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%I #97 Feb 12 2023 12:09:31
%S 1,3,15,21,45,55,91,105,153,171,231,253,325,351,435,465,561,595,703,
%T 741,861,903,1035,1081,1225,1275,1431,1485,1653,1711,1891,1953,2145,
%U 2211,2415,2485,2701,2775,3003,3081,3321,3403,3655,3741,4005,4095,4371,4465,4753,4851
%N Odd triangular numbers.
%C Odd numbers of the form n*(n+1)/2.
%C For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).
%C Even central polygonal numbers minus 1. - _Omar E. Pol_, Aug 17 2011
%C Odd generalized hexagonal numbers. - _Omar E. Pol_, Sep 24 2015
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 68.
%H Vincenzo Librandi, <a href="/A014493/b014493.txt">Table of n, a(n) for n = 1..10000</a>
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0002-9904-1943-07880-9">Recurrence formulas for certain divisor functions</a>, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F From _Ant King_, Nov 17 2010: (Start)
%F a(n) = (2*n-1)*(2*n - 1 - (-1)^n)/2.
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)
%F G.f.: x*(1 + 2*x + 10*x^2 + 2*x^3 + x^4)/((1+x)^2*(1-x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
%F a(n) = A000217(A042963(n)). - _Reinhard Zumkeller_, Feb 14 2012, Oct 04 2004
%F a(n) = A193868(n) - 1. - _Omar E. Pol_, Aug 17 2011
%F Let S = Sum_{n>=0} x^n/a(n), then S = Q(0) where Q(k) = 1 + x*(4*k+1)/(4*k + 3 - x*(2*k+1)*(4*k+3)^2/(x*(2*k+1)*(4*k+3) + (4*k+5)*(2*k+3)/Q(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Feb 27 2013
%F E.g.f.: (2*x^2+x+1)*cosh(x)+x*(2*x-1)*sinh(x)-1. - _Ilya Gutkovskiy_, Apr 24 2016
%F Sum_{n>=1} 1/a(n) = Pi/2 (A019669). - _Robert Bilinski_, Jan 20 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - _Amiram Eldar_, Mar 06 2022
%p [(2*n-1)*(2*n-1-(-1)^n)/2$n=1..50]; # _Muniru A Asiru_, Mar 10 2019
%t Select[ Table[n(n + 1)/2, {n, 93}], OddQ[ # ] &] (* _Robert G. Wilson v_, Nov 05 2004 *)
%t LinearRecurrence[{1,2,-2,-1,1},{1,3,15,21,45},50] (* _Harvey P. Dale_, Jun 19 2011 *)
%o (Magma) [(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2011
%o (PARI) a(n)=(2*n-1)*(2*n-1-(-1)^n)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Sage) [(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # _G. C. Greubel_, Feb 09 2019
%o (GAP) List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # _G. C. Greubel_, Feb 09 2019
%o (Python)
%o def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # _Chai Wah Wu_, Feb 12 2023
%Y Cf. A000217, A000796, A014494, A019669, A042963, A067589, A128880.
%K nonn,easy
%O 1,2
%A _Mohammad K. Azarian_
%E More terms from _Erich Friedman_
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