A188135 -id:A188135 - cp's OEIS frontend

A078140 Convolutory inverse of signed lower Wythoff sequence.

1, 3, 5, 9, 17, 30, 52, 90, 154, 262, 446, 758, 1285, 2176, 3683, 6230, 10533, 17803, 30085, 50831, 85873, 145063, 245037, 413891, 699082, 1180761, 1994293, 3368302, 5688920, 9608292, 16227841, 27407792, 46289925, 78180465, 132041227

Offset: 1

Clark Kimberling, Nov 23 2002

nonn

Suppose that r is a real number in the interval [3/2, 5/3).  Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k).  It appears that c(k) > 0 for all k >= 0.  Indeed, it appears that C(r) is strictly increasing and that the limit L(r) of c(k+1)/c(k) as k -> oo exists.  Following is a guide for selected numbers r.
** r **           C(r)       L(r)
sqrt(7/3)        A188135    A288238
Pi/2             A288229    A288239
sqrt(5/2)        A288230    A288240
4^(1/3)          A288231    A288241
(1 + sqrt(5))/2  A078140    A281112
3e/5             A288232    A288242
sqrt(8/3)        A288233    A288935
-1 + sqrt(7)     A288234    A289003
sqrt(e)          A288235    A289005
-4/5 + sqrt(6)   A288236    A289032
sqrt(11/4)       A288237    A289033

			a(5) = 17 = -[w(5)*a(1)-w(4)*a(2)+w(3)*a(3)-w(2)*a(4)] = -8*1+6*3-4*5+3*9. (a(1),a(2),...,a(n))(*)(w(1),-w(2),w(3),...,-d*w(n)) = (1,0,0,...,0), where (*) denotes convolution, w = lower Wythoff sequence, A000201.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Another question about the golden ratio and other numbers, MathOverflow, Jan 17 2017.

Cf. A000201, A077607, A281112, A279676.

CoefficientList[Series[1/Sum[Floor[GoldenRatio*(k + 1)] (-x)^k, {k, 0, 50}],
{x, 0,50}], x]  (* Clark Kimberling, Dec 12 2016 *)

a(n) = d*[w(n)*a(1)-w(n-1)*a(2)+...+d*w(2)*a(n-1)], where d=(-1)^n, with a(1)=1 and w=floor(n*tau), tau=(1+sqrt(5))/2.

Comments added by Clark Kimberling, Jul 10 2017

A211377 T(n,k) = ((k + n)^2 - 4k + 3 + (-1)^k - (k + n - 2)(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 2, 5, 6, 8, 9, 12, 13, 7, 10, 11, 14, 15, 17, 18, 21, 22, 25, 26, 16, 19, 20, 23, 24, 27, 28, 30, 31, 34, 35, 38, 39, 42, 43, 29, 32, 33, 36, 37, 40, 41, 44, 45, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66, 68

Offset: 1

Boris Putievskiy, Feb 07 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(1,2), T(2,1), T(2,2), T(3,1);
...
T(1,n), T(1,n-1), T(2,n-2), T(2,n-1), T(3,n-2), T(3,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonal - step to the west, step to the southwest, step to the east, step to the southwest and so on.  The length of each step is 1.
Table contains:
row 1 is alternation of elements A130883 and A033816,
row 2 accommodates elements A100037 in odd places;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A071355 and A014106,
column 3 accommodates elements A130861 in even places;
main diagonal accommodates elements A188135 in odd places,
diagonal 1, located above the main diagonal, is alternation of elements A033567 and A033566,
diagonal 2, located above the main diagonal, is alternation of elements A139271 and A033585.

			The start of the sequence as a table:
   1,  3,  2,   8,   7,  17,  16,  30,  29,  47,  46, ...
   4,  5,  9,  10,  18,  19,  31,  32,  48,  49,  69, ...
   6, 12, 11,  21,  20,  34,  33,  51,  50,  72,  71, ...
  13, 14, 22,  23,  35,  36,  52,  53,  73,  74,  98, ...
  15, 25, 24,  38,  37,  55,  54,  76,  75, 101, 100, ...
  26, 27, 39,  40,  56,  57,  77,  78, 102, 103, 131, ...
  28, 42, 41,  59,  58,  80,  79, 105, 104, 134, 133, ...
  43, 44, 60,  61,  81,  82, 106, 107, 135, 136, 168, ...
  45, 63, 62,  84,  83, 109, 108, 138, 137, 171, 170, ...
  64, 65, 85,  86, 110, 111, 139, 140, 172, 173, 209, ...
  66, 88, 87, 113, 112, 142, 141, 175, 174, 212, 211, ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   3,  4;
   2,  5,  6;
   8,  9, 12, 13;
   7, 10, 11, 14, 15;
  17, 18, 21, 22, 25, 26;
  16, 19, 20, 23, 24, 27, 28;
  30, 31, 34, 35, 38, 39, 42, 43;
  29, 32, 33, 36, 37, 40, 41, 44, 45;
  47, 48, 51, 52, 55, 56, 59, 60, 63, 64;
  46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
  ...
The start of the sequence as an array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from row number 2*r-2 of the triangular array above.
Last  2*r-1 numbers are from row number 2*r-1 of the triangular array above.
   1;
   3,  4,  2,  5,  6;
   8,  9, 12, 13,  7, 10, 11, 14, 15;
  17, 18, 21, 22, 25, 26, 16, 19, 20, 23, 24, 27, 28;
  30, 31, 34, 35, 38, 39, 42, 43, 29, 32, 33, 36, 37, 40, 41, 44, 45;
  47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 46, 49, 50, 53, 54, 57, 58, 61, 62, 65, 66;
  ...
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+5, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-1, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000384, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135.

T[n_, k_] := ((k+n)^2 - 4k + 3 + (-1)^k - (k+n-2)(-1)^(k+n))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 29 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3+(-1)**j-t*(-1)**(t+2))/2

As a table:
T(n,k) = ((k + n)^2 - 4*k + 3 + (-1)^k - (k + n - 2)*(-1)^(k + n))/2.
As a linear sequence:
a(n) = ((t + 2)^2 - 4*j + 3 + (-1)^j - t*(-1)^t)/2, where j = (t*t + 3*t + 4)/2 - n and t = int((sqrt(8*n - 7) - 1)/ 2).

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

Original entry on oeis.org

2, 7, 18, 31, 50, 71, 98, 127, 162, 199, 242, 287, 338, 391, 450, 511, 578, 647, 722, 799, 882, 967, 1058, 1151, 1250, 1351, 1458, 1567, 1682, 1799, 1922, 2047, 2178, 2311, 2450, 2591, 2738, 2887, 3042, 3199, 3362, 3527, 3698, 3871, 4050, 4231, 4418, 4607, 4802

Offset: 0

Bruno Berselli, Sep 21 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the numbers in increasing order on the vertical line containing 2 of the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

Also A077591 (without first term) and A157914 interleaved.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195605.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000217, A077591, A157914.
Cf. A047621 (contains first differences), A016754 (contains the sum of any two consecutive terms).
Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195241 [incomplete list].

[(4*n*(n+2)+(-1)^n+3)/2: n in [0..48]];

CoefficientList[Series[(2 + 3 x + 4 x^2 - x^3) / ((1 + x) (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,0,-2,1},{2,7,18,31},50] (* Harvey P. Dale, Jan 21 2017 *)

for(n=0, 48, print1((4*n*(n+2)+(-1)^n+3)/2", "));

G.f.: (2+3*x+4*x^2-x^3)/((1+x)*(1-x)^3).
a(n) = a(-n-2) = 2*a(n-1)-2*a(n-3)+a(n-4).
a(n) = A047524(A000982(n+1)).
Sum_{n>=0} 1/a(n) = 1/2 + Pi^2/16 - cot(Pi/(2*sqrt(2)))*Pi/(4*sqrt(2)). - Amiram Eldar, Mar 06 2023

A193867 Odd central polygonal numbers.

Original entry on oeis.org

1, 7, 11, 29, 37, 67, 79, 121, 137, 191, 211, 277, 301, 379, 407, 497, 529, 631, 667, 781, 821, 947, 991, 1129, 1177, 1327, 1379, 1541, 1597, 1771, 1831, 2017, 2081, 2279, 2347, 2557, 2629, 2851, 2927, 3161, 3241, 3487, 3571, 3829, 3917, 4187, 4279

Offset: 1

Omar E. Pol, Aug 15 2011

Even triangular numbers plus 1.

Union of A188135 and A185438 without repetitions (A188135 is a bisection of this sequence. Another bisection is A185438 but without its initial term).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000124, A014494, A014601, A185438, A188135, A193868.

Select[Accumulate[Range[0,100]],EvenQ]+1 (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,7,11,29,37},50] (* Harvey P. Dale, Nov 29 2014 *)

Vec(-x*(x^2+1)*(x^2+6*x+1) / ((1+x)^2*(x-1)^3) + O(x^100)) \\ Colin Barker, Jan 27 2016

a(n) = A000124(A014601(n-1)).
a(n) = 1 + A014494(n-1).
G.f.: -x*(x^2+1)*(x^2+6*x+1) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Aug 25 2011
From Colin Barker, Jan 27 2016: (Start)
a(n) = (4*n^2+2*(-1)^n*n-4*n-(-1)^n+3)/2.
a(n) = 2*n^2-n+1 for n even.
a(n) = 2*n^2-3*n+2 for n odd. (End)
Sum_{n>=1} 1/a(n) = 2*Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(2*cosh(sqrt(7)*Pi/4) - sqrt(2))). - Amiram Eldar, May 11 2025

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

A188382 Primes of the form 8n^2 + 2n + 1.

Original entry on oeis.org

11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, 2927, 3571, 3917, 4657, 5051, 6329, 8779, 9871, 11027, 14197, 14879, 17021, 20101, 21737, 26107, 27967, 28921, 33931, 34981, 39341, 40471, 41617, 50087, 51361, 59341

Offset: 1

Alonso del Arte, Mar 29 2011

In a variant of the Ulam spiral with only odd numbers, prime numbers can line up in horizontal or vertical lines rather than diagonal lines. These primes are on one such horizontal (or vertical) line.
Primes in A188135. Primes in the 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
Equivalently, primes of the form 2*n^2+n+1. - N. J. A. Sloane, Nov 08 2014

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

Cf. A090684, A187677.

[ a: n in [0..250] | IsPrime(a) where a is 8*n^2 + 2*n + 1 ]; // Vincenzo Librandi, Apr 05 2011

select(isprime,[seq(8*n^2+2*n+1,n=0..86)]); # Peter Luschny, Aug 22 2011

Select[Table[8n^2 + 2n + 1, {n, 100}], PrimeQ]

select(isprime, vector(1000,n,8*n^2+2*n+1)) \\ Charles R Greathouse IV, Jun 14 2011

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

Original entry on oeis.org

1, 2, 3, 23, 59, 111, 179, 263, 363, 479, 611, 759, 923, 1103, 1299, 1511, 1739, 1983, 2243, 2519, 2811, 3119, 3443, 3783, 4139, 4511, 4899, 5303, 5723, 6159, 6611, 7079, 7563, 8063, 8579, 9111, 9659, 10223, 10803, 11399, 12011, 12639, 13283, 13943

Offset: 0

Bruno Berselli, Sep 13 2011 - based on remarks and sequences by Omar E. Pol

Sequence found by reading the line 1, 2, 3, 23,.. in the square spiral whose vertices are the triangular numbers (A000217) - see Pol's comments in other sequences visible in this numerical spiral.

This is a subsequence of A110326 (without signs) and A047838 (apart from the second term, 2).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms: An origin of A195241.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217.

Cf. A033585, A069129, A077221, A102083, A139098, A139271-A139277, A139592, A139593, A188135, A194268, A194431, A195605 [incomplete list].

m:=44; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+19*x^3-3*x^4)/(1-x)^3));

CoefficientList[Series[(1 - x + 19 x^3 - 3 x^4)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{3,-3,1},{1,2,3,23,59},50] (* Harvey P. Dale, Dec 04 2022 *)

makelist(coeff(taylor((1-x+19*x^3-3*x^4)/(1-x)^3, x, 0, n), x, n), n, 0, 43);

Vec((1-x+19*x^3-3*x^4)/(1-x)^3+O(x^44))

G.f.: (1-x+19*x^3-3*x^4)/(1-x)^3.

a(n) = 8*n^2-20*n+11 for n>1; a(0)=1, a(1)=2.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 2, 3, 6, 9, 10, 13, 14, 7, 8, 11, 12, 15, 18, 19, 22, 23, 26, 27, 16, 17, 20, 21, 24, 25, 28, 31, 32, 35, 36, 39, 40, 43, 44, 29, 30, 33, 34, 37, 38, 41, 42, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 69

Offset: 1

Boris Putievskiy, Feb 14 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
T(1,1) = 1;
T(1,3), T(2,2), T(1,2), T(2,1), T(3,1);
. . .
T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);
...
Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.
Table contains the following:
row 1 is alternation of elements A130883 and A100037,
row 2 accommodates elements A033816 in even places;
column 1 is alternation of elements A000384 and A014106,
column 2 is alternation of elements A091823 and A071355,
column 4 accommodates elements A130861 in odd places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,
diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,
diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

			The start of the sequence as a table:
   1   4   2   9   7   8  16 ...
   5   3  10   8  19  17  32 ...
   6  13  11  22  20  35  33 ...
  14  12  23  21  36  34  53 ...
  15  26  24  39  37  56  54 ...
  27  25  40  38  57  55  78 ...
  28  43  41  60  58  81  79 ...
  ...
The start of the sequence as a triangle array read by rows:
   1
   4  5
   2  3  6
   9 10 13 14
   7  8 11 12 15
  18 19 22 23 26 27
  16 17 20 21 24 25 28
  ...
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.
Last 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
   1
   4  5  2  3  6
   9 10 13 14  7  8 11 12 15
  18 19 22 23 26 27 16 17 20 21 24 25 28
  ...
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+6, 2*r*r-5*r+7, ..., 2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A000384, A002260, A003056, A003057, A004736, A014106, A033566, A033567, A033585, A033816, A071355, A091823, A100037, A130861, A130883, A139271, A188135, A194431, A211377.

T:=(n,k)->((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 - (-1)^n - (-1)^(n+k)(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3-(-1)**i-(t+2)*(-1)**t)/2

As a table:
T(n,k) = ((k+n)^2-4*k+3-(-1)^n-(k+n)*(-1)^(k+n))/2.
As a linear sequence:
a(n) = (A003057(n)^2-4*A004736(n)+3-(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3-(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 4, 2, 3, 6, 10, 9, 14, 13, 7, 8, 11, 12, 15, 19, 18, 23, 22, 27, 26, 16, 17, 20, 21, 24, 25, 28, 32, 31, 36, 35, 40, 39, 44, 43, 29, 30, 33, 34, 37, 38, 41, 42, 45, 49, 48, 53, 52, 57, 56, 61, 60, 65, 64, 46, 47, 50, 51, 54, 55, 58, 59, 62, 63, 66, 70

Offset: 1

Boris Putievskiy, Feb 15 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);
. . .
T(1,2*n+1), T(2,2*n), T(2,2*n-1), T(1,2*n), ...T(2*n-1,3), T(2*n,2), T(2*n,1), T(2*n-1,2), T(2*n+1,1);
. . .
Movement along two adjacent antidiagonals - step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.
Table contains:
row  1 accommodates elements A130883 in odd places,
row  2 is alternation of elements A100037 and A033816;
column 1 is alternation of elements A000384 and A091823,
column 2 is alternation of elements A014106 and A071355,
column 3 accommodates elements A130861 in even places;
main diagonal is alternation of elements A188135 and A033567,
diagonal 1, located above the main diagonal accommodates elements A033566 in even places,
diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,
diagonal 3, located above the main diagonal  accommodates of elements A033585.

			The start of the sequence as table:
1....5...2..10...7..19..16...
4....3...9...8..18..17..31...
6...14..11..23..20..36..33...
13..12..22..21..35..34..52...
15..27..24..40..37..57..54...
26..25..39..38..56..55..77...
28..44..41..61..58..82..79...
. . .
The start of the sequence as triangle array read by rows:
1;
5,4;
2,3,6;
10,9,14,13;
7,8,11,12,15;
19,18,23,22,27,26;
16,17,20,21,24,25,28;
. . .
The start of the sequence as array read by rows, the length of row r is 4*r-3.
First 2*r-2 numbers are from the row number 2*r-2 of  triangle array, located above.
Last  2*r-1 numbers are from the row number 2*r-1 of  triangle array, located above.
1;
5,4,2,3,6;
10,9,14,13,7,8,11,12,15;
19,18,23,22,27,26,16,17,20,21,24,25,28;
. . .
Row number r contains permutation 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:
2*r*r-5*r+7, 2*r*r-5*r+6,...2*r*r-r-4, 2*r*r-r-3, 2*r*r-r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A211377, A130883, A100037, A033816, A000384, A091823, A014106, A071355, A130861, A188135, A033567, A033566, A139271, A024847, A033585, A002260, A004736, A003056, A003057.

T:=(n,k)->((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2: seq(seq(T(k,n-k),k=1..n-1),n=1..13); # Muniru A Asiru, Dec 06 2018

T[n_, k_] := ((n+k)^2 - 4k + 3 + (-1)^k - 2(-1)^n - (n+k)(-1)^(n+k))/2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result=((t+2)**2-4*j+3+(-1)**j-2*(-1)**i-(t+2)*(-1)**t)/2

As table
T(n,k) = ((k+n)^2-4*k+3+(-1)^k-2*(-1)^n-(k+n)*(-1)^(k+n))/2.
As linear sequence
a(n) = (A003057(n)^2-4*A004736(n)+3+(-1)^A004736(n)-2*(-1)^A002260(n)-A003057(n)*(-1)^A003056(n))/2;
a(n) = ((t+2)^2-4*j+3+(-1)^j-2*(-1)^i-(t+2)*(-1)^t)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).

A317298 a(n) = (1/2)(1 + (-1)^n + 2n + 4*n^2).

Original entry on oeis.org

1, 3, 11, 21, 37, 55, 79, 105, 137, 171, 211, 253, 301, 351, 407, 465, 529, 595, 667, 741, 821, 903, 991, 1081, 1177, 1275, 1379, 1485, 1597, 1711, 1831, 1953, 2081, 2211, 2347, 2485, 2629, 2775, 2927, 3081, 3241, 3403, 3571, 3741, 3917, 4095, 4279, 4465, 4657

Offset: 0

Stefano Spezia, Jan 22 2019

For n > 0, first differences of A304487.

All the terms of this sequence are odd numbers.

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

Cf. A000290, A004526, A005408, A033567, A033999, A188135, A304487.

Cf. A306362 (prime numbers subsequence).

Flat(List([0..50], n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2)));

[(1/2)*(1+(-1)^n+2*n+4*n^2): n in [0..50]];

a:=n->(1/2)*(1 + (-1)^n + 2*n + 4*n^2): seq(a(n), n=0..50);

a[n_]:=(1/2)*(1 + (-1)^n + 2*n + 4*n^2); Array[a, 50, 0]

makelist((1/2)*(1+(-1)^n+2*n+4*n^2), n, 0, 50);

PARI
```
a(n) = (1/2)*(1+(-1)^n+2*n+4*n^2);
```

[(1+(-1)**n+2*n+4*n**2)/2 for n in range(0,50)]

a(n) = (1/2)*(A033999(n) + A005408(n) + 4*A000290(n)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
a(2*n) = A188135(n).
a(2*n-1) = A033567(n), for n > 0.
O.g.f.: -(1 + x + 5*x^2 + x^3)/(-1 + x)^3*(1 + x).
E.g.f.: (1/2)*exp(-x)*(1 + exp(2*x)*(1 + 6*x + 4*x^2)).
Sum_{n>0} 1/a(n) = (1/4)*(Pi - log(4)) + i*(polygamma(0, 1/8 - i*sqrt(7)/8) - polygamma(0, 1/8 + i*sqrt(7)/8))/(2*sqrt(7)) = 1.603596691017309384564895..., where i is the imaginary unit. - Stefano Spezia, Feb 10 2019
a(n) = 1 + 2*(n^2 + floor(n/2)). - Stefano Spezia, Dec 08 2021

cp's OEIS Frontend

A078140 Convolutory inverse of signed lower Wythoff sequence.

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A211377 T(n,k) = ((k + n)^2 - 4*k + 3 + (-1)^k - (k + n - 2)*(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

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A195605 a(n) = (4*n*(n+2)+(-1)^n+1)/2 + 1.

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A193867 Odd central polygonal numbers.

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

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A188382 Primes of the form 8*n^2 + 2*n + 1.

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A195241 Expansion of (1-x+19*x^3-3*x^4)/(1-x)^3.

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A211377 T(n,k) = ((k + n)^2 - 4k + 3 + (-1)^k - (k + n - 2)(-1)^(k + n))/2; n, k > 0, read by antidiagonals.

A195605 a(n) = (4n(n+2)+(-1)^n+1)/2 + 1.

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

A188382 Primes of the form 8n^2 + 2n + 1.

A195241 Expansion of (1-x+19x^3-3x^4)/(1-x)^3.

A213171 T(n,k) = ((k+n)^2 - 4k + 3 - (-1)^n - (k+n)(-1)^(k+n))/2; n, k > 0, read by antidiagonals.

A213205 T(n,k) = ((k+n)^2-4k+3+(-1)^k-2(-1)^n-(k+n)*(-1)^(k+n))/2; n , k > 0, read by antidiagonals.

A317298 a(n) = (1/2)(1 + (-1)^n + 2n + 4*n^2).