A004766 -id:A004766 - cp's OEIS frontend

A016813 a(n) = 4*n + 1.

1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229, 233, 237

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 23 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 64 ).
Numbers k such that k and (k+1) have the same binary digital sum. - Benoit Cloitre, Jun 05 2002
Numbers k such that (1 + sqrt(k))/2 is an algebraic integer. - Alonso del Arte, Jun 04 2012
Numbers k such that 2 is the only prime p that satisfies the relationship p XOR k = p + k. - Brad Clardy, Jul 22 2012
This may also be interpreted as the array T(n,k) = A001844(n+k) + A008586(k) read by antidiagonals:
    1,   9,  21,  37,  57,  81, ...
    5,  17,  33,  53,  77, 105, ...
   13,  29,  49,  73, 101, 133, ...
   25,  45,  69,  97, 129, 165, ...
   41,  65,  93, 125, 161, 201, ...
   61,  89, 121, 157, 197, 241, ...
   ...
- R. J. Mathar, Jul 10 2013
With leading term 2 instead of 1, 1/a(n) is the largest tolerance of form 1/k, where k is a positive integer, so that the nearest integer to (n - 1/k)^2 and to (n + 1/k)^2 is n^2. In other words, if interval arithmetic is used to square [n - 1/k, n + 1/k], every value in the resulting interval of length 4n/k rounds to n^2 if and only if k >= a(n). - Rick L. Shepherd, Jan 20 2014
Odd numbers for which the number of prime factors congruent to 3 (mod 4) is even. - Daniel Forgues, Sep 20 2014
For the Collatz conjecture, we identify two types of odd numbers. This sequence contains all the descenders: where (3*a(n) + 1) / 2 is even and requires additional divisions by 2. See A004767 for the ascenders. - Fred Daniel Kline, Nov 29 2014 [corrected by Jaroslav Krizek, Jul 29 2016]
a(n-1), n >= 1, is also the complex dimension of the manifold M(S), the set of all conjugacy classes of irreducible representations of the fundamental group pi_1(X,x_0) of rank 2, where S = {a_1, ..., a_{n}, a_{n+1} = oo}, a subset of P^1 = C U {oo}, X = X(S) = P^1 \ S, and x_0 a base point in X. See the Iwasaki et al. reference, Proposition 2.1.4. p. 150. - Wolfdieter Lang, Apr 22 2016
For n > 3, also the number of (not necessarily maximal) cliques in the n-sunlet graph. - Eric W. Weisstein, Nov 29 2017
For integers k with absolute value in A047202, also exponents of the powers of k having the same unit digit of k in base 10. - Stefano Spezia, Feb 23 2021
Starting with a(1) = 5, numbers ending with 01 in base 2. - John Keith, May 09 2022

			From _Leo Tavares_, Jul 02 2021: (Start)
Illustration of initial terms:
                                        o
                        o               o
            o           o               o
    o     o o o     o o o o o     o o o o o o o
            o           o               o
                        o               o
                                        o
(End)

K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 150.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Colin Defant and Noah Kravitz, Loops and Regions in Hitomezashi Patterns, arXiv:2201.03461 [math.CO], 2022. Theorem 1.3.
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 5, 8, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.
Tanya Khovanova, Recursive Sequences
Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
William A. Stein's The Modular Forms Database, PARI-readable dimension tables for Gamma_0(N)
Eric Weisstein's World of Mathematics, Clique
Eric Weisstein's World of Mathematics, Hilbert Number
Eric Weisstein's World of Mathematics, Sunlet Graph
Wikipedia, Interval arithmetic
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A042963 and of A079523.
a(n) = A093561(n+1, 1), (4, 1)-Pascal column.
Cf. A016921, A017281, A017533, A047202, A158057, A161705, A161709, A161714, A128470. - Reinhard Zumkeller, Jun 17 2009
Cf. A004772 (complement).
Cf. A017557.

List([0..70],n->4*n+1); # Muniru A Asiru, Aug 08 2018

a016813 = (+ 1) . (* 4)
a016813_list = [1, 5 ..]  -- Reinhard Zumkeller, Feb 14 2012

Magma
```
[n: n in [1..250 by 4]];
```

seq(4*k+1, k=0..100); # Wesley Ivan Hurt, Sep 28 2013

Range[1, 237, 4] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[4 n + 1, {n, 0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
4 Range[0, 20] + 1 (* Eric W. Weisstein, Nov 29 2017 *)
LinearRecurrence[{2, -1}, {5, 9}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)
CoefficientList[Series[(1 + 3 x)/(-1 + x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=4*n+1 \\ Charles R Greathouse IV, Mar 22 2013

x='x+O('x^100); Vec((1+3*x)/(1-x)^2) \\ Altug Alkan, Oct 22 2015

(0 to 59).map(4 *  + 1) // _Alonso del Arte, Aug 08 2018

a(n) = A005408(2*n).
Sum_{n>=0} (-1)^n/a(n) = (1/(4*sqrt(2)))*(Pi+2*log(sqrt(2)+1)) = A181048 [Jolley]. - Benoit Cloitre, Apr 05 2002 [corrected by Amiram Eldar, Jul 30 2023]
G.f.: (1+3*x)/(1-x)^2. - Paul Barry, Feb 27 2003 [corrected for offset 0 by Wolfdieter Lang, Oct 03 2014]
(1 + 5*x + 9*x^2 + 13*x^3 + ...) = (1 + 2*x + 3*x^2 + ...) / (1 - 3*x + 9*x^2 - 27*x^3 + ...). - Gary W. Adamson, Jul 03 2003
a(n) = A001969(n) + A000069(n). - Philippe Deléham, Feb 04 2004
a(n) = A004766(n-1). - R. J. Mathar, Oct 26 2008
a(n) = 2*a(n-1) - a(n-2); a(0)=1, a(1)=5. a(n) = 4 + a(n-1). - Philippe Deléham, Nov 03 2008
A056753(a(n)) = 3. - Reinhard Zumkeller, Aug 23 2009
A179821(a(n)) = a(A179821(n)). - Reinhard Zumkeller, Jul 31 2010
a(n) = 8*n - 2 - a(n-1) for n > 0, a(0) = 1. - Vincenzo Librandi, Nov 20 2010
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as a(n)^2 - A002943(n)*2^2 = 1. - Vincenzo Librandi, Mar 11 2009 - Nov 25 2012
A089911(6*a(n)) = 8. - Reinhard Zumkeller, Jul 05 2013
a(n) = A004767(n) - 2. - Jean-Bernard François, Sep 27 2013
a(n) = A058281(3n+1). - Eli Jaffe, Jun 07 2016
From Ilya Gutkovskiy, Jul 29 2016: (Start)
E.g.f.: (1 + 4*x)*exp(x).
a(n) = Sum_{k = 0..n} A123932(k).
a(A005098(k)) = x^2 + y^2.
Inverse binomial transform of A014480. (End)
Dirichlet g.f.: 4*Zeta(-1 + s) + Zeta(s). - Stefano Spezia, Nov 02 2018

A228218 T(n,k)=Number of second differences of arrays of length n+2 of numbers in 0..k.

Original entry on oeis.org

5, 9, 15, 13, 49, 31, 17, 103, 199, 63, 21, 177, 625, 665, 127, 25, 271, 1429, 3151, 2059, 255, 29, 385, 2731, 9705, 14053, 6305, 511, 33, 519, 4651, 23351, 58141, 58975, 19171, 1023, 37, 673, 7309, 47953, 176851, 320481, 242461, 58025, 2047, 41, 847, 10825

Offset: 1

R. H. Hardin Aug 16 2013

Table starts
....5......9......13.......17........21.........25.........29..........33
...15.....49.....103......177.......271........385........519.........673
...31....199.....625.....1429......2731.......4651.......7309.......10825
...63....665....3151.....9705.....23351......47953......88215......149681
..127...2059...14053....58141....176851.....439927.....951049.....1854553
..255...6305...58975...320481...1225631....3693505....9399615....21108545
..511..19171..242461..1688101...8006491...29066311...86929081...224817481
.1023..58025..989527..8717049..50556551..219071473..766106895..2276277137
.2047.175099.4017157.44633821.313882531.1609259287.6537612649.22222129177

			Some solutions for n=4 k=4
..4...-5....3...-3....6...-4...-3...-5...-8....1....5...-2....4...-3...-1...-6
.-6....7....1...-2...-6....1....4....5....6...-5...-5....1....0....0...-3....4
..2...-3...-1....5....4....2....0...-2...-2....5....7....1....0...-3....3....1
.-2...-1...-2...-1...-6....0...-4....4....1....2...-7....1....4....6...-4...-6

R. H. Hardin, Table of n, a(n) for n = 1..310

Row 1 is A004766. A228212 (k=2), A228213 (k=3), A228213 (k=4), A228215 (k=5).

Empirical for column k:
k=1: a(n) = 3*a(n-1) -2*a(n-2) for n>3
k=2: a(n) = 5*a(n-1) -6*a(n-2) for n>5
k=3: a(n) = 7*a(n-1) -12*a(n-2) for n>7
k=4: a(n) = 9*a(n-1) -20*a(n-2) for n>9
k=5: a(n) = 11*a(n-1) -30*a(n-2) for n>11
k=6: a(n) = 13*a(n-1) -42*a(n-2) for n>13
k=7: a(n) = 15*a(n-1) -56*a(n-2) for n>15
Empirical for row n:
n=1: a(n) = 4*n + 1
n=2: a(n) = 10*n^2 + 4*n + 1
n=3: a(n) = 20*n^3 + 9*n^2 + 1*n + 1
n=4: a(n) = 35*n^4 + 14*n^3 - 17*n^2 + 30*n + 1
n=5: a(n) = 56*n^5 + 14*n^4 - 108*n^3 + 289*n^2 - 125*n + 1
n=6: a(n) = 84*n^6 - 402*n^4 + 1656*n^3 - 1860*n^2 + 776*n + 1
n=7: a(n) = 120*n^7 - 42*n^6 - 1158*n^5 + 6945*n^4 - 13980*n^3 + 13512*n^2 - 4887*n + 1

A058281 Continued fraction for square root of e.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 9, 1, 1, 13, 1, 1, 17, 1, 1, 21, 1, 1, 25, 1, 1, 29, 1, 1, 33, 1, 1, 37, 1, 1, 41, 1, 1, 45, 1, 1, 49, 1, 1, 53, 1, 1, 57, 1, 1, 61, 1, 1, 65, 1, 1, 69, 1, 1, 73, 1, 1, 77, 1, 1, 81, 1, 1, 85, 1, 1, 89, 1, 1, 93, 1, 1, 97, 1, 1, 101, 1, 1, 105, 1, 1, 109, 1, 1, 113, 1, 1

Offset: 0

Robert G. Wilson v, Dec 07 2000

			sqrt(e) = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(5 + ...)))). - _Harry J. Smith_, May 01 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
Keith Matthews, Finding the continued fraction of e^(l/m).
Thomas J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
Gang Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A004766, A016813.

Cf. A019774 (decimal expansion of sqrt(e)).

ContinuedFraction[ Sqrt[E], 100]
LinearRecurrence[{0,0,2,0,0,-1},{1,1,1,1,5,1},100] (* Harvey P. Dale, Aug 05 2025 *)

PARI
```
contfrac(sqrt(exp(1)))
```

{ allocatemem(932245000); default(realprecision, 60000); x=contfrac(sqrt(exp(1))); for (n=1, 20001, write("b058281.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 01 2009

a(3k+1) = 4k+1, a(i) = 1 otherwise.
G.f.: -(x^2-x+1)*(x^3-2*x^2-2*x-1) / ((x-1)^2*(x^2+x+1)^2). - Colin Barker, Jun 24 2013
E.g.f.: exp(-x/2)*(exp(3*x/2)*(5 + 4*x) + (4 + 8*x)*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, May 05 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*log(sqrt(2)+1)) / (4*sqrt(2)). - Amiram Eldar, May 03 2025

More terms from Jason Earls, Jul 10 2001

A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n.

Original entry on oeis.org

1, 7, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225

Offset: 0

Robert Price, Jun 12 2013

nonn

Apparently a(n) = A004766(n) for n>=2. - R. J. Mathar, May 26 2024

Robert Price, Table of n, a(n) for n = 0..100

Cf. A004766, A226357, A226359, A222947.

f[n_] := Length[Complement[Union[Flatten[Table[If[Abs[i*j*k] ≤ n, {i + j + k}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]], {Null}]]; Table[f[n], {n, 0, 100}]

A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

Original entry on oeis.org

1, 5, 3, 9, 7, 13, 11, 17, 15, 21, 19, 25, 23, 29, 27, 33, 31, 37, 35, 41, 39, 45, 43, 49, 47, 53, 51, 57, 55, 61, 59, 65, 63, 69, 67, 73, 71, 77, 75, 81, 79, 85, 83, 89, 87, 93, 91, 97, 95, 101, 99, 105, 103, 109, 107, 113, 111, 117, 115, 121, 119

Offset: 0

Paul Barry, Jul 26 2003

Partial sums are A086955.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004442, A014681, A065190.

[1] cat [2*n+1-2*(-1)^n: n in [1..70]]; // Vincenzo Librandi, Jun 21 2017

Join[{1}, LinearRecurrence[{1, 1, -1}, {5, 3, 9}, 60]] (* Vincenzo Librandi, Jun 21 2017 *)

Vec((1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2) + O(x^100)) \\ Michel Marcus, Jun 21 2017

G.f.: (1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2).
a(n) = n + abs(2 - (n + 1)*(-1)^n). - Lechoslaw Ratajczak, Dec 09 2016
a(n) = 2*A065190(n+1)-1 and a(n) = 2*A014681(n)+1. - Michel Marcus, Dec 10 2016
From Guenther Schrack, Jun 09 2017: (Start)
a(n) = 2*n + 1 - 2*(-1)^n for n > 0.
a(n) = 2*n + 1 - 2*cos(n*Pi) for n > 0.
a(n) = 4*n - a(n-1) for n > 1.
Linear recurrence: a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
First differences: 2 - 4*(-1)^n for n > 1; -(-1)^n*A010696(n) for n > 1.
a(n) = A065164(n+1) + n for n > 0.
a(A014681(n)) = A005408(n) for n >= 0.
a(A005408(A014681(n)) for n >= 0.
a(n) = A005408(A103889(n)) for n >= 0.
A103889(a(n)) = 2*A065190(n+1) for n >= 0.
a(2*n-1) = A004766(n) for n > 0.
a(2*n+2) = A004767(n) for n >= 0. (End)

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

Original entry on oeis.org

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011
Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - R. J. Mathar, Dec 04 2011
Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

			5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A177712, A136724, A177713.
Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
Contains A004766, A017137 and nonzero terms of A008588.
Disjoint from A002145.
Subsequence of A138591.

f:= proc(n) local r,k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

A253026 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 1 and every value within 1 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 5, 5, 3, 4, 9, 5, 9, 4, 5, 13, 21, 21, 13, 5, 6, 17, 37, 21, 37, 17, 6, 7, 21, 53, 85, 85, 53, 21, 7, 8, 25, 69, 149, 85, 149, 69, 25, 8, 9, 29, 85, 213, 341, 341, 213, 85, 29, 9, 10, 33, 101, 277, 597, 341, 597, 277, 101, 33, 10, 11, 37, 117, 341, 853, 1365

Offset: 1

R. H. Hardin, Dec 26 2014

			Table starts:
.0..1...2...3....4....5.....6.....7.....8.....9.....10.....11.....12......13
.1..1...5...9...13...17....21....25....29....33.....37.....41.....45......49
.2..5...5..21...37...53....69....85...101...117....133....149....165.....181
.3..9..21..21...85..149...213...277...341...405....469....533....597.....661
.4.13..37..85...85..341...597...853..1109..1365...1621...1877...2133....2389
.5.17..53.149..341..341..1365..2389..3413..4437...5461...6485...7509....8533
.6.21..69.213..597.1365..1365..5461..9557.13653..17749..21845..25941...30037
.7.25..85.277..853.2389..5461..5461.21845.38229..54613..70997..87381..103765
.8.29.101.341.1109.3413..9557.21845.21845.87381.152917.218453.283989..349525
.9.33.117.405.1365.4437.13653.38229.87381.87381.349525.611669.873813.1135957
Some solutions for n=4 and k=4:
..0..1..2..2....0..1..1..2....0..0..1..2....0..1..2..2....0..1..1..2
..1..1..2..2....0..1..1..2....0..0..1..2....1..1..2..2....0..1..2..2
..2..2..2..2....1..1..1..2....1..1..1..2....1..2..2..2....1..1..2..2
..2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2....2..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..10018
Aaron Barnoff, Curtis Bright, and Jeffrey Shallit, Using finite automata to compute the base-b representation of the golden ratio and other quadratic irrationals, arXiv:2405.02727 [cs.FL], 2024. See p. 8.
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 21.
Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023, Hardinian Arrays, El. J. Combinat. 31 (2) (2024) #P2.9

Column 1 is A000027(n-1).
Column 2 is A004766(n-2).
Diagonal is A002450(n-1).

T(n,k) = (n-k)*4^(k-1) + (4^(k-1)-1)/3 for all n>=k>=1 (Thm. 2 in the paper of Dougerty-Bliss and Kauers cited above). - Manuel Kauers, Sep 06 2023

T(n,k) = T(k,n) for all n,k.

A294774 a(n) = 2n^2 + 2n + 5.

Original entry on oeis.org

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset: 0

Bruno Berselli, Nov 08 2017

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

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

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.
Second column of Mathar's array in A016813 (Comments section).
Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).
Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Table[2n^2+2n+5,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{5,9,17},50] (* Harvey P. Dale, Sep 18 2023 *)

Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.
E.g.f.: (5 + 4*x + 2*x^2)*exp(x).
a(n) = a(-1-n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).
n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.
a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

cp's OEIS Frontend

A016813 a(n) = 4*n + 1.

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A228218 T(n,k)=Number of second differences of arrays of length n+2 of numbers in 0..k.

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A058281 Continued fraction for square root of e.

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A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |i*j*k| <= n.

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A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

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A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

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A253026 T(n,k) = Number of n X k nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 1 and every value within 1 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

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A222945 Number of distinct sums i+j+k with |i|, |j|, |k|, |ijk| <= n.

A294774 a(n) = 2n^2 + 2n + 5.