A005015 -id:A005015 - cp's OEIS frontend

A112651 Numbers k such that k^2 == k (mod 11).

0, 1, 11, 12, 22, 23, 33, 34, 44, 45, 55, 56, 66, 67, 77, 78, 88, 89, 99, 100, 110, 111, 121, 122, 132, 133, 143, 144, 154, 155, 165, 166, 176, 177, 187, 188, 198, 199, 209, 210, 220, 221, 231, 232, 242, 243, 253, 254, 264, 265, 275, 276, 286, 287, 297, 298

Offset: 1

Jeremy Gardiner, Dec 28 2005

Numbers that are congruent to {0,1} (mod 11). - Philippe Deléham, Oct 17 2011

			12 is a term because 12*12 = 144 == 1 (mod 11) and 12 == 1 (mod 11).

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

Cf. A010880 (n mod 11), A070434 (n^2 mod 11).

Cf. A005015, A030308.

m = 11 for n = 1 to 300 if n^2 mod m = n mod m then print n; next n

Select[Range[0,300],PowerMod[#,2,11]==Mod[#,11]&] (* or *) LinearRecurrence[ {1,1,-1},{0,1,11},60] (* Harvey P. Dale, Apr 19 2015 *)

a(n)=11*n/2-31/4-9*(-1)^n/4 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = 11*n - a(n-1) - 21 (with a(1)=0). - Vincenzo Librandi, Nov 13 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 11*n/2 - 31/4 - 9*(-1)^n/4.
G.f.: x^2*(1+10*x) / ( (1+x)*(x-1)^2 ). (End)
a(n+1) = Sum_{k>=0} A030308(n,k)*A005015(k-1) with A005015(-1)=1. - Philippe Deléham, Oct 17 2011

Edited by N. J. A. Sloane, Aug 19 2010

Definition clarified by Harvey P. Dale, Apr 19 2015

A108732 a(0)=22; if n odd, a(n) = a(n-1)/2, otherwise a(n) = 4*a(n-1).

Original entry on oeis.org

22, 11, 44, 22, 88, 44, 176, 88, 352, 176, 704, 352, 1408, 704, 2816, 1408, 5632, 2816, 11264, 5632, 22528, 11264, 45056, 22528, 90112, 45056, 180224, 90112, 360448, 180224, 720896, 360448, 1441792, 720896

Offset: 0

Alexandre Wajnberg & Guadalupe Garcia, Jun 22 2005

A108213 is a subsequence of this sequence and is also twice this sequence.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2)

Cf. A005015, A108213.

nxt[{n_,a_}]:={n+1,If[EvenQ[n],a/2,4a]}; NestList[nxt,{0,22},40][[All,2]] (* or *) LinearRecurrence[{0,2},{22,11},40] (* Harvey P. Dale, Jul 22 2021 *)

a(2n+1) = a(2n-2).
Recurrence: a(n) = 2a(n-2), a(0)=22, a(1)=11. G.f.: (22x+44)/(1-2x^2). - Ralf Stephan, Jul 16 2013
a(n) = 11 * 2^A028242(n). - Franklin T. Adams-Watters, Mar 29 2006

A140950 a(n) = A140944(n+1) - 3*A140944(n).

Original entry on oeis.org

1, -3, -1, 5, -6, 3, -11, 10, -12, -5, 21, -22, 20, -24, 11, -43, 42, -44, 40, -48, -21, 85, -86, 84, -88, 80, -96, 43, -171, 170, -172, 168, -176, 160, -192, -85, 341, -342, 340, -344, 336, -352, 320, -384, 171, -683, 682, -684, 680, -688

Offset: 0

Paul Curtz, Jul 25 2008

Jacobsthal numbers appear twice: 1) A001045(n+2) signed, terms 0, 1, 3, 6, 10 (A000217); 2) A001045(n+1) signed, terms 0, 2, 5, 9 (n*(n+3)/2=A000096); between them are -3; 5, -6; -11, 10, -12; which appear (opposite sign) by rows in A140503 (1, -1, 2, 3, -2, 4) square.
Consider the permutation of the nonnegative numbers
0, 2, 5,  9, 14, 20, 27,
1, 3, 6, 10, 15, 21, 28,
   4, 7, 11, 16, 22, 29,
      8, 12, 17, 23, 30,
         13, 18, 24, 31,
             19, 25, 32,
                 26, 33,
                     34, etc.
The corresponding distribution of a(n) is
1,  -1,   3,  -5,  11, -21,   43,
-3,  5, -11,  21, -43,  85, -171,
    -6,  10, -22,  42, -86,  170,
        -12,  20, -44,  84, -172,
             -24,  40, -88,  168,
                  -48,  80, -176,
                       -96,  160,
                            -192, etc.
Column sums: -2, -2, -10, -10, -42, -42, -170, ... duplicate of a bisection of -A078008(n+2).
b(n)= 1, -1, 3, -5, 11, 21, ... = (-1)^n*A001045(n+1) = A077925(n). Every row is b(n) or b(n+2) multiplied by 1, -1, -2, -4, -8, -16, ..., essentially -A011782(n).

Cf. A000096, A000217, A005015, A001045, A007283, A020988, A077925, A078008, A140503, A146523, A151575, A175805, A084247.

T[0, 0] = 0; T[1, 0] = T[0, 1] = 1; T[0, n_] := T[0, n] = T[0, n - 1] + 2*T[0, n - 2]; T[d_, d_] = 0; T[d_, n_] := T[d, n] = T[d - 1, n + 1] - T[d - 1, n]; A140944 = Table[T[d, n], {d, 0, 10}, {n, 0, d}] // Flatten; a[n_] := A140944[[n + 2]] - 3*A140944[[n + 1]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 18 2014 *)

More terms and a(19)=-48 instead of 42 corrected by Jean-François Alcover, Dec 22 2014

A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

Original entry on oeis.org

4, 7, 15, 29, 59, 117, 235, 469, 939, 1877, 3755, 7509, 15019, 30037, 60075, 120149, 240299, 480597, 961195, 1922389, 3844779, 7689557, 15379115, 30758229, 61516459, 123032917, 246065835, 492131669, 984263339, 1968526677, 3937053355, 7874106709, 15748213419, 31496426837

Offset: 0

Paul Curtz, Aug 19 2025

Index entries for linear recurrences with constant coefficients, signature (1,2).

Cf. A005015, A052997, A340627.

Bisections give A199210 and A072261.

a[n_] := (11*2^n + (-1)^n)/3; Array[a, 34, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n) = (11*2^n + (-1)^n)/3.
a(n) = A340627(n+1)/2.
a(n) = 2*A052997(n) + 1 for n >= 1.
a(n) = a(n-4) + 55*2^(n-4) for n >= 4.
G.f.: (3*x + 4)/((x + 1)*(1 - 2*x)).
E.g.f: (11*exp(2*x) + exp(-x))/3.

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A112651 Numbers k such that k^2 == k (mod 11).

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A108732 a(0)=22; if n odd, a(n) = a(n-1)/2, otherwise a(n) = 4*a(n-1).

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A140950 a(n) = A140944(n+1) - 3*A140944(n).

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A383953 a(0) = 4, a(n) = 2*a(n-1) + (-1)^n.

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