A249356 -id:A249356 - cp's OEIS frontend

A002265 Nonnegative integers repeated 4 times.

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

N. J. A. Sloane

For n>=1 and i=sqrt(-1) let F(n) the n X n matrix of the Discrete Fourier Transform (DFT) whose element (j,k) equals exp(-2*Pi*i*(j-1)*(k-1)/n)/sqrt(n). The multiplicities of the four eigenvalues 1, i, -1, -i of F(n) are a(n+4), a(n-1), a(n+2), a(n+1), hence a(n+4) + a(n-1) + a(n+2) + a(n+1) = n for n>=1. E.g., the multiplicities of the eigenvalues 1, i, -1, -i of the DFT-matrix F(4) are a(8)=2, a(3)=0, a(6)=1, a(5)=1, summing up to 4. - Franz Vrabec, Jan 21 2005
Complement of A010873, since A010873(n)+4*a(n)=n. - Hieronymus Fischer, Jun 01 2007
For even values of n, a(n) gives the number of partitions of n into exactly two parts with both parts even. - Wesley Ivan Hurt, Feb 06 2013
a(n-4) counts number of partitions of (n) into parts 1 and 4. For example a(11) = 3 with partitions (44111), (41111111), (11111111111). - David Neil McGrath, Dec 04 2014
a(n-4) counts walks (closed) on the graph G(1-vertex; 1-loop, 4-loop) where order of loops is unimportant. - David Neil McGrath, Dec 04 2014
Number of partitions of n into 4 parts whose smallest 3 parts are equal. - Wesley Ivan Hurt, Jan 17 2021

V. Cizek, Discrete Fourier Transforms and their Applications, Adam Hilger, Bristol 1986, p. 61.

Todd Silvestri, Table of n, a(n) for n = 0..999
J. H. McClellan and T. W. Parks, Eigenvalue and Eigenvector Decomposition of the Discrete Fourier Transform, IEEE Trans. Audio and Electroacoust., Vol. AU-20, No. 1, March 1972, pp. 66-74.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A008615, A008621, A249356.
Zero followed by partial sums of A011765.
Partial sums: A130519. Other related sequences: A004526, A010872, A010873, A010874.
Third row of A180969.

[Floor(n/4): n in [0..80]]; // Vincenzo Librandi, Oct 28 2014

A002265:=n->floor(n/4); seq(A002265(n), n=0..100); # Wesley Ivan Hurt, Dec 10 2013

Table[Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 10 2013 *)
Table[{n,n,n,n},{n,0,20}]//Flatten (* Harvey P. Dale, Aug 08 2020 *)

a(n)=n\4 \\ Charles R Greathouse IV, Dec 10 2013

def A002265(n): return n>>2 # Chai Wah Wu, Jul 27 2022

[floor(n/4) for n in range(0,84)] # Zerinvary Lajos, Dec 02 2009

a(n) = floor(n/4), n>=0;
G.f.: (x^4)/((1-x)*(1-x^4)).
a(n) = (2*n-(3-(-1)^n-2*(-1)^floor(n/2)))/8; also a(n) = (2*n-(3-(-1)^n-2*sin(Pi/4*(2*n+1+(-1)^n))))/8 = (n-A010873(n))/4. - Hieronymus Fischer, May 29 2007
a(n) = (1/4)*(n-(3-(-1)^n-2*(-1)^((2*n-1+(-1)^n)/4))/2). - Hieronymus Fischer, Jul 04 2007
a(n) = floor((n^4-1)/4*n^3) (n>=1); a(n) = floor((n^4-n^3)/(4*n^3-3*n^2)) (n>=1). - Mohammad K. Azarian, Nov 08 2007 and Aug 01 2009
For n>=4, a(n) = floor( log_4( 4^a(n-1) + 4^a(n-2) + 4^a(n-3) + 4^a(n-4) ) ). - Vladimir Shevelev, Jun 22 2010
a(n) = A180969(2,n). - Adriano Caroli, Nov 26 2010
a(n) = A173562(n)-A000290(n); a(n+2) = A035608(n)-A173562(n). - Reinhard Zumkeller, Feb 21 2010
a(n+1) = A140201(n) - A057353(n+1). - Reinhard Zumkeller, Feb 26 2011
a(n) = ceiling((n-3)/4), n >= 0. - Wesley Ivan Hurt, Jun 01 2013
a(n) = (2*n + (-1)^n + 2*sin(Pi*n/2) + 2*cos(Pi*n/2) - 3)/8. - Todd Silvestri, Oct 27 2014
E.g.f.: (x/4 - 3/8)*exp(x) + exp(-x)/8 + (sin(x)+cos(x))/4. - Robert Israel, Oct 30 2014
a(n) = a(n-1) + a(n-4) - a(n-5) with initial values a(3)=0, a(4)=1, a(5)=1, a(6)=1, a(7)=1. - David Neil McGrath, Dec 04 2014
a(n) = A004526(A004526(n)). - Bruno Berselli, Jul 01 2016
From Guenther Schrack, May 03 2019: (Start)
a(n) = (2*n - 3 + (-1)^n + 2*(-1)^(n*(n-1)/2))/8.
a(n) = a(n-4) + 1, a(k)=0 for k=0,1,2,3, for n > 3. (End)

A158057 First differences of A051870: 16*n + 1.

Original entry on oeis.org

1, 17, 33, 49, 65, 81, 97, 113, 129, 145, 161, 177, 193, 209, 225, 241, 257, 273, 289, 305, 321, 337, 353, 369, 385, 401, 417, 433, 449, 465, 481, 497, 513, 529, 545, 561, 577, 593, 609, 625, 641, 657, 673, 689, 705, 721, 737, 753, 769, 785, 801, 817, 833, 849

Offset: 0

Vincenzo Librandi, Mar 12 2009

The identity (16*n+1)^2 - (16*n^2+2*n)*(4)^2 = 1 can be written as a(n+1)^2 - A158056(n)*(4)^2 = 1. - Vincenzo Librandi, Feb 09 2012
This sequence gives the 18-gonal (or octadecagonal) gnomonic numbers. Name suggested by Todd Silvestri, Nov 22 2014
All elements are odd and contains subsequence A249356. - Todd Silvestri, Nov 22 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
OEIS Wiki, Gnomonic numbers.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016813, A017077, A051870, A158056, A249356.

List([0..60], n-> 16*n+1); # G. C. Greubel, Sep 18 2019

I:=[1, 17]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]];

seq(16*n+1, n=0..60); # G. C. Greubel, Sep 18 2019

LinearRecurrence[{2,-1}, {1,17}, 60]
Table[16*n+1,{n,0,60}] (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
a[n_Integer/;n>=0]:=16 n+1 (* Todd Silvestri, Nov 22 2014 *)
CoefficientList[Series[(1+15x)/(1-x)^2, {x,0,60}], x] (* Vincenzo Librandi, Nov 23 2014 *)

a(n)=n<<4+1 \\ Charles R Greathouse IV, Dec 23 2011

[16*n+1 for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 16*n + 1.
a(n) = 2*a(n-1) - a(n-2), a(0) = 1, a(1) = 17.
G.f.: (1+15*x)/(1-x)^2. - Vincenzo Librandi, Nov 23 2014
E.g.f.: (1 + 16*x)*exp(x). - G. C. Greubel, Sep 18 2019 [corrected by Elmo R. Oliveira, Apr 12 2025]
a(n) = A017077(2*n) = A016813(4*n). - Elmo R. Oliveira, Apr 12 2025

Name clarified and offset changed by Todd Silvestri, Nov 22 2014
Edited by Vincenzo Librandi Nov 23 2014
Edited: Offset changed to 0 according to the
Todd Silvestri proposal. Name changed. - Wolfdieter Lang, Nov 29 2014

A200975 Numbers on the diagonals in Ulam's spiral.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 17, 21, 25, 31, 37, 43, 49, 57, 65, 73, 81, 91, 101, 111, 121, 133, 145, 157, 169, 183, 197, 211, 225, 241, 257, 273, 289, 307, 325, 343, 361, 381, 401, 421, 441, 463, 485, 507, 529, 553, 577, 601, 625, 651, 677, 703, 729, 757, 785, 813, 841, 871, 901

Offset: 1

Ismael Bouya, Nov 25 2011

All entries are odd.
From Bob Selcoe, Oct 22 2014: (Start)
The following hold:
1. a(n) = (2k + 1)^2  when n = 4k + 1, k >= 0
2. a(n) = 4*k^2 + 1   when n = 4k - 1, k > 0
3  a(n) = k^2 + k + 1 when n = 2k, k > 0.
Conjecture 1: there must be at least one prime in [a(n), a(n+1)] inclusive.
Conjecture 2: generally, when j is in [(2m-1)^2+1, (2m+1)^2] inclusive, there must be at least one prime in [j-2m-1, j] inclusive. If true, then Conjecture 1 is true; also suggests A248623, A248835 and Oppermann's conjecture (see A002620) likely are true. (End)

			The numbers between ** are in this sequence.
.
  *21*--22---23---24--*25*
    |
    |
   20   *7*---8---*9*--10
    |    |              |
    |    |              |
   19    6   *1*---2   11
    |    |         |    |
    |    |         |    |
   18   *5*---4---*3*  12
    |                   |
    |                   |
  *17*--16---15---14--*13*

Todd Silvestri, Table of n, a(n) for n = 1..1000
Project Euler, Problem 28: Number spiral diagonals
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A016754, A054554, A053755, and A054569 interleaved, A002620,

Cf. A121658 (complementary)

Sort@ Flatten@ Table[4n^2 + (2j - 4)n + 1, {j, 0, 3}, {n, 16}] (* Robert G. Wilson v, Jul 10 2014 *)
a[n_Integer/;n>0]:=Quotient[2 n (n+2)+(-1)^n-4 Mod[n^2 (3 n+2),4,-1]+7,8] (* Todd Silvestri, Oct 25 2014 *)

al(n)=local(r=vector(n),j);r[1]=1;for(k=2,n,r[k]=r[k-1]+(k+2)\4*2);r /* Franklin T. Adams-Watters, Nov 26 2011 */

# prints all numbers on the diagonals of a sq*sq spiral
sq = 5
d = 1
while 2*d - 1 < sq:
    print(4*d*d - 4*d +1)
    print(4*d*d - 4*d +1 + 1* 2* d)
    print(4*d*d - 4*d +1 + 2* 2* d)
    print(4*d*d - 4*d +1 + 3* 2* d)
    d += 1
print(sq*sq)

a(4n) = 4n^2 + 2n + 1; a(4n+1) = 4n^2 + 4n + 1; a(4n+2) = 4n^2 + 6n + 3; a(4n+3) = 4n^2 + 8n + 5. [corrected by James Mitchell, Dec 31 2017]
G.f.: -x*(1+x+x^5-x^4) / ( (1+x)*(x^2+1)*(x-1)^3 ). - R. J. Mathar, Nov 28 2011
a(n) = (2*n*(n+2)+(-1)^n-4*sin((Pi*n)/2)+7)/8 = (A249356(n)+7)/8. - Todd Silvestri, Oct 25 2014
a(n) = floor_(n*(n+2)/4) + floor_(n(mod 4)/3) + 1. - Bob Selcoe, Oct 27 2014

Edited with more terms by Franklin T. Adams-Watters, Nov 26 2011

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A002265 Nonnegative integers repeated 4 times.

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A158057 First differences of A051870: 16*n + 1.

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A200975 Numbers on the diagonals in Ulam's spiral.

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Keywords

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Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions