Centered decagonal numbers. - Omar E. Pol, Oct 03 2011 The partial sums of this sequence give A004466. - Leo Tavares, Oct 04 2021 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 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 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 LINKS T. D. Noe, Table of n, a(n) for n = 1..1000 Leo Tavares, Illustration: Pentagonal Stars Leo Tavares, Illustration: Mid-section Stars Leo Tavares, Illustration: Mid-section Star Pillars Leo Tavares, Illustration: Trapezoidal Rays Index entries for sequences related to centered polygonal numbers Index entries for linear recurrences with constant coefficients, signature (3,-3,1). FORMULA a(n) = 5*n*(n-1) + 1. From Gary W. Adamson, Dec 29 2007: (Start) Binomial transform of [1, 10, 10, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 10, 0, 0, 0, ...]. (End) a(n) = 10*n + a(n-1) - 10; a(1)=1. - Vincenzo Librandi, Aug 07 2010 G.f.: x*(1+8*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011 a(n) = A124080(n-1) + 1. - Omar E. Pol, Oct 03 2011 a(n) = A101321(10,n-1). - R. J. Mathar, Jul 28 2016 a(n) = A028387(A016861(n-1))/5 for n > 0. - Art Baker, Mar 28 2019 E.g.f.: (1+5*x^2)*exp(x) - 1. - G. C. Greubel, Mar 30 2019 Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(5))) / sqrt(5). - Vaclav Kotesovec, Jul 23 2019 From Amiram Eldar, Jun 20 2020: (Start) Sum_{n>=1} a(n)/n! = 6*e - 1. Sum_{n>=1} (-1)^n * a(n)/n! = 6/e - 1. (End) a(n) = A005891(n-1) + 5*A000217(n-1). - Leo Tavares, Jul 14 2021 a(n) = A003154(n) - 2*A000217(n-1). See Mid-section Stars illustration. - Leo Tavares, Sep 06 2021 From Leo Tavares, Oct 06 2021: (Start) a(n) = A144390(n-1) + 2*A028387(n-1). See Mid-section Star Pillars illustration. a(n) = A000326(n) + A000217(n) + 3*A000217(n-1). See Trapezoidal Rays illustration. a(n) = A060544(n) + A000217(n-1). (End) From Leo Tavares, Oct 31 2021: (Start) a(n) = A016754(n-1) + 2*A000217(n-1). a(n) = A016754(n-1) + A002378(n-1). a(n) = A069099(n) + 3*A000217(n-1). a(n) = A069099(n) + A045943(n-1). a(n) = A003215(n-1) + 4*A000217(n-1). a(n) = A003215(n-1) + A046092(n-1). a(n) = A001844(n-1) + 6*A000217(n-1). a(n) = A001844(n-1) + A028896(n-1). a(n) = A005448(n) + 7*A000217(n). a(n) = A005448(n) + A024966(n). (End) From Klaus Purath, Oct 30 2022: (Start) a(n) = a(n-2) + 10*(2*n-3). a(n) = 2*a(n-1) - a(n-2) + 10. a(n) = A135705(n-1) + n. a(n) = A190816(n) - n. a(n) = 2*A005891(n-1) - 1. (End) MATHEMATICA FoldList[#1+#2 &, 1, 10Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *) 1+5*Pochhammer[Range[50]-1, 2] (* G. C. Greubel, Mar 30 2019 *) PROG (PARI) j=[]; for(n=1, 75, j=concat(j, (5*n*(n-1)+1))); j (PARI) for (n=1, 1000, write("b062786.txt", n, " ", 5*n*(n - 1) + 1) ) \\ Harry J. Smith, Aug 11 2009 (Magma) [1+5*n*(n-1): n in [1..50]]; // G. C. Greubel, Mar 30 2019 (Sage) [1+5*rising_factorial(n-1, 2) for n in (1..50)] # G. C. Greubel, Mar 30 2019 (GAP) List([1..50], n-> 1+5*n*(n-1)) # G. C. Greubel, Mar 30 2019 CROSSREFS Cf. A001263, A124080, A101321, A028387, A016861, A003154, A005891, A000217, A004466, A144390, A000326, A060544. Cf. also A016754, A002378, A069099, A045943, A003215, A046092, A001844, A028896, A005448, A024966, A082970. Sequence in context: A113747 A202007 A125239 * A090562 A174244 A136061 Adjacent sequences: A062783 A062784 A062785 * A062787 A062788 A062789 KEYWORD easy,nonn AUTHOR Jason Earls, Jul 19 2001 EXTENSIONS Better description from Terrel Trotter, Jr., Apr 06 2002 STATUS approved

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Last modified April 16 01:01 EDT 2024. Contains 371696 sequences. (Running on oeis4.)