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A062786
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Centered 10-gonal numbers.
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32
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001263.
Sequence in context: A113747 A202007 A125239 * A090562 A174244 A136061
Adjacent sequences: A062783 A062784 A062785 * A062787 A062788 A062789
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KEYWORD
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easy,nonn,changed
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AUTHOR
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Jason Earls, Jul 19 2001
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EXTENSIONS
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Better description from Terrel Trotter, Jr., Apr 06 2002
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STATUS
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approved
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