A014493 Odd triangular numbers.

1, 3, 15, 21, 45, 55, 91, 105, 153, 171, 231, 253, 325, 351, 435, 465, 561, 595, 703, 741, 861, 903, 1035, 1081, 1225, 1275, 1431, 1485, 1653, 1711, 1891, 1953, 2145, 2211, 2415, 2485, 2701, 2775, 3003, 3081, 3321, 3403, 3655, 3741, 4005, 4095, 4371, 4465, 4753, 4851

Offset: 1

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Comments

Odd numbers of the form n*(n+1)/2.

For n such that n(n+1)/2 is odd see A042963 (congruent to 1 or 2 mod 4).

Even central polygonal numbers minus 1. - Omar E. Pol, Aug 17 2011

Odd generalized hexagonal numbers. - Omar E. Pol, Sep 24 2015

References

• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 68.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• D. H. Lehmer, Recurrence formulas for certain divisor functions, Bull. Amer. Math. Soc., Vol. 49, No. 2 (1943), pp. 150-156.
• Eric Weisstein's World of Mathematics, Triangular Number.
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A000217, A000796, A014494, A019669, A042963, A067589, A128880.

Programs

GAP

List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # G. C. Greubel, Feb 09 2019

Magma

[(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // Vincenzo Librandi, Aug 18 2011

Maple

[(2*n-1)*(2*n-1-(-1)^n)/2$n=1..50]; # Muniru A Asiru, Mar 10 2019

Mathematica

Select[ Table[n(n + 1)/2, {n, 93}], OddQ[ # ] &] (* Robert G. Wilson v, Nov 05 2004 *)
LinearRecurrence[{1,2,-2,-1,1},{1,3,15,21,45},50] (* Harvey P. Dale, Jun 19 2011 *)

PARI

a(n)=(2*n-1)*(2*n-1-(-1)^n)/2 \\ Charles R Greathouse IV, Sep 24 2015

Python

def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # Chai Wah Wu, Feb 12 2023

Sage

[(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Feb 09 2019

Formula

From Ant King, Nov 17 2010: (Start)

a(n) = (2*n-1)*(2*n - 1 - (-1)^n)/2.

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). (End)

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

a(n) = A000217(A042963(n)). - Reinhard Zumkeller, Feb 14 2012, Oct 04 2004

a(n) = A193868(n) - 1. - Omar E. Pol, Aug 17 2011

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

E.g.f.: (2*x^2+x+1)*cosh(x)+x*(2*x-1)*sinh(x)-1. - Ilya Gutkovskiy, Apr 24 2016

Sum_{n>=1} 1/a(n) = Pi/2 (A019669). - Robert Bilinski, Jan 20 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = log(2). - Amiram Eldar, Mar 06 2022

Extensions

More terms from Erich Friedman