A000034 -id:A000034 - cp's OEIS frontend

A266180 Decimal representation of the n-th iteration of the "Rule 6" elementary cellular automaton starting with a single ON (black) cell.

1, 6, 16, 96, 256, 1536, 4096, 24576, 65536, 393216, 1048576, 6291456, 16777216, 100663296, 268435456, 1610612736, 4294967296, 25769803776, 68719476736, 412316860416, 1099511627776, 6597069766656, 17592186044416, 105553116266496, 281474976710656

Offset: 0

Robert Price, Dec 22 2015

A001025 is a subsequence. - Altug Alkan, Dec 23 2015

Rules 38, 134 and 166 also generate this sequence.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..999
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 22.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (0,16).

Cf. A001025, A266178, A266179, A019590, A003953, A003945, A000034, A032766, A042948, A035608.

rule=6; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)
LinearRecurrence[{0,16},{1,6},30] (* Harvey P. Dale, May 25 2016 *)

print([int(4**(n-1)*(5-(-1)**n)) for n in range(30)]) # Karl V. Keller, Jr., Jun 03 2021

From Colin Barker, Dec 23 2015 and Apr 13 2019: (Start)
a(n) = 4^(n-1)*(5-(-1)^n).
a(n) = 16*a(n-2) for n>1.
G.f.: (1+6*x) / ((1-4*x)*(1+4*x)).
(End)

A028359 Two-bell analog of A028355.

Original entry on oeis.org

1, 2, 12, 121, 212, 1212, 12121, 21212, 121212, 1212121, 2121212, 12121212, 121212121, 212121212, 1212121212, 12121212121, 21212121212, 121212121212, 1212121212121, 2121212121212, 12121212121212, 121212121212121, 212121212121212, 1212121212121212, 12121212121212121

Offset: 1

N. J. A. Sloane

Consider the infinite digits: 121212... . We can break this into a sequence of integers such that the sum of digits in the n-th value is n. - Seiichi Manyama, Oct 31 2018

Seiichi Manyama, Table of n, a(n) for n = 1..1500

Cf. A000034, A028355.

Conjectures from Chai Wah Wu, Apr 18 2024: (Start)
a(n) = 101*a(n-3) - 100*a(n-6) for n > 6.
G.f.: x*(10*x^4 + 20*x^3 + 12*x^2 + 2*x + 1)/(100*x^6 - 101*x^3 + 1). (End)

More terms from Seiichi Manyama, Oct 31 2018

A052997 Expansion of (1+x-x^3)/((1-2x)(1-x^2)).

Original entry on oeis.org

1, 3, 7, 14, 29, 58, 117, 234, 469, 938, 1877, 3754, 7509, 15018, 30037, 60074, 120149, 240298, 480597, 961194, 1922389, 3844778, 7689557, 15379114, 30758229, 61516458, 123032917, 246065834, 492131669, 984263338, 1968526677

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Harvey P. Dale, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1075
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

spec := [S,{S=Prod(Union(Sequence(Prod(Z,Z)),Z),Sequence(Union(Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

f[s_List] := Block[{a = s[[-1]]}, Append[s, If[ OddQ@ Length@ s, 2a +1, 2a]]]; Join[{1},  Nest[f, {3}, 30]] (* or *)
CoefficientList[ Series[(1 + x - x^3)/(1 - 2x - x^2 + 2x^3), {x, 0, 30}], x] (* Robert G. Wilson v, Jul 20 2017 *)
LinearRecurrence[{2,1,-2},{1,3,7,14},40] (* Harvey P. Dale, May 27 2019 *)

G.f.: -(-x+x^3-1)/(-1+x^2)/(-1+2*x).
Recurrence: {a(0)=1, -2*a(n)-a(n+1)+a(n+2)-1, a(1)= 3, a(2)=7, a(3)=14}, 11/6*2^n + Sum(-1/6*(2 + _alpha)*_alpha^(-1-n), _alpha=RootOf(-1 + _Z^2))
a(n) = 2*a(n-1)+1 for even n, otherwise a(n) = 2*a(n-1), with a(0)=1, a(1)=3. [Bruno Berselli, Jun 19 2014]
3*a(n) = 11*2^(n-1)-A000034(n) for n>0. - R. J. Mathar, Feb 27 2019

More terms from James Sellers, Jun 06 2000

A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

Original entry on oeis.org

1, 0, 2, 4, 5, 7, 13, 22, 34, 54, 89, 145, 233, 376, 610, 988, 1597, 2583, 4181, 6766, 10946, 17710, 28657, 46369, 75025, 121392, 196418, 317812, 514229, 832039, 1346269, 2178310, 3524578, 5702886, 9227465, 14930353, 24157817, 39088168

Offset: 0

Creighton Dement, Feb 23 2006

a(n+2) - a(n+1) - a(n) gives match to A000034, apart from signs.

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

Cf. A000034, A000045, A001519, A001906, A006498, A116697, A116698, A116699.

A115008:= func< n | Fibonacci(n+1) - (n mod 2) + 2*0^((n+1) mod 4) >;
[A115008(n): n in [0..50]]; // G. C. Greubel, Aug 24 2025

Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2], {n,0,50}] (* G. C. Greubel, Aug 24 2025 *)

def A115008(n): return fibonacci(n+1) -i**(n-1)*(n%2)
print([A115008(n) for n in range(51)]) # G. C. Greubel, Aug 24 2025

a(2*n) = A000045(2*n+1) = A001519(n).
G.f.: (1-x+2*x^2+x^3)/((1+x^2)*(1-x-x^2)).
a(2*n+1) = (-1)^(n+1) + A001906(n+1) (compare with a similar property for A116697) - Creighton Dement, Mar 31 2006
From G. C. Greubel, Aug 24 2025: (Start)
a(n) = A000045(n+1) - i^(n-1)*(n mod 2).
E.g.f.: exp(x/2)*(cosh(p*x) + (1/(2*p))*sinh(p*x)) - sin(x), where 2*p = sqrt(5). (End)

A115356 Matrix (1,x)+(x,x^2) in Riordan array notation.

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jan 21 2006

As a flat sequence, with starting offset=1, a(n) = 1 if n is a triangular number (A000217) or twice a square (A001105), otherwise 0. - Antti Karttunen, Jan 19 2025

			Triangle begins:
n\k| 0  1  2  3  4  5  6  7  8  9
---+-------------------------------
0  | 1;
1  | 1, 1;
2  | 0, 0, 1;
3  | 0, 1, 0, 1;
4  | 0, 0, 0, 0, 1;
5  | 0, 0, 1, 0, 0, 1;
6  | 0, 0, 0, 0, 0, 0, 1;
7  | 0, 0, 0, 1, 0, 0, 0, 1;
8  | 0, 0, 0, 0, 0, 0, 0, 0, 1;
9  | 0, 0, 0, 0, 1, 0, 0, 0, 0, 1;
(row and column numbering added by _Antti Karttunen_, Jan 19 2025)

Antti Karttunen, Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle
Wikipedia, Riordan array
Index entries for characteristic functions.

Row sums are A000034. Diagonal sums are A115357. Inverse is A115358.
Cf. A000217, A001105, A010054, A379480.
Cf. also A115359.

A115356off1(n) = (ispolygonal(n,3) || (!(n%2) && issquare(n/2))); \\ (This is one-based)
A115356(n) = A115356off1(1+n); \\ (zero-based) - Antti Karttunen, Jan 19 2025

Number triangle T(n, k) = if(n=k, 1, 0) OR if(n=2k+1, 1, 0).

a(n) = A010054(n) + A379480(n). [As a flat sequence with starting offset 1] - Antti Karttunen, Jan 19 2025

A180916 Number of convex polyhedra with n faces that are all regular polygons.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 2, 7, 3, 6, 4, 7, 3, 13, 2, 5, 4, 6, 1, 9, 2, 6, 1, 4, 1, 8, 4, 2, 1, 3, 1, 10, 1, 3, 1, 2, 4, 3, 1, 2, 1, 9, 1, 2, 1, 2, 2, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 1

J. Lowell, Sep 23 2010

For all n > 92, the sequence is identical to A000034 because for large n only prisms (even and odd n) and antiprisms (even n) are convex and have regular polygonal faces. The MathWorld article about Johnson Solids is very informative about this topic.

In a regular-faced polyhedron, any two faces with the same number of edges are congruent. (Proof: As the two faces are regular polygons, it suffices to show their edges have the same length. But as all faces are regular polygons and the polyhedron is connected, all edges have the same length.) - Jonathan Sondow, Feb 11 2018

			a(6) = 3 because the cube, pentagonal pyramid, and triangular bipyramid all qualify. a(7) = 2 because only the pentagonal prism and elongated triangular pyramid qualify; the hexagonal pyramid is impossible with equilateral triangles

Eric W. Weisstein, MathWorld: Cube
Eric W. Weisstein, MathWorld: Pentagonal Pyramid
Eric W. Weisstein, MathWorld: Dipyramid
Eric W. Weisstein, MathWorld: Pentagonal Prism
Eric W. Weisstein, MathWorld: Elongated Triangular Pyramid
Eric W. Weisstein, MathWorld: Hexagonal Pyramid
Eric W. Weisstein, MathWorld: Johnson Solid

Cf. A296602, A296603, A296604.

f = Tally[Join[PolyhedronData["Platonic", "FaceCount"], PolyhedronData["Archimedean", "FaceCount"], PolyhedronData["Johnson", "FaceCount"], {PolyhedronData[{"Prism", 3}, "FaceCount"]}]]; f2 = Transpose[f]; cnt = Table[0, {n, 100}]; cnt[[f2[[1]]]] = f2[[2]]; Do[cnt[[n]]++, {n, 7, 100}] (* add prisms *); Do[ cnt[[n]]++, {n, 10, 100, 2}] (* add antiprisms *); cnt (* T. D. Noe, Mar 04 2011 *)

a(A296602(n)) = 1. - Jonathan Sondow, Jan 29 2018

More terms from J. Lowell, Feb 28 2011

Corrected by T. D. Noe, Mar 04 2011

A191372 The Sierpinski-Stern triangle.

Original entry on oeis.org

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

Offset: 0

Johannes W. Meijer, Jun 05 2011

The knight sums of the first and second kind Kn1y(n) = Kn2y(n), y >= 1, see A180662 for their definitions, of Sierpinski's triangle A047999 lead to the formula Kn1y(n) = A002487(n+(2*y-1)) - AS2S2S2(n,d) where the AS2S2S2(n,d) is the infinite concatenation of a S2(T, d = y-1) sequence; see for the first ten S2(T, d) and the first four Kn1y(n) the examples.
The A191372 sequence is the concatenation of all S2(T, d) sequences, d >= 0. The lengths of the S2(T, d) sequences are 2^ceiling(log(d)/log(2)) for d >= 1 while the length of S2(T, d=0) is 1.
Both the concatenation of the S2(T, d = 2^p) sequences, p >= 0, and the concatenation of the S2(T, d = 2^p-1) sequences, p >= 0, lead to Stern’s diatomic series A002487(n), n >= 2.
The differences of the sequences (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)), T from 0 to (2^(p-1) -1) and 1 <= delta <= (2^(p-1)-1) (take care that p <= pmax), lead to sequences that are snippets of A002487 and, surprisingly, their reverse; see the examples.
The row sums of the Sierpinski-Stern triangle are given by the terms of A191487.

Sam Northshield, Stern's Diatomic Sequence 0,1,1,2,1,3,2,3,1,4,..., Amer. Math. Month., Vol. 117 (7), pp. 581-598, 2010.
Igor Urbiha, Some properties of a function studied by De Rham, Carlitz and Dijkstra and its relation to the (Eisenstein -) Stern's diatomic sequence, Math. Commun. 6, pp. 181-198, 2002.

Cf. A047999 (Sierpinski), A002487 (Stern).

nmax:=2^5; pmax:=log(nmax)/log(2)-1; A047999:=proc(n,k) option remember; A047999(n,k) :=binomial(n,k) mod 2 end: A002487:=proc(n) option remember; if n<=1 then n elif n mod 2=0 then A002487(n/2); else A002487((n-1)/2)+A002487((n+1)/2); fi; end: d:=0: for n from 0 to nmax-d-1 do Kn1(n,d):= add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d): od: for p from 1 to pmax do for d from 2^(p-1) to 2^p do for n from 0 to nmax-d-1 do Kn1(n,d):=add(A047999(n-k+d, k+d),k=0..floor(n/2)): AS2S2S2(n,d):= A002487(n+1+2*d)-Kn1(n,d) od: od: od: S2(0,0):=AS2S2S2(0,0): a(0):=S2(0,0): for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do S2(Tx,d):=AS2S2S2(Tx,d) od: od: Ty:=0: for d from 1 to 2^pmax do for Tx from 0 to 2^ceil(log(d)/log(2))-1 do Ty:=Ty+1: a(Ty):=S2(Tx,d) od: od: S2(0,0); for d from 1 to 2^pmax do seq(S2(Tx,d), Tx=0..2^ceil(log(d)/ log(2))-1) od; seq(a(n),n=0..Ty);

The first few S2(T, d) rows of the Sierpinski-Stern triangle are:
d=0: [0]
d=1: [1]
d=2: [2, 1]
d=3: [2, 1, 3, 2]
d=4: [3, 2, 3, 1]
d=5: [4, 2, 3, 2, 3, 1, 4, 3]
d=6: [4, 2, 3, 1, 4, 3, 5, 2]
d=7: [3, 1, 4, 3, 5, 2, 5, 3]
d=8: [4, 3, 5, 2, 5, 3, 4, 1]
d=9: [6, 3, 6, 4, 5, 2, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4]
The first four Kn1y(n), y = d+1, sequences:
Kn11(n) = A002487(n+1) - A000004(n)
Kn12(n) = A002487(n+3) - A000012(n)
Kn13(n) = A002487(n+5) - A000034(n+1)
Kn14(n) = A002487(n+7) - A157810(n+1)
Three (AS2S2S2(T, 2^p-delta) - AS2S2S2(T, 2^(p-1)-delta)) sequences for p=6:
delta =  1: [1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4]
delta =  8: [4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1]
delta = 16: [5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0]

A207606 Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.

Original entry on oeis.org

1, 2, 3, 2, 4, 7, 2, 5, 16, 11, 2, 6, 30, 36, 15, 2, 7, 50, 91, 64, 19, 2, 8, 77, 196, 204, 100, 23, 2, 9, 112, 378, 540, 385, 144, 27, 2, 10, 156, 672, 1254, 1210, 650, 196, 31, 2, 11, 210, 1122, 2640, 3289, 2366, 1015, 256, 35, 2, 12, 275, 1782, 5148, 8008

Offset: 1

Clark Kimberling, Feb 19 2012

As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 03 2012

			First five rows:
  1;
  2;
  3,  2;
  4,  7,  2;
  5, 16, 11,  2;
Triangle (2, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, ...), 0 <= k <= n, begins:
  1;
  2,   0;
  3,   2,   0;
  4,   7,   2,   0;
  5,  16,  11,   2,   0;
  6,  30,  36,  15,   2,   0;
  7,  50,  91,  64,  19,   2,   0;
  8,  77, 196, 204, 100,  23,   2,   0;

G. C. Greubel, Rows n = 1..101 of the triangle, flattened

Cf. A207607.

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=0 then n+2
    elif k=n then 2
    else 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
      fi; end:
1, seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Mar 15 2020

(* First program *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A207606 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A207607 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, n+2, If[k==n, 2, 2*T[n-1, k] - T[n-2, k] + T[n-1, k-1] ]]]; Join[{1}, Table[T[n, k], {n, 0, 10}, {k, 0, n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else x*u(n - 1, x) + (x + 1)*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==1): return n+1
    elif (k==n): return 2
    else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k)
[1]+[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 15 2020

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = x*u(n-1,x) + (x+1)v(n-1,x), where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0 <= k <= n: g.f.: (1-y*x)/(1-(2+y)*x+x^2). - Philippe Deléham, Mar 03 2012
As triangle T(n,k) with 0 <= k <= n: Sum_{k=0..n} T(n,k)*x^k = A132677(n), A000034(n)*A057077(n), A057079(n), A000027(n+1), A001519(n+1), A001075(n), A002310(n), A038725(n), A172968(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Mar 03 2012
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Mar 03 2012
T(n,k) = C(n+k-1,2*k+1) + 2*C(n+k-1,2*k), where C is binomial. - Yuchun Ji, May 23 2019
T(n,k) = T(n-1,k) + A207607(n-1,k). - Yuchun Ji, May 28 2019

A208950 a(4n) = n(16n^2-1)/3, a(2n+1) = n(n+1)(2n+1)/6, a(4n+2) = (4n+1)(4n+2)(4*n+3)/6.

Original entry on oeis.org

0, 0, 1, 1, 5, 5, 35, 14, 42, 30, 165, 55, 143, 91, 455, 140, 340, 204, 969, 285, 665, 385, 1771, 506, 1150, 650, 2925, 819, 1827, 1015, 4495, 1240, 2728, 1496, 6545, 1785, 3885, 2109, 9139, 2470, 5330, 2870, 12341, 3311, 7095, 3795, 16215, 4324

Offset: 0

Paul Curtz, Mar 03 2012

a(n+2) is divisible by A060819(floor(n/3)).
a(n) is divisible by A176672(floor(n/3)).
Denominator of a(n)/n is of period 24: 1,1,3,4,1,6,1,4,3,1,1,12,1,2,3,4,1,3,1,4,3,2,1,12 (two successive palindromes).
This is the fifth column of the triangle A107711, hence the formula involving gcd(n+2,4) given below follows. - Wolfdieter Lang, Feb 24 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,-6,0,0,0,4,0,0,0,-1).

Cf. A107711 (fifth column).

Cf. A000034, A000292, A002415, A051724, A060819, A061037, A107711, A138190, A145979, A176672, A176895.

[Binomial(n+1,3)*GCD(n+2,4)/4: n in [0..50]]; // G. C. Greubel, Sep 20 2018

CoefficientList[Series[(x^2 + x^3 + 5 x^4 + 5 x^5 + 31 x^6 + 10 x^7 + 22 x^8 + 10 x^9 + 31 x^10 + 5 x^11 + 5 x^12 + x^13 + x^14)/((1 - x)^4 (1 + x)^4 (1 + 4 x^2 + 6 x^4 + 4 x^6 + x^8)), {x, 0, 47}], x] (* Bruno Berselli, Mar 11 2012 *)

A208950(n) := block(
        [a,npr] ,
        if equal(mod(n,4), 0) then (
                a : n/12*(n^2-1)
        ) else if equal(mod(n,2),0) then (
                a : (n-1)*n*(n+1)/6
        ) else (
                npr : (n-1)/2,
                a : npr*(npr+1)*n/6
        ) ,
        return(a)
)$ /* R. J. Mathar, Mar 10 2012 */

vector(50, n, n--; binomial(n+1,3)*gcd(n+2,4)/4) \\ G. C. Greubel, Sep 20 2018

a(n) = 4*a(n-4) - 6*a(n-8) + 4*a(n-12) - a(n-16).
a(n+1) = A002415(n+1)/A145979(n-1).
a(n) = A051724(n-1) * A051724(n) * A051724(n+1).
a(n) = A060819(n-1) * A060819(n) * A060819(n+1) / 3.
a(n) * a(n+4) = A061037(n+1) * A061037(n+2) * A061037(n+3) / 9.
a(n) = A138190(n)/A000034(n) for n > 0.
a(n) = A000292(n-1)/A176895(n+2) for n > 0.
a(n)/a(n+4) = n*(n^2-1)/((n+3)*(n+4)*(n+5)).
a(n)/a(n+12) = (n-1)*n*(n+1)/((n+11)*(n+12)*(n+13)).
G.f.: (x^2 + x^3 + 5*x^4 + 5*x^5 + 31*x^6 + 10*x^7 + 22*x^8 + 10*x^9 + 31*x^10 + 5*x^11 + 5*x^12 + x^13 + x^14) / ((1-x)^4*(1+x)^4*(1 + 4*x^2 + 6*x^4 + 4*x^6 + x^8)). - R. J. Mathar, Mar 10 2012
From Wolfdieter Lang, Feb 24 2014: (Start)
G.f.: (1 + x^12 + x*(1+x^10) + 5*x^2*(1+x^8) + 5*x^3*(1+x^7) + 31*x^4*(1+x^4) + 10*x^5*(1+x^2) + 22*x^6)/(1-x^4)^4. This is the preceding g.f. rewritten.
a(n) = binomial(n+1,3)*gcd(n+2,4)/4, n >= 0. From the g.f., see a comment above on A107711. (End)
a(n) = (n*(n-1)*((n+1)*(4+2*(-1)^n + (1+(-1)^n)*(-1)^((2*n+3+(-1)^n)/4))))/48. - Luce ETIENNE, Jan 01 2015
Sum_{n>=2} 1/a(n) = 12 - 27*log(2)/2. - Amiram Eldar, Aug 12 2022

A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.

Original entry on oeis.org

0, 1, 2, 0, 3, 0, 1, 2, 1, 2, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Omar E. Pol, Jul 11 2016

In the square array we have that:
Antidiagonal sums give A168237.
Odd-indexed rows give A010673.
Even-indexed rows give A010684.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed antidiagonals give the initial terms of A010674.
Even-indexed antidiagonals give the initial terms of A000034.
Main diagonal gives A010674.
This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
In the triangle we have that:
Row sums give A168237.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed diagonals give A010673.
Even-indexed diagonals give A010684.
Odd-indexed rows give the initial terms of A010674.
Even-indexed rows give the initial terms of A000034.
Odd-indexed antidiagonals give the initial terms of A010673.
Even-indexed antidiagonals give the initial terms of A010684.

			The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...

Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.

ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # Robert Israel, Nov 14 2016

Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274913(n) - 1.
From Robert Israel, Nov 14 2016: (Start)
G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).
G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)

cp's OEIS Frontend

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A115356 Matrix (1,x)+(x,x^2) in Riordan array notation.

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