|
%I #88 Oct 17 2023 05:18:39
%S 2,7,12,17,22,27,32,37,42,47,52,57,62,67,72,77,82,87,92,97,102,107,
%T 112,117,122,127,132,137,142,147,152,157,162,167,172,177,182,187,192,
%U 197,202,207,212,217,222,227,232,237,242,247,252,257,262,267,272,277
%N a(n) = 5*n + 2.
%C Numbers ending in 2 or 7. - _Lekraj Beedassy_, Jul 08 2006
%C For n > 2, also the number of (not necessarily maximal) cliques in the n-gear graph. - _Eric W. Weisstein_, Nov 29 2017
%C Also, positive integers k such that 10*k+5 is equal to the product of two integers ending with 5. Proof: if 10*k+5 = (10*a+5) * (10*b+5), then k = 10*a*b + 5*(a+b) + 2 = 5 * (a + b + 2*a*b) + 2, of the form 5m + 2. So, 262 is a term because 2625 = 35 * 75. - _Bernard Schott_, May 15 2019
%C Numbers k such that 2^x + 3^x == 0 mod 31 and 2^x + 3^x == 0 mod 11 with x = 6*k+3. - _Pedro Caceres_, May 18 2022
%H G. C. Greubel, <a href="/A016873/b016873.txt">Table of n, a(n) for n = 0..1000</a>
%H Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/3x5x211k.msnw">solutions to 3^x + 5^x == 2 mod 11</a>. [broken link]
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F a(n) = 10*n - a(n-1) - 1 (with a(0)=2). - _Vincenzo Librandi_, Nov 20 2010
%F G.f.: (2+3*x)/(1-x)^2. - _Colin Barker_, Jan 08 2012
%F E.g.f.: exp(x)*(2 + 5*x). - _Stefano Spezia_, Mar 21 2021
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(2-2/sqrt(5))*Pi/10 + log(phi)/sqrt(5) - log(2)/5, where phi is the golden ratio (A001622). - _Amiram Eldar_, Apr 15 2023
%p a[1]:=2:for n from 2 to 100 do a[n]:=a[n-1]+5 od: seq(a[n], n=1..50); # _Zerinvary Lajos_, Mar 16 2008
%t Range[2, 500, 5] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t (* Programs from _Eric W. Weisstein_, Nov 29 2017 *)
%t 5*Range[0, 70] +2
%t LinearRecurrence[{2, -1}, {7, 12}, {0, 70}]
%t CoefficientList[Series[(2+3*x)/(1-x)^2, {x,0,70}], x] (* End *)
%o (Sage) [i+2 for i in range(300) if gcd(i,5) == 5] # _Zerinvary Lajos_, May 20 2009
%o (PARI) a(n)=5*n+2 \\ _Charles R Greathouse IV_, Jul 10 2016
%o (Magma) [5*n+2: n in [0..80]]; // _G. C. Greubel_, Oct 17 2023
%Y Cf. A001622, A008586, A008587, A016861, A342757.
%Y Cf. A053742 (product of two integers ending with 5).
%Y Cf. A324298 (product of two integers ending with 6).
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
|
