A111956 -id:A111956 - cp's OEIS frontend

A111954 a(n) = A000129(n) + (-1)^n.

1, 0, 3, 4, 13, 28, 71, 168, 409, 984, 2379, 5740, 13861, 33460, 80783, 195024, 470833, 1136688, 2744211, 6625108, 15994429, 38613964, 93222359, 225058680, 543339721, 1311738120, 3166815963, 7645370044, 18457556053, 44560482148, 107578520351

Offset: 0

Creighton Dement, Aug 23 2005

a(n) + a(n+1) = A001333(n+1). Inverse binomial transform of A007070 (with prepended 1). Inverse invert transform of A077995.

Floretion Algebra Multiplication Program, FAMP Code: -4ibasejseq[J*D] with J = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and D = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A111955, A111956, A007070, A077995.

LinearRecurrence[{1,3,1},{1,0,3},40] (* Harvey P. Dale, Nov 24 2014 *)

a(n) = a(n-1) + 3*a(n-2) + a(n-3), n >= 3.
G.f.: (x-1)/((x+1)*(x^2+2*x-1)).
a(n) = (sqrt(2)/4)*((1 + sqrt(2))^n - (1 - sqrt(2))^n) + (-1)^n.
E.g.f.: cosh(x) - sinh(x) + exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

A115312 a(n) = gcd(Lucas(n)-1, Fibonacci(n)+1).

Original entry on oeis.org

2, 2, 3, 2, 2, 1, 14, 2, 5, 2, 18, 1, 26, 2, 47, 2, 34, 1, 246, 2, 89, 2, 322, 1, 466, 2, 843, 2, 610, 1, 4414, 2, 1597, 2, 5778, 1, 8362, 2, 15127, 2, 10946, 1, 79206, 2, 28657, 2, 103682, 1, 150050, 2, 271443, 2, 196418, 1, 1421294, 2, 514229, 2, 1860498, 1

Offset: 1

Giovanni Resta, Jan 20 2006

Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1) + Lucas(n-2). See A000032.
a(n) is prime for n = 1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 16, 20, 21, 22, 26, 28, 32, 33, 34, 38, 40, 44, 45, ... - Vincenzo Librandi, Dec 24 2015
350 of the first 1000 terms are primes. - Harvey P. Dale, Mar 25 2020

			a(15) = 47 since F(15) + 1 =13*47 and L(15) - 1 = 29*47.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000032, A000045, A111956, A115311, A115313, A115314.

[Gcd(Lucas(n)-1, Fibonacci(n)+1): n in [1..60]]; // Vincenzo Librandi, Dec 24 2015

lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]-1, Fibonacci[i]+1], {i, 60}]
GCD[#[[1]]-1,#[[2]]+1]&/@With[{nn=60},Thread[{LucasL[Range[ nn]],Fibonacci[ Range[nn]]}]] (* Harvey P. Dale, Mar 25 2020 *)

a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)-1,fibonacci(n)+1); \\ Altug Alkan, Dec 24 2015

A115314 a(n) = gcd(Lucas(n)+1, Fibonacci(n)-1).

Original entry on oeis.org

2, 4, 1, 2, 4, 1, 6, 4, 11, 2, 8, 1, 58, 4, 21, 2, 76, 1, 110, 4, 199, 2, 144, 1, 1042, 4, 377, 2, 1364, 1, 1974, 4, 3571, 2, 2584, 1, 18698, 4, 6765, 2, 24476, 1, 35422, 4, 64079, 2, 46368, 1, 335522, 4, 121393, 2, 439204, 1, 635622, 4, 1149851, 2, 832040, 1

Offset: 1

Giovanni Resta, Jan 20 2006

Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1) + Lucas(n-2). See A000032.

a(n) is prime for n = 1, 4, 9, 10, 16, 21, 22, 28, 33, 34, 40, 46, 52, 58, 64, 70, 76, 81, 82, 88, 93, 94, ... - Vincenzo Librandi, Dec 24 2015

			a(15) = 21 = 3*7 since F(15) - 1 = 3*7*29 and L(15) + 1 = 3*5*7*13.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000032, A000045, A111956, A115311, A115312, A115313.

[Gcd(Lucas(n)+1, Fibonacci(n)-1): n in [1..60]]; // Vincenzo Librandi, Dec 24 2015

lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]+1, Fibonacci[i]-1], {i, 60}]
Module[{nn=60,l,f},l=LucasL[Range[nn]]+1;f=Fibonacci[Range[nn]]-1;GCD@@@ Thread[ {l,f}]] (* Harvey P. Dale, Apr 29 2020 *)

a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)+1,fibonacci(n)-1); \\ Altug Alkan, Dec 24 2015

A111946 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 2, 1, 1, 8, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 3, 1, 1, 1, 21, 1, 1, 2, 1, 1, 2, 1, 1, 34, 1, 1, 1, 1, 5, 1, 1, 1, 1, 55, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 89, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Nov 28 2005

The function T(n, k) is defined for all integers n, k but only the values for 1 <= k <= n as a triangular array are listed here.

			Triangle begins:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 1, 3;
  1, 1, 1, 1, 5;
  1, 1, 2, 1, 1, 8;
  1, 1, 1, 1, 1, 1, 13;
  1, 1, 1, 3, 1, 1,  1, 21;
  1, 1, 2, 1, 1, 2,  1,  1, 34;
  1, 1, 1, 1, 5, 1,  1,  1,  1, 55;
  ...

T. D. Noe, Rows n = 1..150 of triangle, flattened
P. Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Q. 43 (No. 1, 2005), 3-14.

Cf. A000045, A111956, A111957.

/* As triangle */ [[Gcd(Fibonacci(n), Fibonacci(k)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 20 2015

T[ n_, k_] := Fibonacci @ GCD[ n, k] (* Michael Somos, Jul 18 2011 *)

{T(n, k) = fibonacci( gcd( n, k))} /* Michael Somos, Jul 18 2011 */

T(n, k) = Fibonacci(gcd(n, k)).

T(n, k) = T(k, n) = T(-n, k) = T(n, -k) = T(n, n+k) = T(n+k, k). - Michael Somos, Jul 18 2011

A111957 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Lucas(k)), 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 7, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 18, 1, 1, 4, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Nov 28 2005

			Triangle begins:
1,
1, 1,
1, 1, 2,
1, 3, 1, 1,
1, 1, 1, 1, 1,
1, 1, 4, 1, 1, 2,
1, 1, 1, 1, 1, 1, 1,
1, 3, 1, 7, 1, 3, 1, 1,
1, 1, 2, 1, 1, 2, 1, 1, 2,
1, 1, 1, 1, 11, 1, 1, 1, 1, 1,
=============================

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paulo Ribenboim, FFF (Favorite Fibonacci Flowers), Fib. Quart. 43 (No. 1, 2005), 3-14.

Cf. A000045, A000032, A111946, A111956.

/* As triangle */ [[Gcd(Fibonacci(n), Lucas(k)): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Dec 20 2015

Flatten[Table[GCD[Fibonacci[n], LucasL[k]], {n, 20}, {k, n}]] (* Alonso del Arte, Dec 19 2015 *)

T(n, k) = Lucas(g), where g = gcd(n, k), if n/g is even; = 2 if n/g is odd and 3|g; = 1 otherwise.

A115311 a(n) = gcd(Lucas(n)-1, Fibonacci(n)-1).

Original entry on oeis.org

0, 2, 1, 2, 2, 1, 4, 2, 3, 2, 22, 1, 8, 2, 29, 2, 42, 1, 76, 2, 55, 2, 398, 1, 144, 2, 521, 2, 754, 1, 1364, 2, 987, 2, 7142, 1, 2584, 2, 9349, 2, 13530, 1, 24476, 2, 17711, 2, 128158, 1, 46368, 2, 167761, 2, 242786, 1, 439204, 2, 317811, 2, 2299702, 1

Offset: 1

Giovanni Resta, Jan 20 2006

Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1) + Lucas(n-2). See A000032.

a(n) is prime for n = 2, 4, 5, 8, 9, 10, 14, 15, 16, 20, 22, 26, 27, 28, 32, 34, 38, 39, 40, 44, 46, ... - Vincenzo Librandi, Dec 24 2015

			a(15) = 29 since F(15) - 1 = 3*7*29 and L(15) - 1 = 29*49.

Cf. A000032, A000045, A111956, A115312, A115313, A115314.

[Gcd(Lucas(n)-1, Fibonacci(n)-1): n in [1..60]]; // Vincenzo Librandi, Dec 24 2015

lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]-1, Fibonacci[i]-1], {i, 60}]
Table[GCD[LucasL[n]-1,Fibonacci[n]-1],{n,60}] (* Harvey P. Dale, Sep 25 2017 *)

a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)-1,fibonacci(n)-1); \\ Altug Alkan, Dec 24 2015

A115313 a(n) = gcd(Lucas(n)+1, Fibonacci(n)+1).

Original entry on oeis.org

2, 2, 1, 4, 6, 1, 2, 2, 7, 4, 10, 1, 18, 2, 13, 4, 94, 1, 34, 2, 123, 4, 178, 1, 322, 2, 233, 4, 1686, 1, 610, 2, 2207, 4, 3194, 1, 5778, 2, 4181, 4, 30254, 1, 10946, 2, 39603, 4, 57314, 1, 103682, 2, 75025, 4, 542886, 1, 196418, 2, 710647, 4, 1028458, 1

Offset: 1

Giovanni Resta, Jan 20 2006

Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1)+Lucas(n-2). See A000032.

a(n) is prime for n = 1, 2, 7, 8, 9, 14, 15, 20, 26, 27, 32, 33, 38, 44, 50, 56, 62, 68, 74, 80, 86, 87, ... - Vincenzo Librandi, Dec 24 2015

			a(15) = 13 since F(15) + 1 = 13*47 and L(15) + 1 = 3*5*7*13.

Cf. A000032, A000045, A111956, A115311, A115312, A115314.

[Gcd(Lucas(n)+1, Fibonacci(n)+1): n in [1..60]]; // Vincenzo Librandi, Dec 24 2015

lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]+1, Fibonacci[i]+1], {i, 60}]

a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)+1,fibonacci(n)+1); \\ Altug Alkan, Dec 24 2015

A111955 a(n) = A078343(n) + (-1)^n.

Original entry on oeis.org

0, 1, 4, 7, 20, 45, 112, 267, 648, 1561, 3772, 9103, 21980, 53061, 128104, 309267, 746640, 1802545, 4351732, 10506007, 25363748, 61233501, 147830752, 356895003, 861620760, 2080136521, 5021893804, 12123924127, 29269742060, 70663408245

Offset: 0

Creighton Dement, Aug 25 2005

This sequence is a companion sequence to A111954 (compare formula / program code). Three other companion sequences (i.e., they are generated by the same floretion given in the program code) are A105635, A097076 and A100828.

Floretion Algebra Multiplication Program, FAMP Code: 4kbasejseq[J*D] with J = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and D = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e. (an initial term 0 was added to the sequence)

Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A078343, A000129, A001333, A111954, A111956, A007070, A077995, A100828, A097076, A105635, A048655.

LinearRecurrence[{1,3,1},{0,1,4},40] (* Harvey P. Dale, Mar 12 2015 *)

a(n) + a(n+1) = A048655(n).
a(n) = a(n-1) + 3*a(n-2) + a(n-3), n >= 3; a(n) = (-1/4*sqrt(2)+1)*(1-sqrt(2))^n + (1/4*sqrt(2)+1)*(1+sqrt(2))^n - (-1)^n;
G.f.: -x*(1+3*x) / ( (1+x)*(x^2+2*x-1) ). - R. J. Mathar, Oct 02 2012
E.g.f.: cosh(x) - exp(x)*cosh(sqrt(2)*x) - sinh(x) + 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, May 26 2024

cp's OEIS Frontend

A111954 a(n) = A000129(n) + (-1)^n.

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A115312 a(n) = gcd(Lucas(n)-1, Fibonacci(n)+1).

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A115314 a(n) = gcd(Lucas(n)+1, Fibonacci(n)-1).

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A111946 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.

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A111957 Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Lucas(k)), 1 <= k <= n.

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A115311 a(n) = gcd(Lucas(n)-1, Fibonacci(n)-1).

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A115313 a(n) = gcd(Lucas(n)+1, Fibonacci(n)+1).

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