A067589 -id:A067589 - 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

A067567 Odd numbers with an odd number of partitions.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 23, 29, 33, 35, 37, 39, 41, 43, 49, 51, 53, 61, 63, 67, 69, 71, 73, 77, 81, 83, 85, 87, 89, 91, 93, 95, 99, 105, 107, 111, 115, 119, 121, 123, 127, 139, 143, 145, 155, 157, 159, 161, 165, 169, 173, 177, 181, 183, 185, 189, 193, 195, 199

Offset: 1

Naohiro Nomoto, Jan 30 2002

The original definition was: Numbers n such that A066897(n) is an odd number.
The sequence defined by b(n) = (n/2)*A281708(n) = Sum_{k=1..n} k^3 * p(k) * p(n-k) of Peter Bala appears to have the property that b(n)/n is a positive integer if n is odd, and b(2*n)/n is a positive integer which is odd iff n is a member of this sequence. - Michael Somos, Jan 28 2017
From Peter Bala, Jan 10 2025: (Start)
We generalize the above conjecture as follows.
Define b_m(n) = Sum_{k = 1..n} k^(2*m+1) * p(k) * p(n-k). Then for m >= 1,
i) for odd n, b_m(n)/n is an integer
ii) b(2*n)/n is an integer, which is odd iff n is a term of this sequence.
Cf. A067589.
We further conjecture that A305123(n) is odd iff n is a term of this sequence. (End)

			7 is in the sequence because the number of partitions of 7 is equal to 15 and both 7 and 15 are odd numbers. - _Omar E. Pol_, Mar 18 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000041, A066897, A067589, A163096, A163097, A194798, A209920, A281708, A305123.

# We conjecture that the following program produces the sequence
with(combinat):
b := n -> add(k^3*numbpart(k)*numbpart(n-k), k = 1..n):
c := n -> 2( b(n)/n - floor(b(n)/n) ):
for n from 1 to 400 do
  if c(n) = 1 then print(n/2) end if
end do;
# Peter Bala, Jan 26 2017

Select[Range[1, 200, 2], OddQ[PartitionsP[#]] &] (* T. D. Noe, Mar 18 2012 *)

isok(n) = (n % 2) && (numbpart(n) % 2); \\ Michel Marcus, Jan 26 2017

New name and more terms from Omar E. Pol, Mar 18 2012

A067588 Total number of parts in all partitions of n into odd parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 14, 19, 26, 36, 48, 62, 82, 104, 132, 169, 210, 260, 324, 396, 484, 592, 714, 860, 1036, 1238, 1474, 1756, 2078, 2452, 2894, 3396, 3976, 4654, 5422, 6309, 7332, 8490, 9816, 11338, 13060, 15018, 17254, 19774, 22630, 25878, 29524, 33642

Offset: 0

Naohiro Nomoto, Jan 31 2002

Starting with "1" = triangle A097304 * [1, 2, 3, ...]. - Gary W. Adamson, Apr 09 2010

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, 2016, vol. 170, pp. 17-34.

Cf. A067589, A006128, A066897, A066898.

Cf. A097304. - Gary W. Adamson, Apr 09 2010

G.f.: G(x)*H(x) where G(x) = Sum_{k>=1} x^(2*k-1)/(1-x^(2*k-1)) is g.f. for the number of odd divisors of n (cf. A001227) and H(x) = Product_{k>=1} (1+x^k) is g.f. for the number of partitions of n into odd parts (cf. A000009). Convolution of A001227 and A000009: Sum_{k=0..n} A001227(k)*A000009(n-k). - Vladeta Jovovic, Feb 04 2002
G.f.: Sum_{n>0} n*x^n/Product_{k=1..n} (1-x^(2*k)). - Vladeta Jovovic, Dec 15 2003
a(n) ~ 3^(1/4) * (2*gamma + log(48*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (8*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 25 2018

Corrected by James Sellers, May 31 2007

A306925 Sum over all partitions of n into distinct parts of the bitwise XOR of the parts.

Original entry on oeis.org

0, 1, 2, 6, 6, 11, 16, 35, 36, 46, 50, 84, 94, 130, 158, 285, 338, 424, 460, 616, 672, 810, 816, 1162, 1346, 1680, 1754, 2308, 2562, 3164, 3288, 4486, 5306, 6838, 7522, 9627, 11006, 13496, 14200, 17462, 19682, 24036, 25650, 30842, 33884, 40302, 41644, 48896

Offset: 0

Alois P. Heinz, Mar 16 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Bitwise operation
Wikipedia, Commutative property
Wikipedia, Identity element
Wikipedia, Partition (number theory)
Wikipedia, Truth table

Cf. A000009, A066189, A067588, A067589, A306903, A306923, A306924.

b:= proc(n, i, r) option remember; `if`(i*(i+1)/2 `if`(i b(n$2, 0):
seq(a(n), n=0..51);

a(n) is odd <=> n in { A067589 }.

a(n) is odd <=> A067588(n) is odd.

A193828 Even generalized pentagonal numbers.

Original entry on oeis.org

0, 2, 12, 22, 26, 40, 70, 92, 100, 126, 176, 210, 222, 260, 330, 376, 392, 442, 532, 590, 610, 672, 782, 852, 876, 950, 1080, 1162, 1190, 1276, 1426, 1520, 1552, 1650, 1820, 1926, 1962, 2072, 2262, 2380, 2420, 2542, 2752, 2882, 2926, 3060, 3290, 3432, 3480

Offset: 0

Omar E. Pol, Aug 19 2011

Even numbers in A001318.

Exponents in the expansion of Sum_{n >= 0} q^(2*n)/(Product_{k = 1..2*n} 1 + q^(2*k)) = 1 + q^2 - q^12 - q^22 + q^26 + q^40 - - + + ... (follows from Berndt et al., Theorem 3.3). Cf. A067589. - Peter Bala, Jan 21 2025

G. C. Greubel, Table of n, a(n) for n = 0..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.
Mircea Merca, The bisectional pentagonal number theorem, Journal of Number Theory, Volume 157 (December 2015), Pages 223-232.
Index entries for linear recurrences with constant coefficients, signature (3,-5,7,-7,5,-3,1).

Cf. A001318, A067589, A154293.

CoefficientList[Series[-2*x*(x^2 - x + 1)*(x^2 + 4*x + 1)/((x - 1)^3*(x^2 + 1)^2), {x, 0, 50}], x] (* G. C. Greubel, Jun 06 2017 *)
LinearRecurrence[{3,-5,7,-7,5,-3,1},{0,2,12,22,26,40,70},50] (* Harvey P. Dale, Apr 09 2019 *)

my(x='x+O('x^50)); concat([0], Vec(-2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2))) \\ G. C. Greubel, Jun 06 2017

a(n) = A000217(A108752(n+1))/3 = 2*A154293(n+1).
G.f.: -2*x*(x^2-x+1)*(x^2+4*x+1)/((x-1)^3*(x^2+1)^2). - Colin Barker, Sep 12 2012
Sum_{n>=1} 1/a(n) = 6 - (1+4/sqrt(3))*Pi/2. - Amiram Eldar, Mar 18 2022

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A014493 Odd triangular numbers.

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A067567 Odd numbers with an odd number of partitions.

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A067588 Total number of parts in all partitions of n into odd parts.

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A306925 Sum over all partitions of n into distinct parts of the bitwise XOR of the parts.

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A193828 Even generalized pentagonal numbers.

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