A008590 -id:A008590 - cp's OEIS frontend

A139098 a(n) = 8*n^2.

0, 8, 32, 72, 128, 200, 288, 392, 512, 648, 800, 968, 1152, 1352, 1568, 1800, 2048, 2312, 2592, 2888, 3200, 3528, 3872, 4232, 4608, 5000, 5408, 5832, 6272, 6728, 7200, 7688, 8192, 8712, 9248, 9800, 10368, 10952, 11552, 12168, 12800, 13448, 14112, 14792, 15488, 16200

Offset: 0

Omar E. Pol, Apr 25 2008

Opposite numbers to the centered 16-gonal numbers (A069129) in the square spiral whose vertices are the triangular numbers (A000217).
8 times the squares. - Omar E. Pol, Dec 09 2008
a(n-1) is the molecular topological index of the n-wheel graph W_n. - Eric W. Weisstein, Jul 11 2011
An n X n pandiagonal magic square has a(n) orientations. - Kausthub Gudipati, Sep 15 2011
Area of a square with diagonal 4n. - Wesley Ivan Hurt, Jun 19 2014
Sum of all the parts in the partitions of 4n into exactly two parts. - Wesley Ivan Hurt, Jul 23 2014
Equivalently: integers k such that k$ / (k/2-1)! and k$ / (k/2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information). - Bernard Schott, Dec 02 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Eric Weisstein's World of Mathematics, Molecular Topological Index.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Subsequence of A008586 and of A349081.
Cf. A000217, A000290, A000178, A001105, A016742, A016766, A033582, A069129, A245058.
Cf. A008590, A348692.

[8*n^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

A139098:=n->8*n^2; seq(A139098(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

8 Range[0, 50]^2 (* Wesley Ivan Hurt, Jun 19 2014 *)
LinearRecurrence[{3,-3,1},{0,8,32},50] (* Harvey P. Dale, Oct 05 2023 *)

a(n)=8*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 8*A000290(n) = 4*A001105(n) = 2*A016742(n). - Omar E. Pol, Dec 13 2008
G.f.: -8*x*(1+x)/(x-1)^3. - R. J. Mathar, Nov 27 2015
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/48 (A245058).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/96.
Product_{n>=1} (1 + 1/a(n)) = sqrt(8)*sinh(Pi/sqrt(8))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(8)*sin(Pi/sqrt(8))/Pi. (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Dec 03 2021
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 8*x*(1 + x)*exp(x).
a(n) = n*A008590(n) = A001105(2*n). (End)

A017089 a(n) = 8*n + 2.

Original entry on oeis.org

2, 10, 18, 26, 34, 42, 50, 58, 66, 74, 82, 90, 98, 106, 114, 122, 130, 138, 146, 154, 162, 170, 178, 186, 194, 202, 210, 218, 226, 234, 242, 250, 258, 266, 274, 282, 290, 298, 306, 314, 322, 330, 338, 346, 354, 362, 370, 378, 386, 394, 402, 410, 418, 426

Offset: 0

N. J. A. Sloane, Dec 11 1996

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 33 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 81 ).
First differences of A002939. - Aaron David Fairbanks, May 13 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A002939, A008589, A008590, A016813, A016993, A017005, A017017, A017077.

a017089 = (+ 2) . (* 8)
a017089_list = [2, 10 ..]  -- Reinhard Zumkeller, Jun 07 2015

[8*n+2: n in [0..60]]; // Vincenzo Librandi, May 28 2011

A017089:=n->8*n+2; seq(A017089(n), n=0..50); # Wesley Ivan Hurt, May 13 2014

Range[2, 1000, 8] (* Vladimir Joseph Stephan Orlovsky, May 27 2011 *)

a(n) = 8*n+2; \\ Michel Marcus, Sep 17 2015

a(n) = 8*n+2; a(n) = 2*a(n-1)-a(n-2). - Vincenzo Librandi, May 28 2011
Sum_{n>=0} (-1)^n/a(n) = (Pi + 2*log(cot(Pi/8)))/(8*sqrt(2)). - Amiram Eldar, Dec 11 2021
From Elmo R. Oliveira, Mar 17 2024: (Start)
G.f.: 2*(1+3*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(1 + 4*x).
a(n) = 2*A016813(n) = A008590(n) + 2. (End)

A195161 Multiples of 8 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 8, 3, 16, 5, 24, 7, 32, 9, 40, 11, 48, 13, 56, 15, 64, 17, 72, 19, 80, 21, 88, 23, 96, 25, 104, 27, 112, 29, 120, 31, 128, 33, 136, 35, 144, 37, 152, 39, 160, 41, 168, 43, 176, 45, 184, 47, 192, 49, 200, 51, 208, 53, 216, 55, 224, 57, 232, 59

Offset: 0

Omar E. Pol, Sep 10 2011

A008590 and A005408 interleaved. This is 8*n if n is even, n if n is odd, if n>=0.
Partial sums give the generalized 12-gonal (or dodecagonal) numbers A195162.
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (- 4*log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 12-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 8 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, this sequence, A195312.
Cf. A144433.

&cat[[8*n, 2*n+1]: n in [0..30]]; // Vincenzo Librandi, Sep 27 2011

a := proc(n): (6*(-1)^n+10)*n/4 end: seq(a(n), n=0..59); # Johannes W. Meijer, Jul 03 2016

With[{nn=30},Riffle[8*Range[0,nn],2*Range[0,nn]+1]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,8,3},60] (* Harvey P. Dale, Nov 24 2013 *)

concat(0, Vec(x*(1+8*x+x^2)/((1-x)^2*(1+x)^2) + O(x^99))) \\ Altug Alkan, Jul 04 2016

a(2n) = 8n, a(2n+1) = 2n+1. [corrected by Omar E. Pol, Jul 26 2018]
a(n) = (6*(-1)^n+10)*n/4. - Vincenzo Librandi, Sep 27 2011
a(n) = 2*a(n-2)-a(n-4). G.f.: x*(1+8*x+x^2)/((1-x)^2*(1+x)^2). - Colin Barker, Aug 11 2012
From Ilya Gutkovskiy, Jul 03 2016: (Start)
a(m*2^k) = m*2^(k+2), k>0.
E.g.f.: x*(4*sinh(x) + cosh(x)).
Dirichlet g.f.: 2^(-s)*(2^s + 6)*zeta(s-1). (End)
Multiplicative with a(2^e) = 4*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
a(n) = A144433(n-1) for n > 1. - Georg Fischer, Oct 14 2018

A234275 Expansion of (1+2x+9x^2-4*x^3)/(1-x)^2.

Original entry on oeis.org

1, 4, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432

Offset: 0

N. J. A. Sloane, Dec 24 2013

Also the coordination sequence for a point of degree 4 in the tiling of the Euclidean plane by right triangles (with angles Pi/2, Pi/4, Pi/4). These triangles are fundamental regions for the Coxeter group (2,4,4). In the notation of Conway et al. 2008 this is the tiling *442. The coordination sequence for a point of degree 8 is given by A022144. - N. J. A. Sloane, Dec 28 2015

First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood. Initialized with a single black (ON) cell at stage zero. - Robert Price, May 28 2016

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5. See p. 191.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Colin Barker, Table of n, a(n) for n = 0..1000
J. Choi, N. Pippenger, Counting the Angels and Devils in Escher's Circle Limit IV, arXiv preprint arXiv:1310.1357, 2013.
Tom Karzes, Tiling Coordination Sequences
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A022144, A008590.
For partial sums see A265056.
List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

Join[{1, 4}, LinearRecurrence[{2, -1}, {16, 24}, 60]] (* Jean-François Alcover, Jan 08 2019 *)

Vec(-(4*x^3-9*x^2-2*x-1)/(x-1)^2 + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = A022144(n), n>1. - R. J. Mathar, Jan 11 2014
From Colin Barker, Jul 10 2015: (Start)
a(n) = 8*n, n>1.
a(n) = 2*a(n-1) - a(n-2) for n>3.
G.f.: -(4*x^3-9*x^2-2*x-1) / (x-1)^2.
(End)

A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

Original entry on oeis.org

1, 4, 3, 9, 8, 5, 16, 15, 12, 7, 25, 24, 21, 16, 9, 36, 35, 32, 27, 20, 11, 49, 48, 45, 40, 33, 24, 13, 64, 63, 60, 55, 48, 39, 28, 15, 81, 80, 77, 72, 65, 56, 45, 32, 17, 100, 99, 96, 91, 84, 75, 64, 51, 36, 19, 121, 120, 117, 112, 105, 96, 85, 72, 57, 40, 21

Offset: 1

Reinhard Zumkeller, May 24 2004

(T(n,k) mod 4) <> 2, see A042965, A016825.
All numbers m occur A034178(m) times.
The row polynomials T(n,x) appear in the calculation of the column g.f.s of triangle A120070 (used to find the frequencies of the spectral lines of the hydrogen atom).

			n=3: T(3,x) = 9+8*x+5*x^2.
Triangle begins:
   1;
   4,  3;
   9,  8,  5;
  16, 15, 12,  7;
  25, 24, 21, 16,  9;
  36, 35, 32, 27, 20, 11;
  49, 48, 45, 40, 33, 24, 13;
  64, 63, 60, 55, 48, 39, 28, 15;
  81, 80, 77, 72, 65, 56, 45, 32, 17;
  ... etc. - _Philippe Deléham_, Mar 07 2013

G. C. Greubel, Table of n, a(n) for n = 1..5050

Cf. A000290, A000567, A004736, A005408, A005563, A008586, A008590.
Cf. A008594, A008598, A016825, A016945, A017329, A028347, A028560.
Cf. A028566, A034178, A042965, A094727, A120070, A128624.
Cf. A000384 (alternating row sums), A002412 (row sums).

[n^2-k^2: k in [0..n-1], n in [1..15]]; // G. C. Greubel, Mar 12 2024

Table[n^2 - k^2, {n,12}, {k,0,n-1}]//Flatten (* Michael De Vlieger, Nov 25 2015 *)

flatten([[n^2-k^2 for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 12 2024

Row polynomials: T(n,x) = n^2*Sum_{m=0..n} x^m - Sum_{m=0..n} m^2*x^m = Sum_{k=0..n-1} T(n,k)*x^k, n >= 1.
T(n, k) = A004736(n,k)*A094727(n,k).
T(n, 0) = A000290(n).
T(n, 1) = A005563(n-1) for n>1.
T(n, 2) = A028347(n) for n>2.
T(n, 3) = A028560(n-3) for n>3.
T(n, 4) = A028566(n-4) for n>4.
T(n, n-1) = A005408(n).
T(n, n-2) = A008586(n-1) for n>1.
T(n, n-3) = A016945(n-2) for n>2.
T(n, n-4) = A008590(n-2) for n>3.
T(n, n-5) = A017329(n-3) for n>4.
T(n, n-6) = A008594(n-3) for n>5.
T(n, n-8) = A008598(n-2) for n>7.
T(A005408(k), k) = A000567(k).
Sum_{k=0..n} T(n, k) = A002412(n) (row sums).
From G. C. Greubel, Mar 12 2024: (Start)
Sum_{k=0..n-1} (-1)^k * T(n, k) = A000384(floor((n+1)/2)).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A128624(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = (1/2)*n*(n+1 - (-1)^n*cos(n*Pi/2)). (End)
G.f.: x*(1 - 3*x^2*y + x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Aug 04 2025

A047615 Numbers that are congruent to {0, 5} mod 8.

Original entry on oeis.org

0, 5, 8, 13, 16, 21, 24, 29, 32, 37, 40, 45, 48, 53, 56, 61, 64, 69, 72, 77, 80, 85, 88, 93, 96, 101, 104, 109, 112, 117, 120, 125, 128, 133, 136, 141, 144, 149, 152, 157, 160, 165, 168, 173, 176, 181, 184, 189, 192, 197, 200, 205, 208, 213, 216, 221, 224, 229, 232

Offset: 1

N. J. A. Sloane

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

Union of A008590 and A004770.

Cf. A047398, A047452, A047461, A047470, A047524, A047535, A047617.

Filtered([0..250], n->n mod 8=0 or n mod 8=5); # Muniru A Asiru, Jul 23 2018

[(8*n - 7 + (-1)^n)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 26 2015

a:=n->add(4-(-1)^j, j=1..n): seq(a(n), n=0..59); # Zerinvary Lajos, Dec 13 2008

Table[(8 n - 7 + (-1)^n)/2, {n, 1, 40}] (* Wesley Ivan Hurt, Mar 26 2015 *)
Rest@ CoefficientList[Series[x^2*(5 + 3 x)/((1 - x)^2*(1 + x)), {x, 0, 59}], x] (* Michael De Vlieger, Aug 25 2016 *)
Rest@(Range[0, 60]! CoefficientList[ Series[(6 + Exp[-x] + (8 x - 7)*Exp[x])/2, {x, 0, 60}], x]) (* or *)
LinearRecurrence[{1, 1, -1}, {0, 5, 8}, 60] (* Robert G. Wilson v, Jul 23 2018 *)

forstep(n=0,200,[5,3],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

concat(0, Vec(x^2*(5+3*x)/((1-x)^2*(1+x)) + O(x^100))) \\ Colin Barker, Aug 25 2016

def A047615(n): return (n<<2)-3-(n&1) # Chai Wah Wu, Mar 30 2024

a(n) = 8*n-a(n-1)-11 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=5 and b(k)=2^(k+2) for k>0. - Philippe Deléham, Oct 17 2011
From Wesley Ivan Hurt, Mar 26 2015: (Start)
a(n) = a(n-1)+a(n-2)-a(n-3).
a(n) = (8n - 7 + (-1)^n)/2. (End)
G.f.: x^2*(5+3*x) / ((1-x)^2*(1+x)). - Colin Barker, Aug 25 2016
From Franck Maminirina Ramaharo, Jul 23 2018: (Start)
a(n) = A047470(n) - (-1)^(n - 1) + 1.
E.g.f.: (6 + exp(-x) + (8*x - 7)*exp(x))/2. (End)
Sum_{n>=2} (-1)^n/a(n) = log(2)/2 - (sqrt(2)-1)*Pi/16 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

More terms from Vincenzo Librandi, Aug 06 2010

A168181 Characteristic function of numbers that are not multiples of 8.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 0

Reinhard Zumkeller, Nov 30 2009

Multiplicative with a(p^e) = (if p=2 then A019590(e) else 1), p prime and e>0.

Period 8 Repeat: [0, 1, 1, 1, 1, 1, 1, 1]. - Wesley Ivan Hurt, Jun 21 2014

			G.f. = x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + ...

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A168185, A145568, A168184, A168182, A109720, A097325, A011558, A166486, A011655, A000035, A010877, A244413, A253513.

[Sign(n mod 8) : n in [0..100]]; // Wesley Ivan Hurt, Jun 21 2014

with(numtheory); A168181:=n->signum(n mod 8); seq(A168181(n), n=0..100); # Wesley Ivan Hurt, Jun 21 2014

Table[Sign[Mod[n,8]], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 21 2014 *)

a(n)=n%8 > 0 \\ Felix Fröhlich, Aug 11 2014

def A168181(n): return int(bool(n&7)) # Chai Wah Wu, Jul 09 2022

a(n+8) = a(n);
a(n) = A000007(A010877(n));
a(A047592(n)) = 1; a(A008590(n)) = 0;
A033440(n) = Sum_{k=0..n} a(k)*(n-k).
Dirichlet g.f. (1-1/8^s)*zeta(s). - R. J. Mathar, Feb 19 2011
For the general case: the characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = sign(n mod 8). - Wesley Ivan Hurt, Jun 21 2014
a(n) = sign( 1 - floor(cos(Pi*n/4)) ). - Wesley Ivan Hurt, Jun 21 2014
Euler transform of length 8 sequence [ 1, 0, 0, 0, 0, 0, -1, 1]. - Michael Somos, Jun 24 2014
Moebius transform is length 8 sequence [ 1, 0, 0, 0, 0, 0, 0, -1]. - Michael Somos, Jun 24 2014
G.f.: x * (1 - x^7) / ((1 - x) * (1 - x^8)). - Michael Somos, Jun 24 2014
a(n) = 1-A253513(n). - Antti Karttunen, Oct 08 2017

A220062 Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0

Offset: 0

Alois P. Heinz, Dec 03 2012

Equivalently, the number of walks of length n-1 on the path graph P_k. - Andrew Howroyd, Apr 17 2017

			A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
  1,  1,  1,  1,  1,   1,   1,   1, ...
  0,  1,  2,  3,  4,   5,   6,   7, ...
  0,  0,  2,  4,  6,   8,  10,  12, ...
  0,  0,  2,  6, 10,  14,  18,  22, ...
  0,  0,  2,  8, 16,  24,  32,  40, ...
  0,  0,  2, 12, 26,  42,  58,  74, ...
  0,  0,  2, 16, 42,  72, 104, 136, ...
  0,  0,  2, 24, 68, 126, 188, 252, ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 2-10 give: A000007, A040000, A029744(n+2) for n>0, A006355(n+3) for n>0, A090993(n+1) for n>0, A090995(n-1) for n>2, A129639, A153340, A153362, A153360.
Rows 0-6 give: A000012, A001477, A005843(k-1) for k>0, A016825(k-2) for k>1, A008590(k-2) for k>2, A113770(k-2) for k>3, A063164(k-2) for k>4.
Main diagonal gives: A102699.
Cf. A198632, A188866, A276562, A208727, A208671.

b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-1, j, k), j=1..k),
      `if`(i>1, b(n-1, i-1, k), 0)+
      `if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017

A047451 Numbers that are congruent to {0, 6} mod 8.

Original entry on oeis.org

0, 6, 8, 14, 16, 22, 24, 30, 32, 38, 40, 46, 48, 54, 56, 62, 64, 70, 72, 78, 80, 86, 88, 94, 96, 102, 104, 110, 112, 118, 120, 126, 128, 134, 136, 142, 144, 150, 152, 158, 160, 166, 168, 174, 176, 182, 184, 190, 192, 198, 200, 206, 208, 214, 216, 222, 224, 230

Offset: 1

N. J. A. Sloane

All even numbers m such that Integral_{x=0..2*Pi} Product_{i=1..m/2} cos(2*i*x) dx is nonzero. - William Boyles, Oct 12 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A017137.

Cf. A030308, A047504 (complement).

Array[8 # + {0, 6} &, 29, 0] // Flatten (* or *)
Rest@ CoefficientList[Series[2 x^2*(3 + x)/((1 + x) (x - 1)^2), {x, 0, 58}], x] (* Michael De Vlieger, Nov 18 2019 *)
LinearRecurrence[{1,1,-1},{0,6,8},80] (* Harvey P. Dale, Apr 09 2022 *)

forstep(n=0,200,[6,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = 4*n - 3 + (-1)^n; \\ David Lovler, Jul 25 2022

a(n) = 8*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 06 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 4*n - 3 + (-1)^n.
G.f.: 2*x^2*(3+x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=6 and b(k)=2^(k+2) for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((8/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - Pi/16. - Amiram Eldar, Dec 18 2021
E.g.f.: (4*x + 1)*exp(x) - exp(-x) = 4*x*exp(x) + 2*sinh(x). - David Lovler, Aug 02 2022

A047467 Numbers that are congruent to {0, 2} mod 8.

Original entry on oeis.org

0, 2, 8, 10, 16, 18, 24, 26, 32, 34, 40, 42, 48, 50, 56, 58, 64, 66, 72, 74, 80, 82, 88, 90, 96, 98, 104, 106, 112, 114, 120, 122, 128, 130, 136, 138, 144, 146, 152, 154, 160, 162, 168, 170, 176, 178, 184, 186, 192, 194, 200, 202, 208, 210, 216, 218, 224, 226, 232

Offset: 1

N. J. A. Sloane

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Union of A008590 and A017089.

Cf. A030308, A042948.

{#,#+2}&/@(8*Range[0,30])//Flatten (* or *) LinearRecurrence[{1,1,-1},{0,2,8},60] (* Harvey P. Dale, Nov 30 2019 *)

forstep(n=0,200,[2,6],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

a(n) = 4*n - 5 - (-1)^n; \\ David Lovler, Jul 25 2022

From R. J. Mathar, Sep 19 2008: (Start)
a(n) = 4*n - 5 - (-1)^n = 2*A042948(n-1).
G.f.: 2*x^2*(1+3x)/((1-x)^2*(1+x)). (End)
a(n) = 8*n - a(n-1) - 14 with a(1)=0. - Vincenzo Librandi, Aug 06 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=2 and b(k)=2^(k+2)for k > 0. - Philippe Deléham, Oct 17 2011
a(n) = floor((8/3)*floor(3*n/2)). - Clark Kimberling, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = Pi/16 + 3*log(2)/8. - Amiram Eldar, Dec 18 2021
E.g.f.: 6 + (4*x - 5)*exp(x) - exp(-x). - David Lovler, Jul 22 2022

More terms from Vincenzo Librandi, Aug 06 2010

cp's OEIS Frontend

A139098 a(n) = 8*n^2.

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A017089 a(n) = 8*n + 2.

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A195161 Multiples of 8 and odd numbers interleaved.

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A234275 Expansion of (1+2*x+9*x^2-4*x^3)/(1-x)^2.

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A094728 Triangle read by rows: T(n,k) = n^2 - k^2, 0 <= k < n.

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A047615 Numbers that are congruent to {0, 5} mod 8.

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A234275 Expansion of (1+2x+9x^2-4*x^3)/(1-x)^2.