A128880 -id:A128880 - cp's OEIS frontend

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

Mohammad K. Azarian

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

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

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).

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

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

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

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

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 *)

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

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

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

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

More terms from Erich Friedman

A067589 Numbers k such that A067588(k) is an odd number.

Original entry on oeis.org

1, 5, 7, 15, 35, 51, 57, 77, 117, 145, 155, 187, 247, 287, 301, 345, 425, 477, 495, 551, 651, 715, 737, 805, 925, 1001, 1027, 1107, 1247, 1335, 1365, 1457, 1617, 1717, 1751, 1855, 2035, 2147, 2185, 2301, 2501, 2625, 2667, 2795, 3015, 3151, 3197, 3337

Offset: 1

Naohiro Nomoto, Jan 31 2002

The terms are exactly the odd pentagonal numbers; that is, they are all the odd numbers of the form k*(3*k-1)/2 where k is an integer. - James Sellers, Jun 09 2007
Apparently groups of two odd pentagonal numbers (A000326, A014632) followed by two odd 2nd pentagonal numbers (A005449), which leads to the conjectured generating function x*(x^2+4*x+1)*(x^4-2*x^3+4*x^2-2*x+1)/((x^2+1)^2*(1-x)^3). - R. J. Mathar, Jul 26 2009
Odd generalized pentagonal numbers. - Omar E. Pol, Aug 19 2011
From Peter Bala, Jan 10 2025: (Start)
The sequence terms are the exponents in the expansion of Sum_{n >= 0} x^(2*n+1)/(Product_{k = 1..2*n+1} 1 + x^(2*k+1)) = x + x^5 - x^7 - x^15 + x^35 + x^51 - x^57 - x^77 + + - - ... (follows from Berndt et al., Theorem 3.3). Cf. A193828.
For positive integer m, define b_m(n) = Sum_{k = 1..n} k^(2*m+1)*A000009(k)*A000009(n-k). We conjecture that
i) for odd n, b(n)/ n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a member of this sequence.
Cf. A067567. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.

Cf. A067567, A067588, A000009.
Cf. A001318, A193828.
Cf. A014493, A128880, A154293.

With[{nn=50},Sort[Select[Table[(n(3n-1))/2,{n,-nn,nn}],OddQ]]] (* Harvey P. Dale, Feb 16 2014 *)

Sum_{n>=1} 1/a(n) = Pi/2. - Amiram Eldar, Aug 18 2022

Corrected by T. D. Noe, Oct 25 2006

cp's OEIS Frontend

A014493 Odd triangular numbers.

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