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

1,2

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).

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

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 *)

Prog

(Magma) [(2*n-1)*(2*n-1-(-1)^n)/2: n in [1..50]]; // Vincenzo Librandi, Aug 18 2011
(PARI) a(n)=(2*n-1)*(2*n-1-(-1)^n)/2 \\ Charles R Greathouse IV, Sep 24 2015
(Sage) [(2*n-1)*(2*n-1-(-1)^n)/2 for n in (1..50)] # G. C. Greubel, Feb 09 2019
(GAP) List([1..50], n -> (2*n-1)*(2*n-1-(-1)^n)/2); # G. C. Greubel, Feb 09 2019
(Python)
def A014493(n): return ((n<<1)-1)*(n-(n&1^1)) # Chai Wah Wu, Feb 12 2023

Crossrefs

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

Sequence in context: A216521 A110172 A261274 * A147025 A147017 A171570

Adjacent sequences: A014490 A014491 A014492 * A014494 A014495 A014496

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

More terms from Erich Friedman

Status

approved