A133872 -id:A133872 - cp's OEIS frontend

A255125 Number of times a multiple of four is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x).

1, 0, 1, 1, 1, 3, 6, 13, 26, 47, 81, 140, 253, 482, 949, 1875, 3666, 7088, 13614, 26100, 50082, 96246, 185131, 356123, 684758, 1316197, 2530257, 4868019, 9378335, 18096921, 34974646, 67669905, 130998912, 253565649, 490501587, 947992195, 1830664188, 3533571444

Offset: 0

Antti Karttunen, Feb 18 2015

nonn

Also the number of even numbers in range [A255062(n) .. A255061(n+1)] of A255057 (equally, in A255067). See the sum-formulas.

			For n=5 we start iterating with map m(n) = A236840(n) from the initial value (2^(5+1))-2 = 62. Thus we get m(62) = 60, m(60) = 58, m(58) = 54, m(54) = 50, m(50) = 46, m(46) = 42, m(42) = 36, m(36) = 32 and finally m(32) = 30, which is (2^5)-2. Of the nine numbers encountered, only 60, 36 and 32 are multiples of four, thus a(5) = 3.

A. Karttunen, Ratio a(n)/A255126(n) plotted, with the help of OEIS-server

Cf. A005811, A059841, A133872, A236840, A255056, A255057, A255061, A255062, A255067, A255071, A255126, A255332.

Similar sequences: A218542, A255063.

A005811(n) = hammingweight(bitxor(n, n\2));
write_A255125_and_A255126_and_A255071(n) = { my(k, i, s25, s26); k = (2^(n+1))-2; i = 1; s25 = 0; s26 = 0; while(1, if((k%4),s26++,s25++); k = k - A005811(k); if(!bitand(k+1, k+2), break, i++)); write("b255125.txt", n, " ", s25); write("b255126.txt", n, " ", s26); write("b255071.txt", n, " ", i); };
for(n=1,42,write_A255125_and_A255126_and_A255071(n));

(define (A255125 n) (if (zero? n) 1 (let loop ((i (- (expt 2 (+ 1 n)) 4)) (s 0)) (cond ((pow2? (+ 2 i)) s) (else (loop (- i (A005811 i)) (+ s (A133872 i))))))))
;; Alternatively:
(define (A255125 n) (add (COMPOSE A059841 A255057) (A255062 n) (A255061 (+ 1 n))))
(define (A255125 n) (add (COMPOSE A059841 A255067) (A255062 n) (A255061 (+ 1 n))))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))

a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255057(k)).
a(n) = Sum_{k = A255062(n) .. A255061(n+1)} A059841(A255067(k)).
a(n) = A255071(n) - A255126(n).

A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

0, 4, 6, 8, 10, 12, 14, 12, 12, 18, 22, 19, 26, 22, 19, 16, 34, 22, 38, 22, 23, 32, 46, 23, 20, 36, 18, 26, 58, 37, 62, 20, 34, 46, 29, 29, 74, 50, 38, 31, 82, 38, 86, 36, 30, 58, 94, 30, 28, 32, 46, 40, 106, 30, 37, 37, 50, 70, 118, 41, 122, 74, 36, 24, 41, 48, 134, 50, 58, 50

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(6)=12, since A133900(6)=72=2*2*2*3*3 and 2+2+2+3+3=12.
a(12)=19, since A133900(12)=864=2*2*2*2*2*3*3*3 and 2+2+2+2+2+3+3+3=19.

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A133911.

A286074 Number of permutations of [n] with nondecreasing cycle sizes.

Original entry on oeis.org

1, 1, 2, 4, 13, 45, 250, 1342, 10085, 76165, 715588, 6786108, 78636601, 896672473, 12112535378, 163963519810, 2534311844905, 39211836764457, 688584972407680, 12003902219337760, 234324625117308533, 4571805253649981173, 98183221386947058286

Offset: 0

Alois P. Heinz, May 01 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

a(n) mod 2 = A133872(n).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Cf. A133872, A275311, A286071, A286072, A286073, A286075, A286076, A286077.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      (j-1)!*b(n-j, j)*binomial(n-1, j-1), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, i, n}]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 25 2018, translated from Maple *)

A182432 Recurrence a(n)a(n-2) = a(n-1)(a(n-1) + 3) with a(0) = 1, a(1) = 4.

Original entry on oeis.org

1, 4, 28, 217, 1705, 13420, 105652, 831793, 6548689, 51557716, 405913036, 3195746569, 25160059513, 198084729532, 1559517776740, 12278057484385, 96664942098337, 761041479302308, 5991666892320124, 47172293659258681, 371386682381749321

Offset: 0

Peter Bala, Apr 30 2012

The non-linear recurrence equation a(n)*a(n-2) = a(n-1)*(a(n-1) + r) with initial conditions a(0) = 1, a(1) = 1 + r has the solution a(n) = 1/2 + (1/2)*Sum_{k = 0..n} (2*r)^k*binomial(n+k,2*k) = 1/2 + b(n,2*r)/2, where b(n,x) are the Morgan-Voyce polynomials of A085478. The recurrence produces sequences A101265 (r = 1), A011900 (r = 2) and A054318 (r = 4), as well as signed versions of A133872 (r = -1), A109613 (r = -2), A146983 (r = -3) and A084159(r = -4).
Also the indices of centered pentagonal numbers (A005891) which are also centered triangular numbers (A005448). - Colin Barker, Jan 01 2015
Also positive integers y in the solutions to 3*x^2 - 5*y^2 - 3*x + 5*y = 0. - Colin Barker, Jan 01 2015

Seiichi Manyama, Table of n, a(n) for n = 0..1116 (first 201 terms from Vincenzo Librandi)
Giovanni Lucca, Circle Chains Inscribed in Symmetrical Lenses and Integer Sequences, Forum Geometricorum, Volume 16 (2016) 419-427.
Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Cf. A011900, A054318, A070997, A101265, A109613, A133872, A146983, A211955.

I:=[1, 4, 28]; [n le 3 select I[n] else 9*Self(n-1)-9*Self(n-2)+Self(n-3): n in [1..25]]; // Vincenzo Librandi, May 18 2012

RecurrenceTable[{a[0]==1,a[1]==4,a[n]==(a[-1+n] (3+a[-1+n]))/a [-2+n]}, a[n],{n,30}] (* or *) LinearRecurrence[{9,-9,1},{1,4,28},30] (* Harvey P. Dale, May 14 2012 *)

Vec((1-5*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^100)) \\ Colin Barker, Jan 01 2015

a(n) = 1/2 + (1/2)*Sum_{k = 0..n} 6^k*binomial(n+k,2*k).
a(n) = R(n,3) where R(n,x) denotes the row polynomials of A211955.
a(n) = (1/u)*T(n,u)*T(n+1,u) with u = sqrt(5/2) and T(n,x) the n-th Chebyshev polynomial of the first kind.
Recurrence equation: a(n) = 8*a(n-1) - a(n-2) - 3 with a(0) = 1 and a(1) = 4.
O.g.f.: (1 - 5*x + x^2)/((1 - x)*(1 - 8*x + x^2)) = 1 + 4*x + 28*x^2 + ....
Sum_{n >= 0} 1/a(n) = sqrt(5/3); 5 - 3*(Sum_{k = 0..2*n} 1/a(k))^2 = 2/A070997(n)^2.
a(0) = 1, a(1) = 4, a(2) = 28, a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3). - Harvey P. Dale, May 14 2012

A219977 Expansion of 1/(1+x+x^2+x^3).

Original entry on oeis.org

1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0

Offset: 0

Harvey P. Dale, Dec 02 2012

			G.f. = 1 - x + x^4 - x^5 + x^8 - x^9 + x^12 - x^13 + x^16 - x^17 + x^20 - x^21 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785, 2016
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1).

Cf. A036765, A038505, A049347, A077962, A133872.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1+x+x^2+x^3))); // Vincenzo Librandi, Apr 22 2015

CoefficientList[Series[1/(1+x+x^2+x^3),{x,0,80}],x] (* or *) PadRight[{},120,{1,-1,0,0}]
LinearRecurrence[{-1,-1,-1},{1,-1,0},80] (* Harvey P. Dale, May 22 2021 *)

{a(n) = [1, -1, 0, 0][n%4 + 1]} /* Michael Somos, Dec 12 2012 */

Vec(1/(1+x+x^2+x^3) + O(x^100)) \\ Michel Marcus, Jan 28 2016

G.f.: 1/(1 +x +x^2 +x^3).
Euler transform of length 4 sequence [ -1, 0, 0, 1]. - Michael Somos, Dec 12 2012
a(n) = a(n+4) = -a(1-n). |a(n)| = A133872(n). REVERT transform is A036765. INVERT transform is A077962. - Michael Somos, Dec 12 2012
A038505(n+2) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Dec 12 2012
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) +a(n-1) +a(n-2) +a(n-3) = 0.
a(n) = (-1)^n/2 +(-1)^(n/2 +1/4 -(-1)^n/4)/2. (End)

A133882 a(n) = binomial(n+2,n) mod 2^2.

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 2^3 = 8.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1, 1, -1, 1).

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133872, A133880, A133890, A133900, A133910.
For the sequence regarding "binomial(n+2, n) mod 2" see A133872.
A105198 shifted once left.

Table[Mod[Binomial[n+2,n],4],{n,0,120}]  (* Harvey P. Dale, Apr 16 2011 *)

A133882(n) = (binomial(n+2,n) % 4); \\ Antti Karttunen, Aug 10 2017

a(n) = binomial(n+2,2) mod 2^2.
G.f.: (1 + 3*x + 2*x^2 + 2*x^3 + 3*x^4 + x^5)/(1-x^8).
G.f.: (1+x)*(1+2*x+2*x^3+x^4)/(1-x^8) = (1+2*x+2*x^3+x^4)/((1-x)*(1+x^2)*(1+x^4)).
a(n) = A105198(n+1). - R. J. Mathar, Jun 08 2008

A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 3, 3, 11, 2, 13, 3, 3, 2, 17, 2, 19, 2, 4, 4, 23, 2, 5, 5, 3, 3, 29, 3, 31, 2, 5, 5, 5, 2, 37, 6, 6, 2, 41, 3, 43, 4, 3, 7, 47, 2, 7, 3, 7, 4, 53, 2, 7, 3, 8, 8, 59, 2, 61, 9, 4, 2, 8, 3, 67, 5, 9, 3, 71, 2, 73, 9, 4, 5, 8, 4, 79, 2, 3, 9, 83, 3, 9, 11, 10, 3, 89, 2, 9, 6, 11, 12

Offset: 1

Hieronymus Fischer, Oct 23 2007

nonn

			a(6)=2, since floor(A134331(6)/A133911(6))=floor(12/5)=2.
a(7)=7, since floor(A134331(7)/A133911(7))=floor(14/2)=7.

Cf. A000040, A100118, A046363, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133910, A133911, A134331.

a(n)=floor(A134331(n)/A133911(n)) for n>1, defining a(1):=1.

a(n)=n, if n is a prime or 1.

A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 4, 6, 12, 15, 24, 28, 40, 45, 60, 66, 84, 91, 112, 120, 144, 153, 180, 190, 220, 231, 264, 276, 312, 325, 364, 378, 420, 435, 480, 496, 544, 561, 612, 630, 684, 703, 760, 780, 840, 861, 924, 946, 1012, 1035, 1104, 1128, 1200, 1225, 1300, 1326, 1404, 1431

Offset: 0

Robert Price, Dec 05 2015

Take the first 2n positive integers and choose n of them such that their sum: a) is divisible by n, and b) is minimal. It seems their sum equals a(n). - Ivan N. Ianakiev, Feb 16 2019

			From _Michael De Vlieger_, Dec 14 2015: (Start)
First 12 rows, replacing "0" with "." for better visibility of ON cells, followed by the total number of 1's per row, and the running total up to that row:
                        1                          =  1 ->  1
                      1 1 1                        =  3 ->  4
                    1 . . . 1                      =  2 ->  6
                  1 1 1 . 1 1 1                    =  6 -> 12
                1 . . . 1 . . . 1                  =  3 -> 15
              1 1 1 . 1 1 1 . 1 1 1                =  9 -> 24
            1 . . . 1 . . . 1 . . . 1              =  4 -> 28
          1 1 1 . 1 1 1 . 1 1 1 . 1 1 1            = 12 -> 40
        1 . . . 1 . . . 1 . . . 1 . . . 1          =  5 -> 45
      1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1        = 15 -> 60
    1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1      =  6 -> 66
  1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1 . 1 1 1    = 18 -> 84
1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1 . . . 1  =  7 -> 91
(End)

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

Robert Price, Table of n, a(n) for n = 0..999
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche (2021) Vol. 76, No. 1, see p. 301.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A071030, A118108, A118109, A133872.

A265225:=n->1/4*(n+1)*(2*n-(-1)^n+5): seq(A265225(n), n=0..60); # Wesley Ivan Hurt, Dec 25 2016

rule = 54; rows = 30; Table[Total[Take[Table[Total[Table[Take[CellularAutomaton[rule,{{1},0},rows-1,{All,All}][[k]],{rows-k+1,rows+k-1}],{k,1,rows}][[k]]],{k,1,rows}],k]],{k,1,rows}]
Accumulate[Total /@ CellularAutomaton[54, {{1}, 0}, 52]]

Conjectures from Colin Barker, Dec 08 2015 and Apr 20 2019: (Start)
a(n) = (n+1)*(2*n -(-1)^n +5)/4.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
G.f.: (1+3*x) / ((1-x)^3*(1+x)^2).
(End)
a(n) = n + 1 + (n+1) * floor((n+1)/2), conjectured. - Wesley Ivan Hurt, Dec 25 2016
a(n) = A093353(n) + n + 1, conjectured. - Matej Veselovac, Jan 21 2020

A133906 Least number m such that binomial(n+m, m) mod m = 1.

Original entry on oeis.org

2, 3, 5, 2, 2, 7, 9, 2, 2, 3, 3, 2, 2, 17, 17, 2, 2, 3, 3, 2, 2, 23, 25, 2, 2, 4, 3, 2, 2, 31, 37, 2, 2, 8, 8, 2, 2, 3, 41, 2, 2, 4, 4, 2, 2, 3, 3, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 4, 4, 2, 2, 67, 3, 2, 2, 44, 44, 2, 2, 16, 16, 2, 2, 3, 4, 2, 2, 5, 5, 2, 2, 3, 3, 2, 2, 89, 9, 2, 2, 3, 3, 2, 2, 97, 97, 2, 2, 7

Offset: 1

Hieronymus Fischer, Oct 20 2007

nonn

			a(1)=2, since binomial(1+2, 2) mod 2 = 3 mod 2 = 1 and 2 is the minimal number with this property.
a(7)=9 because of binomial(7+9, 9) = 11440 = 1271*9 + 1, but binomial(7+k, k) mod k <> 1 for all numbers < 9.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A000040, A133620, A133621, A133623, A133630, A133635.

Cf. A133872, A133880, A133890, A133900, A133907, A133910.

Table[Block[{m = 1}, While[Mod[Binomial[n + m, m], m] != 1, m++]; m], {n, 98}] (* Michael De Vlieger, Jul 30 2018 *)

a(n) = {my(m = 1, ok = 0); until (ok, if (binomial(n+m, m) % m == 1, ok = 1, m++);); return (m);} \\ Michel Marcus, Jul 15 2013

A168511 Triangle T(n,k), read by rows, given by [0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,...] DELTA [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 4, 8, 0, 0, 4, 10, 12, 16, 0, 0, 8, 24, 36, 32, 32, 0, 0, 16, 56, 101, 112, 80, 64, 0, 0, 32, 128, 270, 360, 320, 192, 128, 0, 0, 64, 288, 696, 1086, 1160, 864, 448, 256, 0, 0, 128, 640, 1744, 3120, 3900, 3488, 2240, 1024, 512

Offset: 0

Philippe Deléham, Nov 28 2009

			Triangle begins:
  1;
  0, 1;
  0, 0,  2;
  0, 0,  1,  4;
  0, 0,  2,  4,   8;
  0, 0,  4, 10,  12,  16;
  0, 0,  8, 24,  36,  32, 32;
  0, 0, 16, 56, 101, 112, 80, 64;
  ...

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

Cf. A021913, A133872.

Sum_{k=0..n} T(n,k)*x^(n-k) = A001405(n), A011782(n), A000108(n), A168490(n), A168492(n) for x = -1,0,1,2,3 respectively.

Corrected and extended by Philippe Deléham, Mar 20 2013

cp's OEIS Frontend

A255125 Number of times a multiple of four is encountered when iterating from 2^(n+1)-2 to (2^n)-2 with the map x -> x - (number of runs in binary representation of x).

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PARI

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A134331 Sum of prime factors (counted with multiplicity) of the period numbers defined by A133900.

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A286074 Number of permutations of [n] with nondecreasing cycle sizes.

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Maple

Mathematica

A182432 Recurrence a(n)*a(n-2) = a(n-1)*(a(n-1) + 3) with a(0) = 1, a(1) = 4.

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Magma

Mathematica

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A219977 Expansion of 1/(1+x+x^2+x^3).

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Magma

Mathematica

PARI

PARI

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A133882 a(n) = binomial(n+2,n) mod 2^2.

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Mathematica

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A134332 Integer part of the arithmetic mean of the prime factors (counted with multiplicity) of the period numbers defined by A133900.

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A265225 Total number of ON (black) cells after n iterations of the "Rule 54" elementary cellular automaton starting with a single ON (black) cell.

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A182432 Recurrence a(n)a(n-2) = a(n-1)(a(n-1) + 3) with a(0) = 1, a(1) = 4.