A004768 -id:A004768 - cp's OEIS frontend

A017077 a(n) = 8*n + 1.

1, 9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 97, 105, 113, 121, 129, 137, 145, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 241, 249, 257, 265, 273, 281, 289, 297, 305, 313, 321, 329, 337, 345, 353, 361, 369, 377, 385, 393, 401, 409, 417, 425, 433

Offset: 0

N. J. A. Sloane

Cf. A007519 (primes), subsequence of A047522.
a(n-1), n >= 1, gives the first column of the triangle A238475 related to the Collatz problem. - Wolfdieter Lang, Mar 12 2014
First differences of A054552. - Wesley Ivan Hurt, Jul 08 2014
An odd number is congruent to a perfect square modulo every power of 2 iff it is in this sequence. Sketch of proof: Suppose the modulus is 2^k with k at least three and note that the only odd quadratic residue (mod 8) is 1. By application of difference of squares and the fact that gcd(x-y,x+y)=2 we can show that for odd x,y, we have x^2 and y^2 congruent mod 2^k iff x is congruent to one of y, 2^(k-1)-y, 2^(k-1)+y, 2^k-y. Now when we "lift" to (mod 2^(k+1)) we see that the degeneracy between a^2 and (2^(k-1)-a)^2 "breaks" to give a^2 and a^2-2^ka+2^(2k-2). Since a is odd, the latter is congruent to a^2+2^k (mod 2^(k+1)). Hence we can form every binary number that ends with '001' by starting modulo 8 and "lifting" while adding digits as necessary. But this sequence is exactly the set of binary numbers ending in '001', so our claim is proved. - Rafay A. Ashary, Oct 23 2016
For n > 3, also the number of (not necessarily maximal) cliques in the n-antiprism graph. - Eric W. Weisstein, Nov 29 2017
Bisection of A016813. - L. Edson Jeffery, Apr 26 2022

			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 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 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       o
--------------------------------------------------------------
.  1       9          17              25                  33
- _Bruno Berselli_, Feb 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Antiprism Graph.
Eric Weisstein's World of Mathematics, Clique.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002189 (subsequence), A004768, A007519, A010731 (first differences), A016813, A047522, A054552.
Column 1 of A093565. Column 5 of triangle A130154. Second leftmost column of triangle A281334.
Row 1 of the arrays A081582, A238475, A371095, and A371096.
Row 2 of A257852.
Apart from the initial term, row sums of triangle A278480.

a017077 = (+ 1) . (* 8)
a017077_list = [1, 9 ..]  -- Reinhard Zumkeller, Dec 28 2012

I:=[1,9]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..60]]; // Vincenzo Librandi, Mar 14 2014

[8*n+1 : n in [0..50]]; // Wesley Ivan Hurt, Jul 08 2014

A017077:=n->8*n+1: seq(A017077(n), n=0..50); # Wesley Ivan Hurt, Jul 08 2014

Table[8 n + 1, {n, 0, 6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
CoefficientList[Series[(1 + 7 x)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Mar 14 2014 *)
8 Range[0, 50] + 1 (* Wesley Ivan Hurt, Jul 08 2014 *)
LinearRecurrence[{2, -1}, {9, 17}, {0, 20}] (* Eric W. Weisstein, Nov 29 2017 *)

a(n)=8*n+1 \\ Charles R Greathouse IV, Jul 10 2016

G.f.: (1+7*x)/(1-x)^2.
a(n+1) = A004768(n). - R. J. Mathar, May 28 2008
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Mar 14 2014
E.g.f.: exp(x)*(1 + 8*x). - Stefano Spezia, May 13 2021
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = a(n-1) + 8 with a(0)=1.
a(n) = A016813(2*n). (End)

A278480 Number of neighbors of the n-th term in a full right triangle read by rows.

Original entry on oeis.org

2, 4, 5, 5, 7, 5, 5, 8, 7, 5, 5, 8, 8, 7, 5, 5, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5, 5, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 5

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the first column and the first two diagonals, the rest of the elements are 8's.

For the same idea but for an isosceles triangle see A278481; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as a right triangle begins:
2;
4, 5;
5, 7, 5;
5, 8, 7, 5;
5, 8, 8, 7, 5;
5, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 7, 5;
5, 8, 8, 8, 8, 8, 8, 8, 7, 5;
...

Row sums give 2 together with the elements > 1 of A017077.
Also, row sums give 2 together with A004768.
Cf. A010722, A010731, A278317, A278481, A278545.

A138393 Numbers of form 8k+1 which are not squares.

Original entry on oeis.org

17, 33, 41, 57, 65, 73, 89, 97, 105, 113, 129, 137, 145, 153, 161, 177, 185, 193, 201, 209, 217, 233, 241, 249, 257, 265, 273, 281, 297, 305, 313, 321, 329, 337, 345, 353, 369, 377, 385, 393, 401, 409, 417, 425, 433, 449, 457, 465, 473, 481, 489, 497, 505

Offset: 1

Artur Jasinski, Mar 18 2008

nonn

Cf. A004768, A017077.

a = {}; Do[If[Sqrt[8k + 1] == Floor[Sqrt[8k + 1]],[null],AppendTo[a, 8k + 1]], {k, 0, 100}]; a
Select[8*Range[80]+1,!IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, May 03 2018 *)

A146302 a(n) = (8n+5)(8*n+9).

Original entry on oeis.org

45, 221, 525, 957, 1517, 2205, 3021, 3965, 5037, 6237, 7565, 9021, 10605, 12317, 14157, 16125, 18221, 20445, 22797, 25277, 27885, 30621, 33485, 36477, 39597, 42845, 46221, 49725, 53357, 57117, 61005, 65021, 69165, 73437, 77837, 82365

Offset: 0

Miklos Kristof, Oct 29 2008

From Miklos Kristof, Nov 03 2008: (Start)
f(y) = y^4*(1 + y^4) = y^4 - y^8 + y^12 - y^16 + y^20 - y^24 + ...
Integral_{y} f(y) dy = y^5/5 - y^9/9 + y^13/13 - y^17/17 + y^21/21 - y^25/25 + ...
Integral_{y=0..1} f(y) dy = 1/5 - 1/9 + 1/13 - 1/17 + 1/21 - 1/25 + ...
                  = (9 - 5)/(5*9) + (17 - 13)/(13*17) + (25 - 21)/(21*25) + ...
                  = 4/(5*9) + 4/(13*17) + 4/(21*25) + ...
Integral_{y=0..1} f(y) dy = Sum_{m>=0} 4/((8*m+5)*(8*m+9))
                  = -(1/8)*sqrt(2)*Pi + 1 - (1/4)*sqrt(2)*log(1+sqrt(2))
                  = 0.13302701266008896241... (End)

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

seq((8*m+5)*(8*m+9),m=0..40); # Miklos Kristof, Nov 03 2008

Table[(8n+5)(8n+9),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{45,221,525},40] (* Harvey P. Dale, Oct 10 2015 *)

a(n)=(8*n+5)*(8*n+9) \\ Charles R Greathouse IV, Jun 17 2017

G.f: (45 + 86*x - 3*x^2)/(1-x)^3.
E.g.f.: (45 + 176*x + 64*x^2)*exp(x).
a(n) = A004770(n) * A004768(n). - Reinhard Zumkeller, Oct 30 2008

cp's OEIS Frontend

A017077 a(n) = 8*n + 1.

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A278480 Number of neighbors of the n-th term in a full right triangle read by rows.

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A138393 Numbers of form 8k+1 which are not squares.

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A146302 a(n) = (8*n+5)*(8*n+9).

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Maple

Mathematica

PARI

Formula

A146302 a(n) = (8n+5)(8*n+9).