A010731 -id:A010731 - 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)

A010709 Constant sequence: the all 4's sequence.

Original entry on oeis.org

4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 25 2010: (Start)
Continued fraction expansion of 2+sqrt(5).
Decimal expansion of 4/9.
Inverse binomial transform of A020707. (End)

Tom Edgar, Proof without words: sum of powers of 4/9, Math. Mag. 89 no. 3 (2016) 191.
Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1012
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

From Klaus Brockhaus, May 25 2010: (Start)
Equals 4*A000012, 2*A007395, A010731/2, A010855/4, A010871/8.
Cf. A098317 (decimal expansion of 2+sqrt(5)), A020707 (2^(n+2)). (End)

makelist(4, n, 0, 30); /* Martin Ettl, Nov 09 2012 */

a(n) = 4 \\ Charles R Greathouse IV, Apr 07 2012

def A010709(n): return 4 # Chai Wah Wu, Mar 22 2023

From Klaus Brockhaus, May 25 2010: (Start)
a(n) = 4.
G.f.: 4/(1-x). (End)
E.g.f.: 4*e^x. - Vincenzo Librandi, Jan 29 2012

A010689 Periodic sequence: Repeat 1, 8.

Original entry on oeis.org

1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1

Offset: 0

N. J. A. Sloane

Also the digital root of 8^n. Also the decimal expansion of 2/11 = 0.181818181818... - Cino Hilliard, Dec 31 2004
Interleaving of A000012 and A010731. - Klaus Brockhaus, Apr 02 2010
Continued fraction expansion of (2 + sqrt(6))/4. - Klaus Brockhaus, Apr 02 2010
Digital root of the powers of any number congruent to 8 mod 9. - Alonso del Arte, Jan 26 2014

			0.18181818181818181818181818181818181818181...

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000012 (all 1's sequence), A010731 (all 8's sequence), A174925 (decimal expansion of (2 + sqrt(6))/4). [Klaus Brockhaus, Apr 02 2010]
Cf. Digital roots of powers of c mod 9: c = 2, A153130; c = 4, A100402; c = 5, A070366; c = 7, A070403.
Cf. sequences listed in Comments section of A283393.
Cf. A010888.

&cat[ [1, 8]: n in [0..52] ]; // Klaus Brockhaus, Apr 02 2010

&cat [[1,8]^^60]; // Bruno Berselli, Mar 10 2017

Table[Mod[8^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)
PadRight[{},120,{1,8}] (* Harvey P. Dale, Jun 03 2015 *)

A010689(n):=if evenp(n) then 1 else 8$
makelist(A010689(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=1; if(n%2==1, 8, 1) \\ Felix Fröhlich, Aug 11 2014

[power_mod(8,n,9)for n in range(0,105)] # Zerinvary Lajos, Nov 27 2009

From Paul Barry, Sep 16 2004: (Start)
G.f.: (1 + 8*x)/((1 - x)*(1 + x)).
a(n) = (9 - 7*(-1)^n)/2.
a(n) = 8^(ceiling(n/2) - floor(n/2)).
a(n) = gcd((n-1)^3, (n+1)^3). (End)
E.g.f.: cosh(x) + 8*sinh(x). - Stefano Spezia, Feb 09 2025
a(n) = A010888(8*a(n-1)). - Stefano Spezia, Mar 20 2025

Definition edited and keywords cons, cofr added by Klaus Brockhaus, Apr 02 2010

A220521 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A220520.

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 8, 8, 4, 4, 8, 12, 16, 12, 16, 16, 4, 4, 8, 12, 16, 20, 24, 26, 24, 12, 20, 32, 40, 28, 32, 32, 4, 4, 8, 12, 16, 20, 24, 26, 24, 20, 32, 44, 64, 52, 48, 54, 40, 12, 20, 36, 48, 56, 64, 74, 76, 30, 44, 72, 88, 60, 64, 64, 4, 4, 8, 12

Offset: 1

Omar E. Pol, Dec 15 2012

From Omar E. Pol, Apr 26 2020: (Start)
The cellular automaton described in A220520 has word "ab", so the structure of this triangle is as follows:
a,b;
a,b;
a,b,a,b;
a,b,a,b,a,b,a,b;
a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b;
...
The row lengths are the terms of A011782 multiplied by 2, equaling the column 2 of the square array A296612: 2, 2, 4, 8, 16, ...
This arrangement has the property that the odd-indexed columns (a) contain numbers of the toothpicks of length 1, and the even-indexed columns (b) contain numbers of the D-toothpicks.
For further information about the "word" of a cellular automaton see A296612. (End)

			Written as an irregular triangle the sequence begins:
1,2;
4,4;
4,4,8,8;
4,4,8,12,16,12,16,16;
4,4,8,12,16,20,24,26,24,12,20,32,40,28,32,32;
4,4,8,12,16,20,24,26,24,20,32,44,64,52,48,54,40,12,20,...
Triangle reformatted by _Omar E. Pol_, Apr 26 2020

First differences of A220520.
First differs from A194441 at a(14).
Columns 1-3: A123932, A040002, A010731.
Cf. A139250, A139251, A194443, A220501, A220523, A296612.

0 removed and offset changed by Omar E. Pol, Apr 26 2020

A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.

Original entry on oeis.org

1, 5, 21, 57, 121, 221, 365, 561, 817, 1141, 1541, 2025, 2601, 3277, 4061, 4961, 5985, 7141, 8437, 9881, 11481, 13245, 15181, 17297, 19601, 22101, 24805, 27721, 30857, 34221, 37821, 41665, 45761, 50117, 54741, 59641, 64825, 70301, 76077, 82161, 88561, 95285, 102341, 109737, 117481, 125581

Offset: 0

Paul Curtz, Oct 14 2009

Atomic number of first transition metal of period 2n (n>3) or of the element after n-th alkaline earth metal. This can be calculated by finding the sum of the first n even squares plus 1. - Natan Arie Consigli, Jul 03 2016

JANET,Charles, La structure du Noyau de l'atome,consideree dans la Classification periodique,des elements chimiques,1927 (Novembre),N. 2,BEAUVAIS,67 pages,3 leaflets.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002492, A010731, A016742, A017113, A018227.

[(3+2*n+6*n^2+4*n^3)/3: n in [0..60]]; // G. C. Greubel, Jul 27 2024

Table[(3+2*n+6*n^2+4*n^3)/3, {n,0,60}] (* G. C. Greubel, May 15 2016 *)

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

[(3+2*n+6*n^2+4*n^3)//3 for n in range(61)] # G. C. Greubel, Jul 27 2024

a(n) - a(n-1) = 4*(n+1)^2 = A016742(n+1).
a(n) - 2*a(n-1) + a(n-2) = -4 + 8*n = A017113(n+1).
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 8 = A010731(n).
a(n) - 4*a(n-1) + 6*a(n-2) - 4*a(n-3) + a(n-4) = 0.
Binomial transform of quasi-finite sequence 1,4,12,8,0,(0 continued).
G.f.: (1+x+7*x^2-x^3)/(1-x)^4. - R. J. Mathar, Feb 15 2010
From Natan Arie Consigli, Jul 03 2016: (Start)
a(n) = A018227(2*n) + 3.
a(n) = A002492(n) + 1. (End)
E.g.f.: (1/3)*(3 + 12*x + 18*x^2 + 4*x^3)*exp(x). - G. C. Greubel, Jul 27 2024

Edited by N. J. A. Sloane, Oct 17 2009

More terms a(11)-a(35) from Vincenzo Librandi, Oct 17 2009

A176458 Decimal expansion of 4+sqrt(17).

Original entry on oeis.org

8, 1, 2, 3, 1, 0, 5, 6, 2, 5, 6, 1, 7, 6, 6, 0, 5, 4, 9, 8, 2, 1, 4, 0, 9, 8, 5, 5, 9, 7, 4, 0, 7, 7, 0, 2, 5, 1, 4, 7, 1, 9, 9, 2, 2, 5, 3, 7, 3, 6, 2, 0, 4, 3, 4, 3, 9, 8, 6, 3, 3, 5, 7, 3, 0, 9, 4, 9, 5, 4, 3, 4, 6, 3, 3, 7, 6, 2, 1, 5, 9, 3, 5, 8, 7, 8, 6, 3, 6, 5, 0, 8, 1, 0, 6, 8, 4, 2, 9, 6, 6, 8, 4, 5, 4

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of 4+sqrt(17) is A010731.

This is the shape of an 8-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			4+sqrt(17) = 8.12310562561766054982...

Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2

Cf. A010473 (decimal expansion of sqrt(17)), A010731 (all 8's sequence).

Cf. A049310.

r=8; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]

4+sqrt(17) \\ Charles R Greathouse IV, Jul 24 2013

a(n) = A010473(n) for n > 1.
Equals exp(arcsinh(4)), since arcsinh(x)=log(x+sqrt(x^2+1)). - Stanislav Sykora, Nov 01 2013
Equals lim_{n->infinity} S(n, 2*sqrt(17))/S(n-1, 2*sqrt(17)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A164897 a(n) = 4n(n+1) + 3.

Original entry on oeis.org

3, 11, 27, 51, 83, 123, 171, 227, 291, 363, 443, 531, 627, 731, 843, 963, 1091, 1227, 1371, 1523, 1683, 1851, 2027, 2211, 2403, 2603, 2811, 3027, 3251, 3483, 3723, 3971, 4227, 4491, 4763, 5043, 5331, 5627, 5931, 6243, 6563, 6891, 7227, 7571, 7923, 8283, 8651, 9027, 9411

Offset: 0

Paul Curtz, Aug 30 2009

One-fourth the sum of the three terms produced by the division of complex numbers (2*n-3+(2*n-1)*i)/(2*n+1+(2*n+3)*i). For (b+c*i)/(d+e*i) the three terms in parentheses are ((b*d+c*e)+(c*d-b*e)*i)/(d^2+e^2). By substituting b=2*n-3, c=2*n-1, d=2*n+1, and e=2*n+3 one gets a(n). - J. M. Bergot, Sep 10 2015

The continued fraction expansion of sqrt(a(n)) is [2n+1; {2n+1, 4n+2}]. - Magus K. Chu, Sep 08 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..900
R. M. Green and Tianyuan Xu, 2-roots for simply laced Weyl groups, arXiv:2204.09765 [math.RT], 2022.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A008590, A010731, A016743, A069894.

Odd-indexed terms of A059100.

[4*n*(n+1)+3: n in [0..50]]; // Vincenzo Librandi, Apr 24 2011

A164897:=n->4*n*(n+1)+3: seq(A164897(n), n=0..100); # Wesley Ivan Hurt, Sep 10 2015

Table[4 n (n + 1) + 3, {n, 0, 50}] (* Harvey P. Dale, Jan 23 2011 *)

a(n)=4*n*(n+1)+3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A000124(2*n) + A000124(2*n+1) = A069894(n)+1.
a(n+1) - a(n) = 8n+8 = A008590(n+1) (first differences).
a(n+1) - 2*a(n) + a(n-1) = 8 = A010731(n) (second differences).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>2.
G.f.: (3+2*x+3*x^2) / (1-x)^3.
Sum_{k=n+1..2*n+1} a(k) - Sum_{k=0..n} a(k) = (2*n+2)^3. - Bruno Berselli, Jan 24 2011
E.g.f.: (4x^2 + 8x + 1)*exp(x). - G. C. Greubel, Sep 22 2015
a(n)^2 = A222465(n)*A222465(n+1) - 12. - Ezhilarasu Velayutham, Mar 18 2020
Sum_{n>=0} 1/a(n) = tanh(Pi/sqrt(2))*Pi/(4*sqrt(2)). - Amiram Eldar, Aug 21 2022
a(n) = A059100(2*n+1). - Dimitri Papadopoulos, Nov 21 2023

Definition simplified by R. J. Mathar, Sep 16 2009

A023007 Number of partitions of n into parts of 8 kinds.

Original entry on oeis.org

1, 8, 44, 192, 726, 2464, 7704, 22528, 62337, 164560, 417140, 1020416, 2418710, 5573568, 12520744, 27484160, 59068372, 124505880, 257770964, 524871424, 1052316364, 2079491744, 4053978040, 7803219968, 14840711765, 27907041392, 51917588800, 95608651776

Offset: 0

David W. Wilson

nonn

a(n) is Euler transform of A010731. - Alois P. Heinz, Oct 17 2008

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. 8th column of A144064. - Alois P. Heinz, Oct 17 2008

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*8, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^8,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Feb 28 2015 *)

a(n) ~ exp(4 * Pi * sqrt(n/3)) / (sqrt(2) * 3^(9/4) * n^(11/4)). - Vaclav Kotesovec, Feb 28 2015
a(0) = 1, a(n) = (8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

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.

A278481 Number of neighbors of the n-th term in a full isosceles triangle read by rows.

Original entry on oeis.org

2, 4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 6, 6, 6, 4, 4, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4

Offset: 1

Omar E. Pol, Nov 23 2016

Apart from the left border and the right border, the rest of the elements are 6's.

For the same idea but for a right triangle see A278480; for a square array see A278545, for a square spiral see A010731; and for a hexagonal spiral see A010722.

			The sequence written as an isosceles triangle begins:
.
.                     2;
.                   4,  4;
.                 4,  6,  4;
.               4,  6,  6,  4;
.             4,  6,  6,  6,  4;
.           4,  6,  6,  6,  6,  4;
.         4,  6,  6,  6,  6,  6,  4;
.       4,  6,  6,  6,  6,  6,  6,  4;
.     4,  6,  6,  6,  6,  6,  6,  6,  4;
.   4,  6,  6,  6,  6,  6,  6,  6,  6,  4;
...

Row sums give A016933.
Left border gives A040002, the same as the right border.
Middle column gives the elements > 1 of A134201, also twice A122553.
Cf. A010722, A010731, A275015, A278317, A278480, A278545.

cp's OEIS Frontend

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

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A010709 Constant sequence: the all 4's sequence.

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Maxima

PARI

Python

Formula

A010689 Periodic sequence: Repeat 1, 8.

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Magma

Magma

Mathematica

Maxima

PARI

Sage

Formula

Extensions

A220521 Number of toothpicks or D-toothpicks added at n-th stage in the toothpick structure of A220520.

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A166464 a(n) = (3 + 2*n + 6*n^2 + 4*n^3)/3.

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Magma

Mathematica

PARI

SageMath

Formula

Extensions

A176458 Decimal expansion of 4+sqrt(17).

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A166464 a(n) = (3 + 2n + 6n^2 + 4*n^3)/3.

A164897 a(n) = 4n(n+1) + 3.