A017653 -id:A017653 - cp's OEIS frontend

A089911 a(n) = Fibonacci(n) mod 12.

0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0, 1, 1

Offset: 0

Casey Mongoven, Nov 14 2003

From Reinhard Zumkeller, Jul 05 2013: (Start)
Sequence has been applied by several composers to 12-tone equal temperament pitch structure. The complete Fibonacci mod 12 system (a set of 10 periodic sequences) exhausts all possible ordered dyads; that is, every possible combination of two pitches is found in these sets.
a(A008594(n)) = 0;
a(A227144(n)) = 1;
a(3*A047522(n)) = 2;
a(A017569(n)) = a(2*A016933(n)) = a(4*A016777(n)) = 3;
a(2*A017629(n)) = a(3*A017137(n)) = a(6*A004767(n)) = 4;
a(A227146(n)) = 5;
a(nonexistent) = 6;
a(2*A017581(n)) = 7;
a(2*A017557(n)) = a(4*A016813(n)) = 8;
a(A017617(n)) = a(2*A016957(n)) = a(4*A016789(n)) = 9;
a(3*A047621(n)) = 10;
a(2*A017653(n)) = 11. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..1199
Ron Knott, Fibonacci Numbers and the Golden Section
M. Renault, The Fibonacci Sequence Modulo M
A. P. Shah, Fibonacci Sequence Modulo m, Fibonacci Quarterly, Vol.6, No.2 (1968), 139-141.
D. D. Wall, Fibonacci series modulo m, Amer. Math. Monthly, 67 (1960), 525-532.
Index entries for linear recurrences with constant coefficients, signature (1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1, 0, -1, 1).

Cf. A000045, A001175, A003893, A010881.

a089911 n = a089911_list !! n
a089911_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 12)
                       (tail a089911_list) a089911_list
-- Reinhard Zumkeller, Jul 01 2013

[Fibonacci(n) mod 12: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

with(combinat,fibonacci); A089911 := proc(n) fibonacci(n) mod 12; end;

Table[Mod[Fibonacci[n], 12], {n, 0, 100}] (* Vincenzo Librandi, Feb 04 2014 *)

a(n)=fibonacci(n)%12 \\ Charles R Greathouse IV, Feb 03 2014

Has period of 24, restricted period 12 and multiplier 5.

a(n) = (a(n-1) + a(n-2)) mod 12, a(0) = 0, a(1) = 1.

More terms from Ray Chandler, Nov 15 2003

A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

Original entry on oeis.org

0, 2, 9, 11, 12, 14, 21, 23, 24, 26, 33, 35, 36, 38, 45, 47, 48, 50, 57, 59, 60, 62, 69, 71, 72, 74, 81, 83, 84, 86, 93, 95, 96, 98, 105, 107, 108, 110, 117, 119, 120, 122, 129, 131, 132, 134, 141, 143, 144, 146, 153, 155, 156, 158, 165, 167, 168, 170, 177, 179, 180

Offset: 1

Jim McCann (jmccann(AT)umich.edu), Aug 04 2002

The "piece" in question is also called a "tribone" [Ardila and Stanley]. - N. J. A. Sloane, Feb 27 2014

			A possible side-9 arrangement:
          A
         A A
        B B C
       D B C C
      D D E E F
     G H H E F F
    G G H I I J J
   K L L M I N J O
  K K L M M N N O O

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), 183-208. [From _N. J. A. Sloane_, Jul 04 2011]
Jim McCann, Triangle Sequence
N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.
Torsten Sillke, A Word Problem
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Union of A008594, A017545, A017629 and A017653.

a072065 n = a072065_list !! n
a072065_list = filter ((`elem` [0,2,9,11]) . (`mod` 12)) [0..]
-- Reinhard Zumkeller, Jan 09 2013

f:=r-> {seq(12*i+r,i=0..100)}; t1:= f(0) union f(2) union f(9) union f(11); t2:=sort(convert(t1,list)); # N. J. A. Sloane, Jul 04 2011

Select[Range[0,200],MemberQ[{0,2,9,11},Mod[#,12]]&] (* Harvey P. Dale, Dec 15 2011 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,9,11,12},70] (* Harvey P. Dale, Jan 30 2015 *)

concat(0, Vec(x^2*(2+7*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 12 2015

A number n is in the sequence iff n == 0, 2, 9 or 11 (mod 12). See Conway-Lagarias or the Sillke link. - Sascha Kurz, Mar 04 2003
a(1)=0, a(2)=2, a(3)=9, a(4)=11, a(5)=12, a(n) = a(n-1)+a(n-4)-a(n-5). - Harvey P. Dale, Jan 30 2015
From Colin Barker, Dec 12 2015: (Start)
a(n) = (3/4+(3*i)/4)*(i^n-i*(-i)^n)-(-1)^n/2+3*(n+1)-5 where i = sqrt(-1).
G.f.: x^2*(2+7*x+2*x^2+x^3) / ((1-x)^2*(1+x)*(1+x^2)). (End)
E.g.f.: (2 + 3*cos(x) + (6*x - 5)*cosh(x) - 3*sin(x) + (6*x - 3)*sinh(x))/2. - Stefano Spezia, May 05 2022
a(n) = (6*n-4-(-1)^n+3*(-1)^((2*n+1-(-1)^n)/4))/2. - Wesley Ivan Hurt, Nov 09 2023

Offset corrected by Reinhard Zumkeller, Jan 09 2013

A369462 Number of representations of 12n-1 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 0, 1, 2, 0, 2, 1, 2, 0, 1, 1, 3, 1, 1, 2, 5, 0, 1, 0, 2, 2, 2, 1, 4, 1, 3, 0, 3, 1, 2, 2, 3, 0, 2, 1, 8, 1, 1, 1, 4, 2, 2, 3, 3, 0, 4, 0, 4, 1, 1, 4, 3, 1, 3, 1, 6, 2, 3, 0, 5, 3, 1, 2, 6, 2, 6, 2, 2, 0, 1, 1, 5, 1, 2, 1, 10, 1, 3, 1, 3, 4, 2, 1, 6, 3, 6, 1, 4, 1, 3, 1, 5, 2, 3, 0

Offset: 1

Antti Karttunen, Jan 23 2024

nonn

See A369452 for the cumulative sum, and comments there.

Question: Is there only a finite number of 0's in this sequence? See discussion at A369055 and see A369463 for empirical data.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Trisection of A369055.

Cf. A017653, A369054, A369252, A369452 (partial sums), A369460, A369461, A369463 (= (12*i)-1, where i are the indices of zeros in this sequence).

A369054(n) = if(3!=(n%4),0, my(v = [3,3], ip = #v, r, c=0); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r),c++)); if(!ip, return(c)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
A369462(n) = A369054((12*n)-1);

a(n) = A369054(A017653(n-1)) = A369054(12*n - 1).

a(n) = A369055(3*n).

A001538 a(n) = (12n+1)(12*n+11).

Original entry on oeis.org

11, 299, 875, 1739, 2891, 4331, 6059, 8075, 10379, 12971, 15851, 19019, 22475, 26219, 30251, 34571, 39179, 44075, 49259, 54731, 60491, 66539, 72875, 79499, 86411, 93611, 101099, 108875, 116939, 125291, 133931, 142859, 152075, 161579, 171371, 181451, 191819

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001533, A017533, A017653.

Table[(12n+1)(12n+11),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{11,299,875},40] (* Harvey P. Dale, Jul 22 2024 *)

a(n)=(12*n+1)*(12*n+11) \\ Charles R Greathouse IV, Jun 16 2017

a(n) = (9*A001533(n) - 19)/4.
a(n) = 288*n + a(n-1) with a(0)=11. - Vincenzo Librandi, Nov 12 2010
G.f.: -(11 + 266*x + 11*x^2)/(x-1)^3. - R. J. Mathar, Jun 30 2020
From Amiram Eldar, Feb 20 2023: (Start)
a(n) = A017533(n)*A017653(n).
Sum_{n>=0} 1/a(n) = (sqrt(3)+2)*Pi/120.
Sum_{n>=0} (-1)^n/a(n) = (4*log(sqrt(2)+1) + sqrt(3)*log(5+2*sqrt(6)))/(60*sqrt(2)).
Product_{n>=0} (1 - 1/a(n)) = (2*sqrt(2)/(sqrt(3)-1))*cos(sqrt(13/2)*Pi/6).
Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(2+sqrt(3))*cos(Pi/sqrt(6)). (End)
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(11 + 144*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

Original entry on oeis.org

1, 3, 11, 7, 23, 11, 35, 15, 47, 19, 59, 23, 71, 27, 83, 31, 95, 35, 107, 39, 119, 43, 131, 47, 143, 51, 155, 55, 167, 59, 179, 63, 191, 67, 203, 71, 215, 75, 227, 79, 239, 83, 251, 87, 263, 91, 275, 95, 287, 99, 299, 103, 311, 107, 323, 111, 335, 115, 347, 119, 359

Offset: 1

Walter Carlini, Dec 23 2008

1 followed by interleaving of A004767 and A017653. - Klaus Brockhaus, Jan 04 2009

			a(1) = 1;
a(2) = 2^2 - a(1) = 4 - 1 = 3;
a(3) = 3^2 + a(2) - a(1) = 9 + 3 - 1 = 11;
a(4) = 4^2 - 11 + 3 - 1 = 7;
a(5) = 25 + 7 - 11 + 3 - 1 = 23;
a(6) = 36 - 23 + 7 - 11 + 3 - 1 = 11; etc.

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

The second of a family of sequences that includes A153284 and A153286

Cf. A004767 (4n+3), A017653 (12n+11). - Klaus Brockhaus, Jan 04 2009

S:=[ 1 ]; for n in [2..61] do Append(~S, n^2 + &+[ (-1)^j*S[j]: j in [1..n-1] ]); end for; S; // Klaus Brockhaus, Jan 04 2009

(define (A153285 n) (cond ((= 1 n) n) ((even? n) (+ n n -1)) (else (+ (* 6 n) -7)))) ;; Antti Karttunen, Aug 10 2017

a(n) = 2n-1 if n is 1 or an even number;
a(n) = 6n-7 if n is an odd number other than 1.
G.f.: x*(1 + 3*x + 9*x^2 + x^3 + 2*x^4)/((1+x)^2*(1-x)^2). - Klaus Brockhaus, Oct 15 2009
a(n) = 4*(n-1) - (2*n-3)*(-1)^n for n>1, a(1)=1. - Bruno Berselli, Sep 14 2011

Extended beyond a(30) by Klaus Brockhaus, Jan 04 2009

A369463 Numbers of the form 12m-1 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.

Original entry on oeis.org

11, 23, 35, 47, 59, 83, 107, 143, 179, 227, 323, 347, 443, 515, 659, 683, 827, 947, 1259, 1523, 1763, 1787, 2075, 2267, 2675, 2963, 3023, 3203, 3275, 3347, 3467, 3635, 4523, 4643, 4859, 5003, 5147, 5747, 5819, 6395, 6803, 6827, 7235, 8003, 8123, 8171, 8747, 8963, 9323, 9659, 9827, 10367, 10427, 12347, 12923, 13187

Offset: 1

Antti Karttunen, Jan 23 2024

nonn

Equal to (12*i)-1, where i are the positions of 0's in A369462.
Terms of the form 3k+2 in A369056. These seem to be much more rare than terms of A369248.
Question: Is this a finite sequence, with the last term a(285) = 50688947 = (12*4224079)-1? See conjecture in A369055.
If it exists, a(286) > 201326603 (= (12*(2^24))+11).

Antti Karttunen, Table of n, a(n) for n = 1..285 (please update, also comments, if more terms are found!)

Intersection of A016789 and A369056 (and of A369464).
Subsequence of A017653.
Cf. A369054, A369055, A369462.
Cf. also A369248.

isA369251(n) =  if(3!=(n%4),0, my(v = [3,3], ip = #v, r); while(1, r = (n-(v[1]*v[2])) / (v[1]+v[2]); if(r < v[2], ip--, ip = #v; if(1==denominator(r) && isprime(r), return(1))); if(!ip, return(0)); v[ip] = nextprime(1+v[ip]); for(i=1+ip,#v,v[i]=v[i-1])));
isA369463(n) = ((11==(n%12)) && !isA369251(n));

A017659 a(n) = (12n+11)^7.

Original entry on oeis.org

19487171, 3404825447, 64339296875, 506623120463, 2488651484819, 9095120158391, 27136050989627, 69833729609375, 160578147647843, 337931541778439, 662062621900811, 1222791080775407, 2149422977421875

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A001015, A017653.

[(12*n+11)^7: n in [0..15]]; // Vincenzo Librandi, May 21 2012

(12Range[0,20]+11)^7 (* Harvey P. Dale, May 16 2012 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{19487171,3404825447,64339296875,506623120463,2488651484819,9095120158391,27136050989627,69833729609375},20] (* Harvey P. Dale, May 16 2012 *)

a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)- 70*a(n-4)+ 56*a(n-5)- 28*a(n-6)+ 8*a(n-7)-a(n-8). - Harvey P. Dale, May 16 2012

a(n) = A001015(A017653(n)). - Michel Marcus, Oct 29 2022

A110596 Balanced numbers n such that n mod 12 = 11.

Original entry on oeis.org

35, 124355, 1739507, 3281663, 3852155, 7649915, 9815195, 10434515, 13321295, 19154135, 19296035, 32807555, 36664595, 41523911, 50329955, 60668135, 69664595, 83338199, 107008811, 123543695, 145960451, 275361359, 321198059, 365269355, 393656879, 407002211

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

For the first 26 terms, the quotient (sigma(n)/phi(n)) is 2 or 3.

Amiram Eldar, Table of n, a(n) for n = 1..6654 (terms below 6.5*10^14, calculated using data from Jud McCranie)

Intersection of A017653 and A020492.

Cf. A000010, A000203.

with(numtheory); BNM11:=[]: for z from 1 to 1 do for n from 1 to 500000 do m:=12*n+11; if sigma(m) mod phi(m) = 0 then BNM11:=[op(BNM11),m] fi; od; od; BNM11;

Select[Range[10^7], Mod[#, 12] == 11 && Divisible[DivisorSigma[1, #], EulerPhi[#]] &] (* Amiram Eldar, Dec 04 2019 *)

a(6)-a(26) from Donovan Johnson, Aug 30 2012

A294614 Sum of the divisors of 12*n - 1, divided by 12, minus n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 2, 0, 2, 3, 0, 0, 0, 3, 4, 0, 0, 0, 0, 8, 4, 3, 0, 3, 6, 0, 0, 5, 0, 7, 4, 0, 0, 0, 18, 0, 0, 0, 0, 9, 4, 12, 4, 0, 14, 0, 0, 5, 8, 11, 0, 0, 6, 0, 12, 9, 0, 5, 0, 13, 6, 5, 10, 7, 14, 0, 0, 5, 0, 31, 0, 5, 0, 7, 30, 0, 12, 0, 0, 17, 6, 0, 0, 13, 18, 9, 8

Offset: 1

Omar E. Pol and Robert G. Wilson v, Nov 04 2017

a(n) = 0 iff n is in A138620.

First occurrence of k > -1: 1, 3, 8, 13, 18, 31, 28, 33, 23, 43, 66, 53, 45, 63, 48, 101, 166, etc.

			a(13) = 3 since d(12*13-1)/12 - 13 = 192/12 - 13 = 16 - 13 = 3.

Inspired by A291900.

Cf. A000203, A017653, A068231, A086463, A138620.

a[n_] := DivisorSigma[1, 12 n - 1]/12 - n; Array[a, 90]

PARI
```
a(n) = sigma(12*n-1)/12 - n;
```

a(n) = sigma(12*n-1)/12 - n = A000203(A017653(n-1))/12 - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A347838 Positive numbers that are congruent to 2, 5, or 11 modulo 12.

Original entry on oeis.org

2, 5, 11, 14, 17, 23, 26, 29, 35, 38, 41, 47, 50, 53, 59, 62, 65, 71, 74, 77, 83, 86, 89, 95, 98, 101, 107, 110, 113, 119, 122, 125, 131, 134, 137, 143, 146, 149, 155, 158, 161, 167, 170, 173, 179, 182, 185, 191, 194, 197, 203, 206, 209, 215, 218, 221, 227, 230, 233, 239

Offset: 1

Wolfdieter Lang, Oct 21 2021

This sequence follows from the first column sequence of the array A347834, namely A047529 ({1,3,7} (mod 8)), as given in the formula below.

Together with A017617, the positive integers congruent to 8 modulo 12, one obtains A016789, the positive integers congruent to 2 modulo 3. See the array A347839.

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

Cf. A002264, A016921, A016789, A017617, A017653, A040117(k+1), A047261, A047529, A102283, A347834, A347839.

Map[(3 # + 1)/2 &, LinearRecurrence[{1, 0, 1, -1}, {1, 3, 7, 9}, 60]] (* Michael De Vlieger, Oct 21 2021 *)

a(n) = (3*A047529(n) + 1)/2.
Trisection: a(3*k+1) = 2 + 12*k, a(3*k+2) = 5 + 12*k, a(3*k+3) = 11 + 12*k, or with a(3*k) = -1 + 12*k for k >= 0.
O.g.f. with a(0) =-1: G(x) = (-1 + 3*x + 3*x^2 + 7*x^3)/((1 - x)*(1 - x^3)) = -6/(1-x) + 4/(1-x)^2 + (1 + x)/(1 + x + x^2). Note that (1 - x)*(1 - x^3) = (1-x)^2*(1 + x + x^2) =  1 - x - x^3 + x^4.
a(n) = a(n-1) + a(n-3) - a(n-4), for n >= 4, given a(n) for 0..3, with a(0) = -1.
a(n) = 2*b(n) + 3*b(n-1) + 6*b(n-2) + b(n-3), with b(n) = floor((n+2)/3) = A002264(n+2).
a(n) = -1 + 3*n + 3*floor(n/3) (from the partial fraction decomposition of G).
E.g.f.: 1 + 2*exp(x)*(2*x - 1) + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Dec 08 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = ((sqrt(2)+1)*Pi + sqrt(3)*log(sqrt(3)+2) + sqrt(6)*log(5-2*sqrt(6)))/12. - Amiram Eldar, Dec 30 2021

cp's OEIS Frontend

A089911 a(n) = Fibonacci(n) mod 12.

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A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

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A369462 Number of representations of 12n-1 as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A001538 a(n) = (12*n+1)*(12*n+11).

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A153285 a(1)=1; for n > 1, a(n) = n^2 + Sum_{j=1..n-1} (-1)^j*a(j).

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A369463 Numbers of the form 12*m-1 for which there is no representation as a sum (p*q + p*r + q*r) with three odd primes p <= q <= r.

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A017659 a(n) = (12n+11)^7.

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A369462 Number of representations of 12n-1 as a sum (pq + pr + q*r) with three odd primes p <= q <= r.

A001538 a(n) = (12n+1)(12*n+11).

A369463 Numbers of the form 12m-1 for which there is no representation as a sum (pq + pr + qr) with three odd primes p <= q <= r.