A084849 -id:A084849 - cp's OEIS frontend

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

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

A007606 Take 1, skip 2, take 3, etc.

Original entry on oeis.org

1, 4, 5, 6, 11, 12, 13, 14, 15, 22, 23, 24, 25, 26, 27, 28, 37, 38, 39, 40, 41, 42, 43, 44, 45, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 137, 138

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

List the natural numbers: 1, 2, 3, 4, 5, 6, 7, ... . Keep the first number (1), delete the next two numbers (2, 3), keep the next three numbers (4, 5, 6), delete the next four numbers (7, 8, 9, 10) and so on.
a(A000290(n)) = A000384(n). - Reinhard Zumkeller, Feb 12 2011
A057211(a(n)) = 1. - Reinhard Zumkeller, Dec 30 2011
Numbers k with the property that the smallest Dyck path of the symmetric representation of sigma(k) has a central valley. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
Union of nonzero terms of A000384 and A317304. - Omar E. Pol, Aug 29 2018
The values of k such that, in a listing of all congruence classes of positive integers, the k-th congruence class contains k. Here the class r mod m (with r in {1,...,m}) precedes the class r' mod m' (with r' in {1,...,m'}) iff mA360418. - James Propp, Feb 10 2023

			From _Omar E. Pol_, Aug 29 2018: (Start)
Written as an irregular triangle in which the row lengths are the odd numbers the sequence begins:
    1;
    4,   5,   6;
   11,  12,  13,  14,  15;
   22,  23,  24,  25,  26,  27,  28;
   37,  38,  39,  40,  41,  42,  43,  44,  45;
   56,  57,  58,  59,  60,  61,  62 , 63,  64,  65,  66;
   79,  80,  81,  82 , 83,  84,  85,  86,  87,  88,  89,  90,  91;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120;
...
Row sums give A005917.
Column 1 gives A084849.
Column 2 gives A096376, n >= 1.
Right border gives A000384, n >= 1.
(End)

C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I, Erhus Publ., Glendale, 1994.
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 177.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
F. Smarandache, Properties of Numbers, 1972.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
Index entries for sequences generated by sieves

Complement of A007607.
Cf. A007950, A007951, A007952, A048859, A004201.
Cf. A000384, A005917, A084849, A096376.

a007606 n = a007606_list !! (n-1)
a007606_list = takeSkip 1 [1..] where
   takeSkip k xs = take k xs ++ takeSkip (k + 2) (drop (2*k + 1) xs)
-- Reinhard Zumkeller, Feb 12 2011

Flatten[ Table[i, {j, 1, 17, 2}, {i, j(j - 1)/2 + 1, j(j + 1)/2}]] (* Robert G. Wilson v, Mar 11 2004 *)
Join[{1},Flatten[With[{nn=20},Range[#[[1]],Total[#]]&/@Take[Thread[ {Accumulate[ Range[nn]]+1,Range[nn]}],{2,-1,2}]]]] (* Harvey P. Dale, Jun 23 2013 *)
With[{nn=20},Take[TakeList[Range[(nn(nn+1))/2],Range[nn]],{1,nn,2}]]//Flatten (* Harvey P. Dale, Feb 10 2023 *)

for(n=1,66,m=sqrtint(n-1);print1(n+m*(m+1),","))

a(n) = n + m*(m+1) where m = floor(sqrt(n-1)). - Klaus Brockhaus, Mar 26 2004

a(n+1) = a(n) + if n=k^2 then 2*k+1 else 1; a(1) = 1. - Reinhard Zumkeller, May 13 2009

A100035 a(n+1) occurs not earlier as a neighbor of terms = a(n): either it is the greatest number < a(n) or, if no such number exists, the smallest number > a(n); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 1, 4, 3, 5, 4, 2, 5, 1, 6, 5, 7, 6, 4, 7, 3, 6, 2, 7, 1, 8, 7, 9, 8, 6, 9, 5, 8, 4, 9, 3, 8, 2, 9, 1, 10, 9, 11, 10, 8, 11, 7, 10, 6, 11, 5, 10, 4, 11, 3, 10, 2, 11, 1, 12, 11, 13, 12, 10, 13, 9, 12, 8, 13, 7, 12, 6, 13, 5, 12, 4, 13, 3, 12, 2, 13, 1, 14, 13, 15, 14, 12, 15, 11, 14, 10

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

The natural numbers (A000027) occur infinitely many times as disjoint subsequences, see the example below and A100036, A100037, A100038 and A100039: exactly one k exists for all x < y such that a(k) = x and (a(k-1) = y or a(k+1) = y).
a(2*k^2 + k + 1) = a(A084849(k)) = 1 for k >= 0;
a(2*k^2 - 3*k) = a(A014107(k)) = 2 for k > 1;
a(2*k^2 + 5*k) = a(A033537(k)) = 3 for k > 1;
a(2*k^2 + k - 5) = a(A100040(k)) = 4 for k > 2;
a(2*k^2 + k - 7) = a(A100041(k)) = 5 for k > 3.

			First terms (10 = A, 11 = B, 12 = C) and some subsequences = A000027:
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1CBD
123.4.5....6.7........8.9............A.B................C.D.
...1....2........3............4................5..........
..........1........2............3................4......
.....................1............2................3....

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000027, A014107, A033537, A100036, A100037, A100038, A100039, A100040, A100041.

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 7, 7, 4, 1, 1, 6, 9, 13, 9, 4, 1, 1, 7, 11, 21, 16, 11, 4, 1, 1, 8, 13, 31, 25, 22, 13, 4, 1, 1, 9, 15, 43, 36, 37, 28, 15, 5, 1, 1, 10, 17, 57, 49, 56, 49, 40, 17, 5, 1, 1, 11, 19, 73, 64, 79, 76, 85, 43, 19, 5, 1, 1, 12, 21, 91, 81, 106, 109, 156, 89, 49, 21, 5, 1

Offset: 1

Omar E. Pol, Mar 05 2015

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A070941: 1, 2, 3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5,   5,   5
A005408: 1, 3, 5,  7,  9, 11, 13,  15,  17,  19,  21,  23,  25,  27,  29
A151788: 1, 4, 7, 13, 16, 22, 28,  40,  43,  49,  55,  67,  73,  85,  97
A147562: 1, 5, 9, 21, 25, 37, 49,  85,  89, 101, 113, 149, 161, 197, 233
A151790: 1, 6,11, 31, 36, 56, 76, 156, 161, 181, 201, 281, 301, 381, 461
A151781: 1, 7,13, 43, 49, 79,109, 259, 265, 295, 325, 475, 505, 655, 805
A151792: 1, 8,15, 57, 64,106,148, 400, 407, 449, 491, 743, 785,1037,1289
A151793: 1, 9,17, 73, 81,137,193, 585, 593, 649, 705,1097,1153,1545,1937
A255764: 1,10,19, 91,100,172,244, 820, 829, 901, 973,1549,1621,2197,2773
A255765: 1,11,21,111,121,211,301,1111,1121,1211,1301,2111,2201,3011,3821
A255766: 1,12,23,133,144,254,364,1464,1475,1585,1695,2795,2905,4005,5105
...

Index entries for sequences related to cellular automata

Rows 1-12: A000012, A070941, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255764, A255765, A255766.
Columns 1-10: A000012, A000027, A005408, A002061, A000290, A084849, A056107, A053698, A100705, A100104.
Cf. A000120, A255740.

A100036 a(n) = smallest m such that A100035(m) = n.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 14, 23, 25, 38, 40, 57, 59, 80, 82, 107, 109, 138, 140, 173, 175, 212, 214, 255, 257, 302, 304, 353, 355, 408, 410, 467, 469, 530, 532, 597, 599, 668, 670, 743, 745, 822, 824, 905, 907, 992, 994, 1083, 1085, 1178, 1180, 1277, 1279, 1380

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

Smallest positions of occurrences of the natural numbers as subsequence in A100035;
A100035(a(n)) = n and A100035(m) <> n for m < a(n);
a(n) < A100037(n) < A100038(n) < A100039(n).

			First terms (10=A,11=B,12=C) of A100035(a(n)):
123.4.5....6.7........8.9............A.B................C.
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1CBD;
a(1) = A084849(1) = 1, A100035(1) = 1;
a(2) = A014107(1) = 2, A100035(2) = 2;
a(3) = A033537(1) = 3, A100035(3) = 3;
a(4) = A100040(1) = 5, A100035(5) = 4;
a(5) = A100041(1) = 7, A100035(7) = 5.

Conjecture: a(n) = partial sums of sequence [1,1,1,2,2,5,2,9,2,13,2,17,2,21,2,25,2,29,2,33,...2,n/2-7,2,...]. In other words, a(n) consists of the numbers 1,2,3 and the sequences A096376 and A096376+2 interspersed. - Ralf Stephan, May 15 2007

A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

Original entry on oeis.org

1, 2, 1, 0, 4, 1, 0, 0, 6, 1, 0, 0, 0, 8, 1, 0, 0, 0, 0, 10, 1, 0, 0, 0, 0, 0, 12, 1, 0, 0, 0, 0, 0, 0, 14, 1, 0, 0, 0, 0, 0, 0, 0, 16, 1, 0, 0, 0, 0, 0, 0, 0, 0, 18, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

A134082 * [1,2,3,...] = A084849: (1, 4, 11, 22, 37, ...).
Binomial transform of A134082 = A134083.
A112295 replaces subdiagonal with (-1,-3,-5, ...).

			First few rows of the triangle:
  1;
  2,  1;
  0,  4,  1;
  0,  0,  6,  1;
  0,  0,  0,  8,  1;
  0,  0,  0,  0, 10,  1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005408, A084849, A112295, A134083.

A134082:= func< n,k | k eq n select 1 else k eq n-1 select 2*n else 0 >;
[A134082(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 17 2021

T[n_, k_]:= If[k==n, 1, If[k==n-1, 2*n, 0]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

def A134082(n,k): return 1 if k==n else 2*n if k==n-1 else 0
flatten([[A134082(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Feb 17 2021

Triangle read by rows, (n-1) zeros followed by (2n, 1). As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal and (2,4,6,8,...) in the subdiagonal.
From formalism in A132382, e.g.f. = I_o[2*(u*x)^(1/2)] (1+2x) where I_o is the zeroth modified Bessel function of the first kind, i.e., I_o[2*(u*x)^(1/2)] = Sum_{j>=0} u^j/j! * x^j/j!. - Tom Copeland, Dec 07 2007
Row polynomial e.g.f.: exp(x*y)(1+2x). - Tom Copeland, Dec 03 2013
Sum_{k=0..n} T(n,k) = 2*n+1 = A005408(n). - G. C. Greubel, Feb 17 2021

More terms added by G. C. Greubel, Feb 17 2021

A177342 a(n) = (4n^3-3n^2+5*n-3)/3.

Original entry on oeis.org

1, 9, 31, 75, 149, 261, 419, 631, 905, 1249, 1671, 2179, 2781, 3485, 4299, 5231, 6289, 7481, 8815, 10299, 11941, 13749, 15731, 17895, 20249, 22801, 25559, 28531, 31725, 35149, 38811, 42719, 46881, 51305, 55999, 60971, 66229, 71781, 77635

Offset: 1

Bruno Berselli, May 06 2010 - Nov 27 2010

This sequence is related to the fourth powers (A000583) by n^4 = n*a(n) - Sum_{i=1..n-1} a(i) - (n-1), with n>1.

Also, n*a(n) - Sum_{i=1..n-1} a(i) provides the first column of A162624 and the second column of A162622 (or A162623). - Bruno Berselli, revised Dec 14 2012

B. Berselli, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

First differences: 2*A084849.
Partial sums: A178073.
Cf. A174723, A162622-A162624.

[(4*n^3-3*n^2+5*n-3)/3: n in [1..39]]; // Bruno Berselli, Aug 24 2011

I:=[1,9,31,75]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Aug 19 2013

CoefficientList[Series[(1 + 5 x + x^2 + x^3) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 19 2013 *)
Table[(4 n^3 - 3 n^2 + 5 n - 3)/3, {n, 1, 40}] (* Bruno Berselli, Feb 17 2015 *)
LinearRecurrence[{4,-6,4,-1},{1,9,31,75},40] (* Harvey P. Dale, Jul 31 2021 *)

a(n)=(4*n^3-3*n^2+5*n-3)/3 \\ Charles R Greathouse IV, Jun 23 2011

G.f.: x*(1 + 5*x + x^2 + x^3)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) - a(-n) = 2*A004006(2n).
a(n) + a(-n) = -A002522(n).
a(n) = 1 + (n-1)*(4*n^2+n+6)/3 = 2*A174723(n)-1.

Formulae added and revised by Bruno Berselli, Feb 17 2015

A174723 a(n) = n(4n^2 - 3*n + 5)/6.

Original entry on oeis.org

1, 5, 16, 38, 75, 131, 210, 316, 453, 625, 836, 1090, 1391, 1743, 2150, 2616, 3145, 3741, 4408, 5150, 5971, 6875, 7866, 8948, 10125, 11401, 12780, 14266, 15863, 17575, 19406, 21360, 23441, 25653, 28000, 30486, 33115, 35891, 38818, 41900, 45141

Offset: 1

Michel Lagneau, Mar 28 2010

We prove that a(n) = Sum_{k=1..n^2} floor(sqrt(k)): a(n) = Sum_{k=1..3} 1 + Sum_{k=4..8} 2 + ... + Sum_{k=(n-1)^2..n^2 - 1} (n-1) + n = 3*1 + 5*2 + 7*3 + ... + (2n-1)(n-1)+ n = Sum_{k=1..n} (2k-1)*(k-1) + n = 2*Sum_{k=1..n} k^2 - 3*Sum_{k=1..n} k + 2n = 2n(n+1)(2n+1)/6 - 3n(n+1)/2 + 2n = n*(4n^2 - 3n + 5) / 6.

Notice that a(4) = 4 + 3*5 + 2*6 + 1*7 and a(8) = 8 + 7*9 + 6*10 + 5*11 + 4*12 + 3*13 + 2*14 + 1*15. In general, a(n) = n + Sum_{k=1..n-1} (n-k)*(n+k). - J. M. Bergot, Jul 31 2013

			From _Bruno Berselli_, Feb 17 2015: (Start)
Third differences:  1, 2,  4,  4,   4,   4,   4, (repeat 4) ... (A151798)
Second differences: 1, 3,  7, 11,  15,  19,  23,  27,   31, ... (A131098)
First differences:  1, 4, 11, 22,  37,  56,  79, 106,  137, ... (A084849)
-------------------------------------------------------------------------
This sequence:      1, 5, 16, 38,  75, 131, 210, 316,  453, ...
-------------------------------------------------------------------------
Partial sums:       1, 6, 22, 60, 135, 266, 476, 792, 1245, ... (A071239)
(End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed., Oxford Univ. Press, 1954.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000196, A071239, A084849, A131098, A151798, A022554.

I:=[1, 5, 16, 38]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 04 2012

A174723 := proc(n)
        n*(4*n^2-3*n+5)/6 ;
end proc:
seq( A174723(n),n=1..20) ; # R. J. Mathar, Nov 07 2011

Table[n (4n^2-3n+5)/6,{n,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,5,16,38},50] (* Harvey P. Dale, Jan 16 2012 *)

a(n)=n*(4*n^2-3*n+5)/6 \\ Charles R Greathouse IV, Oct 07 2015

G.f. x*(1 + x + 2*x^2) / (x-1)^4. - R. J. Mathar, Nov 07 2011
a(1)=1, a(2)=5, a(3)=16, a(4)=38; for n > 4, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jan 16 2012
a(n) = A022554(n^2). - Ridouane Oudra, Jun 13 2025

A370980 If n is even, (n^2-2*n+2)/2, otherwise (n^2-n+2)/2.

Original entry on oeis.org

1, 1, 1, 4, 5, 11, 13, 22, 25, 37, 41, 56, 61, 79, 85, 106, 113, 137, 145, 172, 181, 211, 221, 254, 265, 301, 313, 352, 365, 407, 421, 466, 481, 529, 545, 596, 613, 667, 685, 742, 761, 821, 841, 904, 925, 991, 1013, 1082, 1105, 1177, 1201, 1276, 1301, 1379, 1405, 1486, 1513, 1597, 1625, 1712, 1741, 1831, 1861, 1954

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Mar 23 2024

nonn

Total number of circles in A371373 and A371253, if in the later all the circular arcs are completed to form full circles.

The sequence also gives the number of vertices created from circle intersections when a circle of radius r is drawn around each of n equally spaced points on the circumference of a circle of radius r. The number of regions in these constructions is A093005(n) and the number of edges is A183207(n). See the attached images. - Scott R. Shannon, Jul 06 2024.

			a(n) = 1+n*floor((n-1)/2) = 1+n*A004526(n-1). - _Chai Wah Wu_, Mar 23 2024

Scott R. Shannon, Image for n = 3. In this and other images the center of each circle of shown as a white dot.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 20.

Bisections are A001844 and A084849.
Cf. A371373, A371374, A371375.
Cf. A004526, A371253, A371254, A371253, A183297, A093005, A183207.

def A370980(n): return n*(n-1>>1)+1 # Chai Wah Wu, Mar 23 2024

a(n) = A183207(n) - A093005(n) + 1, by Euler's formula. - Scott R. Shannon, Jul 07 2024

A317304 Numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have a central valley.

Original entry on oeis.org

4, 5, 11, 12, 13, 14, 22, 23, 24, 25, 26, 27, 37, 38, 39, 40, 41, 42, 43, 44, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

Omar E. Pol, Aug 27 2018

Also triangle read by rows which gives the even-indexed rows of triangle A014132.
There are no triangular number (A000217) in this sequence.
For more information about the symmetric representation of sigma see A237593 and its related sequences.
Equivalently, numbers k with the property that both Dyck paths of the symmetric representation of sigma(k) have an even number of peaks. - Omar E. Pol, Sep 13 2018

			Written as an irregular triangle in which the row lengths are the positive even numbers, the sequence begins:
    4,   5;
   11,  12,  13,  14;
   22,  23,  24,  25,  26,  27;
   37,  38,  39,  40,  41,  42,  43,  44;
   56,  57,  58,  59,  60,  61,  62,  63,  64,  65;
   79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90;
  106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119;
...
Illustration of initial terms:
-------------------------------------------------
   k  sigma(k)  Diagram of the symmetry of sigma
-------------------------------------------------
                       _ _           _ _ _ _
                      | | |         | | | | |
                     _| | |         | | | | |
                 _ _|  _|_|         | | | | |
   4      7     |_ _ _|             | | | | |
   5      6     |_ _ _|             | | | | |
                                 _ _|_| | | |
                               _|    _ _|_| |
                             _|     |  _ _ _|
                            |      _|_|
                 _ _ _ _ _ _|  _ _|
  11     12     |_ _ _ _ _ _| |  _|
  12     28     |_ _ _ _ _ _ _| |
  13     14     |_ _ _ _ _ _ _| |
  14     24     |_ _ _ _ _ _ _ _|
.
For the first six terms of the sequence we can see in the above diagram that both Dyck path (the smallest and the largest) of the symmetric representation of sigma(k) have a central valley.
Compare with A317303.

Row sums give A084367. n >= 1.
Column 1 gives A084849, n >= 1.
Column 2 gives A096376, n >= 1.
Right border gives the nonzero terms of A014106.
The union of A000217, A317303 and this sequence gives A001477.
Some other sequences related to the central peak or the central valley of the symmetric representation of sigma are A000217, A000384, A007606, A007607, A014105, A014132, A162917, A161983, A317303. See also A317306.
Cf. A000203, A005843, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A237271, A239660, A239931, A239932, A239933, A239934, A244050, A245092, A249351, A262626.

cp's OEIS Frontend

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

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A007606 Take 1, skip 2, take 3, etc.

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A100035 a(n+1) occurs not earlier as a neighbor of terms = a(n): either it is the greatest number < a(n) or, if no such number exists, the smallest number > a(n); a(1) = 1.

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A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

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A100036 a(n) = smallest m such that A100035(m) = n.

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A134082 Triangle read by rows, (n-1) zeros followed by (2n, 1).

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A177342 a(n) = (4*n^3-3*n^2+5*n-3)/3.

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A174723 a(n) = n*(4*n^2 - 3*n + 5)/6.

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

A177342 a(n) = (4n^3-3n^2+5*n-3)/3.

A174723 a(n) = n(4n^2 - 3*n + 5)/6.