A193867 -id:A193867 - cp's OEIS frontend

A014494 Even triangular numbers.

0, 6, 10, 28, 36, 66, 78, 120, 136, 190, 210, 276, 300, 378, 406, 496, 528, 630, 666, 780, 820, 946, 990, 1128, 1176, 1326, 1378, 1540, 1596, 1770, 1830, 2016, 2080, 2278, 2346, 2556, 2628, 2850, 2926, 3160, 3240, 3486, 3570, 3828, 3916, 4186, 4278, 4560

Offset: 0

Mohammad K. Azarian

Even numbers of the form n*(n+1)/2.
Even generalized hexagonal numbers. - Omar E. Pol, Apr 24 2016
The sequence terms occur as the exponents in the expansion of (1 - q^6) * Product_{n >= 1} (1 - q^(16*n-6))*(1 - q^(16*n))*(1 - q^(16*n+6)) = Sum_{n in Z} (-1)^n * q^(2*n*(4*n+1)) = 1 - q^6 - q^10 + q^28 + q^36 - q^66 - q^78 + + - - . - Peter Bala, Dec 23 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A000796, A014493, A056575, A074378, A193867.

See also similar sequences listed in A299645.

[1/2*(2*n+1)*(2*n+1-(-1)^n): n in [0..50]]; // Vincenzo Librandi, Aug 18 2011

Table[2Ceiling[n/2]*(2n + 1), {n, 0, 47}] (* Robert G. Wilson v, Nov 05 2004 *)
1/2 (2#+1)(2#+1-(-1)^#) &/@Range[0,47] (* Ant King, Nov 18 2010 *)
Select[1/2 #(#+1) &/@Range[0,95],EvenQ] (* Ant King, Nov 18 2010 *)

a(n)=(2*n+1)*(2*n+1-(-1)^n)/2 \\ Charles R Greathouse IV, Oct 07 2015

def A014494(n): return (2*n+1)*(n+n%2) # Chai Wah Wu, Mar 11 2022

From Ant King, Nov 18 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.: -2*x*(3*x^2+2*x+3)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009
a(n) = A000217(A014601(n)). - Reinhard Zumkeller, Oct 04 2004
a(n) = A014493(n+1)-(2n+1)*(-1)^n. - R. J. Mathar, Sep 15 2009
a(n) = A193867(n+1) - 1. - Omar E. Pol, Aug 17 2011
Sum_{n>=1} 1/a(n) = 2 - Pi/2. - Robert Bilinski, Jan 20 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2)-2. - Amiram Eldar, Mar 06 2022
E.g.f.: x*(5 + 2*x)*cosh(x) + (1 + x)*(1 + 2*x)*sinh(x). - Stefano Spezia, Dec 24 2024

More terms from Erich Friedman

A188135 a(n) = 8n^2 + 2n + 1.

Original entry on oeis.org

1, 11, 37, 79, 137, 211, 301, 407, 529, 667, 821, 991, 1177, 1379, 1597, 1831, 2081, 2347, 2629, 2927, 3241, 3571, 3917, 4279, 4657, 5051, 5461, 5887, 6329, 6787, 7261, 7751, 8257, 8779, 9317, 9871, 10441, 11027, 11629, 12247, 12881, 13531, 14197, 14879, 15577, 16291, 17021, 17767

Offset: 0

Paul Curtz, Mar 30 2011

Bisection of A193867. - Omar E. Pol, Aug 16 2011

Sequence found by reading the line from 1, in the direction 1, 11, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Sep 04 2011

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A004770, A084849, A113770, A185438, A193867.

[1 + 2*n + 8*n^2: n in [0..50]]; // Vincenzo Librandi, Mar 30 2011

a(n)=8*n^2+2*n+1 \\ Charles R Greathouse IV, Oct 07 2015

First differences: a(n) - a(n-1) = 16*n - 6 = A113770(n) = 2*A004770(n).
Second differences: a(n) - 2*a(n-1) + a(n-2) = 16.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From R. J. Mathar, Apr 06 2011: (Start)
G.f.: -(1+x)*(7*x+1)/(x-1)^3.
a(n) = A084849(2*n). (End)
E.g.f.: exp(x)*(1 + 10*x + 8*x^2). - Elmo R. Oliveira, Oct 19 2024

a(41)-a(47) from Elmo R. Oliveira, Oct 19 2024

A185438 a(n) = 8n^2 - 2n + 1.

Original entry on oeis.org

1, 7, 29, 67, 121, 191, 277, 379, 497, 631, 781, 947, 1129, 1327, 1541, 1771, 2017, 2279, 2557, 2851, 3161, 3487, 3829, 4187, 4561, 4951, 5357, 5779, 6217, 6671, 7141, 7627, 8129, 8647, 9181, 9731, 10297, 10879, 11477, 12091, 12721, 13367, 14029, 14707, 15401, 16111, 16837, 17579

Offset: 0

Paul Curtz, Feb 03 2011

Odd numbers (A005408) written clockwise as a square spiral:
.
  41--43--45--47--49--51
   |                   |
  39  13--15--17--19  53
   |   |           |   |
  37  11   1---3  21  55
   |   |       |   |   |
  35   9---7---5  23  57
   |               |   |
  33--31--29--27--25  59
                       |
  71--69--67--65--63--61
.
Walking in straight lines away from the center:
  1, 17, 49, ... = A069129(n+1) =  1 - 8*n + 8*n^2,
  1,  3, 21, ... = A033567(n)   =  1 - 6*n + 8*n^2,
  1, 15, 45, ... = A014634(n)   =  1 + 6*n + 8*n^2,
  1,  5, 25, ... = A080856(n)   =  1 - 4*n + 8*n^2,
  1, 13, 41, ... = A102083(n)   =  1 + 4*n + 8*n^2,
  1,  7, 29, ... = a(n)         =  1 - 2*n + 8*n^2,
  1, 11, 37, ... = A188135(n)   =  1 + 2*n + 8*n^2,
  1,  9, 33, ... = A081585(n)   =  1       + 8*n^2,
  5, 29, 69, ... = A108928(n+1) = -3       + 8*n^2,
  7, 31, 71, ... = A157914(n+1) = -1       + 8*n^2,
  9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.
All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).
For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011
Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A005408, A014634, A014635, A033566, A033567, A033951, A054552, A064038.

Cf. A069129, A080856, A081585, A102083, A108928, A157914, A181407, A188135, A193867.

[1-2*n+8*n^2: n in [0..50]]; // Vincenzo Librandi, Feb 03 2011

Table[1 - 2n + 8n^2, {n, 0, 39}] (* Alonso del Arte, Feb 03 2011 *)
CoefficientList[Series[(-1 - 4 x - 11 x^2)/(x - 1)^3, {x, 0, 47}], x] (* Michael De Vlieger, Aug 01 2016 *)

a(n)=8*n^2-2*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 16*n - 10 (n > 0).
a(n) = 2*a(n-1) - a(n-2) + 16 (n > 1).
a(n) = 3*(n-1) - 3*a(n-2) + a(n-3) (n > 2).
G.f.: (-1 - 4*x - 11*x^2)/(x-1)^3. - R. J. Mathar, Feb 03 2011
a(n) = A014635(n) + 1. - Bruno Berselli, Apr 09 2011
E.g.f.: exp(x)*(1 + 6*x + 8*x^2). - Elmo R. Oliveira, Nov 17 2024

A193868 Even central polygonal numbers.

Original entry on oeis.org

2, 4, 16, 22, 46, 56, 92, 106, 154, 172, 232, 254, 326, 352, 436, 466, 562, 596, 704, 742, 862, 904, 1036, 1082, 1226, 1276, 1432, 1486, 1654, 1712, 1892, 1954, 2146, 2212, 2416, 2486, 2702, 2776, 3004, 3082, 3322, 3404, 3656, 3742, 4006, 4096, 4372

Offset: 1

Omar E. Pol, Aug 15 2011

Odd triangular numbers plus 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000124, A014493, A174114, A193867.

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

Table[(3 + (-1)^n - 2 (2 + (-1)^n) n + 4 n^2)/2, {n, 50}] (* or *)
Select[PolygonalNumber@ Range@ 100, OddQ] + 1 (* Version 10.4, or *)
Table[If[EvenQ@ n, 2 n^2 - 3 n + 2, 2 n^2 - n + 1], {n, 50}] (* or *)
Rest@ CoefficientList[Series[-2 x (1 + x + 4 x^2 + x^3 + x^4)/((1 + x)^2 (x - 1)^3), {x, 0, 50}], x] (* Michael De Vlieger, Jun 30 2016 *)
LinearRecurrence[{1,2,-2,-1,1},{2,4,16,22,46},50] (* Harvey P. Dale, Sep 13 2020 *)

a(n)=(2*n-1)*(2*n-1-(-1)^n)/2+1 \\ Charles R Greathouse IV, Jun 11 2015

Vec(2*x*(1+x+4*x^2+x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = A000124(A042963(n-1)).
a(n) = 1 + A014493(n).
a(n) = 2*A174114(n).
G.f.: -2*x*(1+x+4*x^2+x^3+x^4) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (3+(-1)^n-2*(2+(-1)^n)*n+4*n^2)/2.
a(n) = 2*n^2-3*n+2 for n even.
a(n) = 2*n^2-n+1 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(sqrt(2) + 2*cosh(sqrt(7)*Pi/4))). - Amiram Eldar, May 11 2025

cp's OEIS Frontend

A014494 Even triangular numbers.

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A188135 a(n) = 8*n^2 + 2*n + 1.

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A185438 a(n) = 8*n^2 - 2*n + 1.

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A193868 Even central polygonal numbers.

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A188135 a(n) = 8n^2 + 2n + 1.

A185438 a(n) = 8n^2 - 2n + 1.