A010709 -id:A010709 - cp's OEIS frontend

A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

1, 2, 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

David Applegate, Jun 29 2009

A010709 preceded by 1, 2.
Partial sums give A131098.
The INVERT transform gives A077996 without A077996(0). The Motzkin transform gives A105696 without A105696(0). Decimal expansion of 28/225=0.12444... . - R. J. Mathar, Jun 29 2009
Continued fraction expansion of 1 + sqrt(1/5). - Arkadiusz Wesolowski, Mar 30 2012
The number of solutions x (mod 2^(n+1)) of x^2 = 1 (mod 2^(n+1)), namely x = 1 (n=0), x = -1, 1 (n=1) and x = -1, 1, 2^n-1, 2^n+1 (n at least 2). - Christopher J. Smyth, May 15 2014
Also, the number of n-step self-avoiding walks on the L-lattice with no non-contiguous adjacencies (see A322419 for details of L-lattice). - Sean A. Irvine, Jul 29 2020

David Applegate, The movie version
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A010709, A131098, A077996, A105696.

[ n le 1 select n+1 else 4: n in [0..104] ];

f[n_] := Fold[#2*Floor[#1/#2 + 1/2] &, n, Reverse@ Range[n - 1]]; Array[f, 55]

Vec((1+x+2*x^2)/(1-x) + O(x^100)) \\ Altug Alkan, Jan 19 2016

G.f.: (1+x+2*x^2)/(1-x).
E.g.f. A(x)=x*B(x) satisfies the differential equation B'(x)=1+x+x^2+B(x). - Vladimir Kruchinin, Jan 19 2011
E.g.f.: 4*exp(x) - 2*x - 3. - Elmo R. Oliveira, Aug 06 2024

A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

Original entry on oeis.org

3, 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

Natan Arie Consigli, Feb 25 2015

See A054871 for definitions and key links.
Also, decimal expansion of 31/90. - Bruno Berselli, Mar 18 2015
Essentially the same as A010709, A040002, A113311, A123932, and A151798. - R. J. Mathar, Mar 20 2015
Remainder of the Euclidean division when 10^(10^n) is divided by 7 (proof by induction for n >= 1) [see reference Julien Freslon & Jérôme Poineau]; example: 10^(10^1) = 1428571428 * 7 + 4. - Bernard Schott, Aug 28 2020

			a(0) = H_0(2,2) = 2+1 = 3.
a(1) = H_1(2,2) = 2+2 = 4.
a(2) = H_2(2,2) = 2*2 = 4.
a(3) = H_3(2,2) = 2^2 = 4.
a(n) = H_n(2,2) = H_{n-1}(2,H_n(2,1)) = H_{n-1}(2,2) = 4, for n>1.

Julien Freslon & Jérôme Poineau, Les 100 exercices-types de mathématiques: MPSI/PCSI/PTSI, EdiScience, 2007, Exercice 11.2, page 242.

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

Cf. A054871.

Cf. A040000, A157532.

G.f.: (3 + x)/(1 - x). - Bruno Berselli, Mar 18 2015

a(n) = 10^(10^n) mod 7. - Bernard Schott, Aug 28 2020

Edited by Danny Rorabaugh, Oct 20 2015

A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

Original entry on oeis.org

1, 1, 1, 4, -1, 1, 0, 4, -5, 1, 24, -16, 4, -13, 1, -8, 40, -48, 4, -29, 1, 104, -88, 72, -112, 4, -61, 1, -48, 184, -248, 136, -240, 4, -125, 1, 352, -400, 344, -568, 264, -496, 4, -253, 1, 80, 544, -1104, 664, -1208, 520, -1008, 4, -509, 1, 1424, -784, 928, -2512, 1304, -2488, 1032, -2032, 4, -1021, 1

Offset: 1

Antti Karttunen, Nov 22 2019

			The top left corner of the array:
   n   p_n |k=1,     2, 3,      4,     5,      6,     7,       8,      9,      10
  ---------+----------------------------------------------------------------------
   1 ->  2 |  1,     1, 4,      0,    24,     -8,   104,     -48,    352,      80,
   2 ->  3 |  1,    -1, 4,    -16,    40,    -88,   184,    -400,    544,    -784,
   3 ->  5 |  1,    -5, 4,    -48,    72,   -248,   344,   -1104,    928,   -2512,
   4 ->  7 |  1,   -13, 4,   -112,   136,   -568,   664,   -2512,   1696,   -5968,
   5 -> 11 |  1,   -29, 4,   -240,   264,  -1208,  1304,   -5328,   3232,  -12880,
   6 -> 13 |  1,   -61, 4,   -496,   520,  -2488,  2584,  -10960,   6304,  -26704,
   7 -> 17 |  1,  -125, 4,  -1008,  1032,  -5048,  5144,  -22224,  12448,  -54352,
   8 -> 19 |  1,  -253, 4,  -2032,  2056, -10168, 10264,  -44752,  24736, -109648,
   9 -> 23 |  1,  -509, 4,  -4080,  4104, -20408, 20504,  -89808,  49312, -220240,
  10 -> 29 |  1, -1021, 4,  -8176,  8200, -40888, 40984, -179920,  98464, -441424,
  11 -> 31 |  1, -2045, 4, -16368, 16392, -81848, 81944, -360144, 196768, -883792,

Cf. A000079, A000203, A000225, A075708, A182944, A329610, A329644, A329890.
Rows 1-2: A329891, A329892 (from n>=1).
Column 1: A000012, Column 2: -A036563(n) (from n>=1), Column 3: A010709.

up_to = 105;
A329890(n) = if(1==n,1,sigma((2^n)-1)-sigma((2^(n-1))-1));
A329637sq(n,k) = ((2^(n+k-1)) - (((2^n)-1) * A329890(k)));
A329637list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A329637sq(col,(a-(col-1))))); (v); };
v329637 = A329637list(up_to);
A329637(n) = v329637[n];

A(n, k) = A329644(A182944(n, k)).

A(n, k) = A000079(n+k-1) - (A000225(n) * A329890(k)).

A177499 Period 4: repeat [1, 16, 4, 16].

Original entry on oeis.org

1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16

Offset: 0

Paul Curtz, May 10 2010

From Klaus Brockhaus, May 14 2010: (Start)
Interleaving of A000012, A010855, A010709 and A010855.
Continued fraction expansion of (44+sqrt(2442))/88. (End)

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

Cf. A010674, A177838. [Klaus Brockhaus, May 14 2010]

Cf. A000012, A010709, A010855, A176895.

&cat[[1, 16, 4, 16]^^26]; // Klaus Brockhaus, May 14 2010

seq(op([1, 16, 4, 16]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

Table[{1, 16, 4, 16}, {21}] // Flatten (* Jean-François Alcover, May 24 2013 *)

From Klaus Brockhaus, May 14 2010: (Start)
a(n+2) - a(n) = A010674(n).
a(n) = a(n-4) for n > 3.
G.f.: (1+16*x+4*x^2+16*x^3)/(1-x^4). (End)
a(n) = A176895(n)^2. - Paul Curtz, Mar 21 2011
a(n) = (37 - 6*cos(n*Pi/2) - 27*cos(n*Pi) - 27*I*sin(n*Pi))/4. - Wesley Ivan Hurt, Jul 09 2016

A226936 Number T(n,k) of squares of size k^2 in all tilings of an n X n square using integer-sided square tiles; triangle T(n,k), n >= 1, 1 <= k <= n, read by rows.

Original entry on oeis.org

1, 4, 1, 29, 4, 1, 312, 69, 4, 1, 5598, 1184, 153, 4, 1, 176664, 40078, 4552, 373, 4, 1, 9966344, 2311632, 285414, 18160, 917, 4, 1, 1018924032, 241967774, 30278272, 2128226, 74368, 2321, 4, 1, 190191337356, 45914039784, 5860964300, 411308056, 16210982, 311784, 5933, 4, 1

Offset: 1

Alois P. Heinz, Jun 22 2013

			For n=3 there are [29, 4, 1] squares of sizes [1^2, 2^2, 3^3] in all tilings of a 3 X 3 square:
._._._.  ._._._.  ._._._.  ._._._.  ._._._.  ._._._.
|     |  |   |_|  |_|_|_|  |_|   |  |_|_|_|  |_|_|_|
|     |  |___|_|  |   |_|  |_|___|  |_|   |  |_|_|_|
|_____|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |_|_|_|.
Triangle T(n,k) begins:
n \ k        1          2         3        4      5     6   7   8
--:----------------------------------------------------------------
1 :          1;
2 :          4,         1;
3 :         29,         4,        1;
4 :        312,        69,        4,       1;
5 :       5598,      1184,      153,       4,     1;
6 :     176664,     40078,     4552,     373,     4,    1;
7 :    9966344,   2311632,   285414,   18160,   917,    4,  1;
8 : 1018924032, 241967774, 30278272, 2128226, 74368, 2321,  4,  1;

Alois P. Heinz, Rows n = 1..15, flattened

Row sums give: A226554.
Main diagonal and lower diagonals give: A000012, A010709, A226892.
Cf. A045846.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0$2] elif n=0 then [1, 0]
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:= s+(h->h+
           [0, h[1]*x^(1+i-k)])(b(n, [l[j]$j=1..k-1,
           1+i-k$j=k..i, l[j]$j=i+1..nops(l)])) od; s
      fi
    end:
T:= n-> seq(coeff(b(n, [0$n])[2],x,k), k=1..n):
seq(T(n), n=1..10);

$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0, {1, 0}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1, 1][[1, 1]]; s = {0, 0}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h + {0, h[[1]]*x^(1+i-k)}][b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ] ] ]; s] ]; T[n_] := Table[Coefficient[b[n, Array[0&, n]][[2]], x, k], {k, 1, n}]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

Sum_{k=1..n} T(n,k) = A226554(n).

Sum_{k=1..n} k^2 * T(n,k) = n^2 * A045846(n).

A168309 Period 2: repeat 4,-3.

Original entry on oeis.org

4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3, 4, -3

Offset: 1

Klaus Brockhaus, Nov 22 2009

Interleaving of A010709 and -3*A000012.
Binomial transform of 4 followed by a signed version of A005009.
Inverse binomial transform of 4 followed by A000079.
a(n+1) - a(n) = 7*(-1)^n.
A168230 without initial term 0 gives partial sums.
Nonsimple continued fraction expansion of 2+2*sqrt(2/3) = 3.6329931618... - R. J. Mathar, Mar 08 2012

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

Cf. A010709 (all 4's sequence), A000012 (all 1's sequence), A010727 (all 7's sequence), A168230, A005009 (7*2^n), A000079 (powers of 2).

&cat[ [4, -3]: n in [1..42] ];
[ n eq 1 select 4 else -Self(n-1)+1: n in [1..84] ];

LinearRecurrence[{0,1},{4, -3}, 50] (* or *) Table[(1 - 7*(-1)^n)/2,{n,0,25}] (* G. C. Greubel, Jul 17 2016 *)
PadRight[{},120,{4,-3}] (* Harvey P. Dale, Oct 20 2018 *)

a(n) = (1 - 7*(-1)^n)/2.
a(n) = -a(n-1) + 1 for n > 1; a(1) = 4.
a(n) = a(n-2) for n > 2; a(1) = 4, a(2) = -3.
G.f.: x*(4 - 3*x)/((1-x)*(1+x)).
E.g.f.: (1/2)*(-1 + exp(x))*(7 + exp(x))*exp(-x). - G. C. Greubel, Jul 17 2016

A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

Original entry on oeis.org

0, -1, -1, -4, -5, -2, -7, -8, -3, -10, -11, -4, -13, -14, -5, -16, -17, -6, -19, -20, -7, -22, -23, -8, -25, -26, -9, -28, -29, -10, -31, -32, -11, -34, -35, -12, -37, -38, -13, -40, -41, -14, -43, -44, -15, -46, -47, -16, -49, -50, -17, -52, -53, -18, -55, -56, -19, -58, -59, -20

Offset: 0

Paul Curtz, Sep 07 2011

0,  -1, -1, -4/3, -5/3, -2, -7/3, -8/3, -3, -10/3, -11/3, -4, -13/4, -14/3, -5,    = a(n)/b(n),
1,    0,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
1,   -2, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
3,   -2,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
5,   -6, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
11, -10,  1,  4/3,  5/3,  2,  7/3,  8/3,  3,
21, -22, -1, -4/3, -5/3, -2, -7/3, -8/3, -3,
Vertical: A001045(n), -A078008(n), (-1)^(n+1)*A000012(n), (-1)^(n+1)*A010709(n)/A010701(n), (-1)^(n+1)*A010716(n+1)/A010701(n), A007395(n), .. .
a(n)=0, 1 before (-A145064(n+1)=-A051176(n+3).
b(n)=1, 1 before A169609(n). b(n)=1, 1, 1 before A144437(n+1).
a(n+5)-a(n+2)=b(n+5)  (=-1,-3,-3,=-A169609(n)).

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. Merlini, R. Sprugnoli, and M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

a[0]=0; a[1]=-1; a[n_] := (-n-1)/Max[1, 2*Mod[n, 3]-1]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Sep 18 2012 *)

a(3*n)=-3*n-1 except a(0)=0; a(3*n+1)=-3*n-2 except a(1)=-1; a(3*n+2)=-n-1.
From Chai Wah Wu, May 07 2024: (Start)
a(n) = 2*a(n-3) - a(n-6) for n > 7.
G.f.: x*(x^6 + x^5 - 3*x^3 - 4*x^2 - x - 1)/(x^6 - 2*x^3 + 1). (End)

A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2n + 5, a(4n+2) = 2n + 3.

Original entry on oeis.org

1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 3, 17, 9, 19, 5, 21, 11, 23, 5, 25, 13, 27, 7, 29, 15, 31, 7, 33, 17, 35, 9, 37, 19, 39, 9, 41, 21, 43, 11, 45, 23, 47, 11, 49, 25, 51, 13, 53, 27, 55, 13, 57, 29, 59, 15, 61, 31, 63, 15, 65, 33, 67

Offset: 0

Paul Curtz, Sep 21 2014

Essentially a permutation of A129756 (odd numbers repeated four times).
a(-1) = 3, a(-2) = a(-3) = 1.
Distance between the first two (2*k+1)'s: 2*k+1 terms. Distance between the last two (2*n+1)'s: 4 terms. Essentially same distances as in -a(-n) = -1, -3, -1, -1, 1, 1, 1, 3, 1, 5, 3, 7, 3, 9, 5, 11, 3, 13, 7, 15, 5, 17, 9, 19, 5, 21, 11, 23, 7, 25, 13, 27, 7, ... .

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,0,0,1,0,0,0,-1).

Cf. A010709, A061037, A121262, A129756, A176895, A246416.

I:=[1,5,3,7,1,9,5,11,3,13,7,15]; [n le 12 select I[n] else Self(n-4)+Self(n-8)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Oct 15 2014

A247617:=n->(n+4)*(1-ceil((2-n)/4)-ceil((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2*(-1)^(n/4))*(ceil(n/4)-floor(n/4)-1)/4: seq(A247617(n), n=0..50); # Wesley Ivan Hurt, Sep 21 2014

Table[(n + 4) (1 - Ceiling[(2 - n)/4] - Ceiling[(n - 2)/4])/2 + (n + 4) (1 + Floor[(1 - n)/2] + Floor[(n - 1)/2]) - (n + 2 + 2 (-1)^(n/4)) (Ceiling[n/4] - Floor[n/4] - 1)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Sep 21 2014 *)

Vec(-(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1)/((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)) + O(x^100)) \\ Colin Barker, Sep 21 2014

a(n) = a(n-4) + a(n-8) - a(n-12).
a(n) * A246416(n) = A061037(n+2).
A246416(n+4) - a(n) = sequence of period 4: [1, 0, 0, 0].
a(n+4) - a(n) = sequence of period 8: [0, 4, 2, 4, 2, 4, 2, 4].
G.f.: -(3*x^11+x^10+x^9-x^8-4*x^7-2*x^6-4*x^5-7*x^3-3*x^2-5*x-1) / ((x-1)^2*(x+1)^2*(x^2+1)^2*(x^4+1)). - Colin Barker, Sep 21 2014
a(n) = a(n-8) + sequence of period 4: [2, 8, 4, 8] (= 2*A176895(n)).
a(-n) * A246416(-n) = A061037(n-2).
a(n) = (n+4)*(1-ceiling((2-n)/4)-ceiling((n-2)/4))/2+(n+4)*(1+floor((1-n)/2)+floor((n-1)/2))-(n+2+2(-1)^(n/4))*(ceiling(n/4)-floor(n/4)-1)/4. - Wesley Ivan Hurt, Sep 21 2014

A328839 Smallest prime not dividing n times smallest prime not dividing A276087(n).

Original entry on oeis.org

4, 15, 4, 21, 4, 25, 4, 21, 4, 33, 4, 15, 4, 33, 4, 21, 4, 25, 4, 39, 4, 39, 4, 15, 4, 39, 4, 51, 4, 21, 4, 21, 4, 15, 4, 25, 4, 33, 4, 39, 4, 15, 4, 39, 4, 39, 4, 25, 4, 51, 4, 51, 4, 15, 4, 51, 4, 33, 4, 21, 4, 33, 4, 21, 4, 25, 4, 39, 4, 39, 4, 15, 4, 51, 4, 51, 4, 25, 4, 51, 4, 57, 4, 15, 4, 39, 4, 57, 4, 21, 4, 39, 4, 51, 4

Offset: 1

Antti Karttunen, Oct 29 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to primorial numbers

Cf. A010709 (odd bisection), A053669, A276086, A276087, A328578, A328579.

Cf. A328585 (positions of the prime squares).

A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328579(n) = A053669(A276086(A276086(n)));
A328839(n) = (A053669(n) * A328579(n));

a(n) = A053669(n) * A328579(n).

A268741 a(n) = 2*a(n - 2) - a(n - 1) for n>1, a(0) = 4, a(1) = 5.

Original entry on oeis.org

4, 5, 3, 7, -1, 15, -17, 47, -81, 175, -337, 687, -1361, 2735, -5457, 10927, -21841, 43695, -87377, 174767, -349521, 699055, -1398097, 2796207, -5592401, 11184815, -22369617, 44739247, -89478481, 178956975, -357913937, 715827887, -1431655761, 2863311535

Offset: 0

Ilya Gutkovskiy, Feb 12 2016

In general, the ordinary generating function for the recurrence relation b(n) = 2*b(n - 2) - b(n - 1) with n>1 and b(0)=k, b(1)=m, is (k + (k + m)*x)/(1 + x - 2*x^2). This recurrence gives the closed form a(n) = ((-2)^n*(k - m) + 2*k + m).

			a(0) = (5 + 3)/2 = 4  because a(1) = 5, a(2) = 3;
a(1) = (3 + 7)/2 = 5  because a(2) = 3, a(3) = 7;
a(2) = (7 - 1)/2 = 3  because a(3) = 7, a(4) = -1, etc.

Ilya Gutkovskiy, Extended graphical example
Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A084247, A140683, A140966, A154570, A171382.

Cf. A001045, A010709, A077925.

[(13-(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Feb 13 2016

Table[(13 - (-2)^n)/3, {n, 0, 33}]
LinearRecurrence[{-1, 2}, {4, 5}, 34]
RecurrenceTable[{a[1] == 4, a[2] == 5, a[n] == 2*a[n-2] - a[n-1]}, a, {n, 50}] (* Vincenzo Librandi, Feb 13 2016 *)

Vec((4 + 9*x)/(1 + x - 2*x^2) + O(x^40)) \\ Michel Marcus, Feb 25 2016

G.f.: (4 + 9*x)/(1 + x - 2*x^2).
a(n) = (13 - (-2)^n)/3.
a(n) = A084247(n) + 3.
a(n) = (-1)^n*A154570(n+1) + 1.
a(n) = (-1)^(n-1)*A171382(n-1) + 2.
Limit_{n -> oo} a(n)/a(n + 1) = -1/2.
a(n) = 4 - (-1)^n *A001045(n). - Paul Curtz, Feb 26 2024

cp's OEIS Frontend

A151798 a(0)=1, a(1)=2, a(n)=4 for n>=2.

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A255176 a(n) = H_n(2,2) where H_n is the n-th hyperoperator.

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A329637 Square array A(n, k) = A329644(prime(n)^k), read by falling antidiagonals: (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

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A177499 Period 4: repeat [1, 16, 4, 16].

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A226936 Number T(n,k) of squares of size k^2 in all tilings of an n X n square using integer-sided square tiles; triangle T(n,k), n >= 1, 1 <= k <= n, read by rows.

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A168309 Period 2: repeat 4,-3.

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A194880 The numerators of the inverse Akiyama-Tanigawa algorithm from A001045(n).

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A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2*n + 5, a(4n+2) = 2*n + 3.

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A247617 a(4n) = n + 1/2 - (-1)^n/2 + (-1)^n, a(2n+1) = 2n + 5, a(4n+2) = 2n + 3.