A022097 -id:A022097 - cp's OEIS frontend

A022369 Fibonacci sequence beginning 2, 14.

2, 14, 16, 30, 46, 76, 122, 198, 320, 518, 838, 1356, 2194, 3550, 5744, 9294, 15038, 24332, 39370, 63702, 103072, 166774, 269846, 436620, 706466, 1143086, 1849552, 2992638, 4842190, 7834828, 12677018

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

Cf. A022097.

a={};b=2;c=14;AppendTo[a, b];AppendTo[a, c];Do[b=b+c;AppendTo[a, b];c=b+c;AppendTo[a, c], {n, 4!}];a (* Vladimir Joseph Stephan Orlovsky, Sep 18 2008 *)
Transpose[NestList[{Last[#],Total[#]}&,{2,14},40]][[1]]  (* Harvey P. Dale, Apr 13 2011 *)
LinearRecurrence[{1,1},{2,14},40] (* Harvey P. Dale, Oct 05 2019 *)

for(n=0,50, print1(2*(Fibonacci(n+2) + 5*Fibonacci(n)), ", ")) \\ G. C. Greubel, Aug 27 2017

G.f.: (2+12*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = 2*(Fibonacci(n+2) + 5*Fibonacci(n)). - G. C. Greubel, Aug 27 2017
a(n) = 2 * A022097(n). - Alois P. Heinz, Aug 27 2017

A099255 Expansion of g.f. (7+6x-6x^2-3x^3)/((x^2+x-1)(x^2-x-1)).

Original entry on oeis.org

7, 6, 15, 15, 38, 39, 99, 102, 259, 267, 678, 699, 1775, 1830, 4647, 4791, 12166, 12543, 31851, 32838, 83387, 85971, 218310, 225075, 571543, 589254, 1496319, 1542687, 3917414, 4038807, 10255923, 10573734, 26850355, 27682395, 70295142, 72473451

Offset: 0

Creighton Dement, Oct 09 2004

One of two sequences involving the Lucas/Fibonacci numbers.

This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs. "pos((Ex)^n)" sums up over all floretion basis vectors with positive coefficients for each n. The following relations appear to hold: a(2n) - (a(2n-1) + a(2n-2)) = 2*Luc(2n) a(2n+1) - a(2n) = Fib(2n), apart from initial term a(2n+1)/a(2n-1) -> 2 + golden ratio phi a(2n)/a(2n-2) -> 2 + golden ratio phi An identity: (1/2)a(n) - (1/2)A099256(n) = ((-1)^n)A000032(n)

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

Cf. A099256, A000032.

LinearRecurrence[{0,3,0,-1},{7,6,15,15},40] (* Harvey P. Dale, Dec 29 2012 *)

a(n) = 2*pos((Ex)^n)
a(0) = 7, a(1) = 6, a(2) = a(3) = 15, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022097(2n+1), a(2n+1) = A022086(2n+3).
a(n) = A061084(n+1)+A013655(n+2). [R. J. Mathar, Nov 30 2008]

More terms from Creighton Dement, Apr 19 2005

A099256 Expansion of g.f. (3-x)(1+3x+x^2)/((1-x-x^2)*(1+x-x^2)).

Original entry on oeis.org

3, 8, 9, 23, 24, 61, 63, 160, 165, 419, 432, 1097, 1131, 2872, 2961, 7519, 7752, 19685, 20295, 51536, 53133, 134923, 139104, 353233, 364179, 924776, 953433, 2421095, 2496120, 6338509, 6534927, 16594432, 17108661, 43444787, 44791056, 113739929, 117264507, 297775000, 307002465, 779585071

Offset: 0

Creighton Dement, Oct 18 2004

One of two sequences involving the Lucas/Fibonacci numbers. This sequence consists of pairs of numbers more or less close to each other with "jumps" in between pairs.
a(n+3) + a(n) - a(n+2) appears to be mysteriously connected with a(n+1).
Both this sequence and A099255 were created using "Floretion dynamical symmetries" (see link for further details).

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000045, A099255, A000032, A055273 (bisection), A097134 (bisection).

LinearRecurrence[{0,3,0,-1},{3,8,9,23},40] (* Harvey P. Dale, Apr 22 2012 *)

a(2n+2) - a(2n+1) = Fibonacci(2n-1).
A099255(n)/2 - a(n)/2 = (-1)^n*A000032(n)
a(0) = 3, a(1) = 8, a(2) = 9, a(3) = 23, a(n+4) = 3a(n+2) - a(n).
a(2n) = A022086(2n+2), a(2n+1) = A022097(2n+2).
a(n) = A013655(n+2)-A061084(n+1).

Definition corrected, extended. - R. J. Mathar, Nov 13 2008

A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 11, 9, 10, 8, 18, 15, 17, 12, 13, 29, 24, 27, 19, 14, 21, 47, 39, 44, 31, 23, 16, 34, 76, 63, 71, 50, 37, 25, 20, 55, 123, 102, 115, 81, 60, 41, 33, 22, 89, 199, 165, 186, 131, 97, 66, 53, 35, 26, 144, 322, 267, 301, 212, 157, 107, 86, 57, 43, 28

Offset: 1

Casey Mongoven, Nov 07 2011

The rows of the array can be seen to have the form A(n, k) = p(n)*Fibonacci(k) + q(n)*Fibonacci(k+1) where p(n) is the sequence {0, 1, 3, 3, 3, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, ...}{n >= 1} and q(n) is the sequence {1, 3, 3, 7, 2, 9, 9, 13, 13, 17, 17, 19, 19, 23, 23, 25, ...}{n >= 1}. - G. C. Greubel, Jun 23 2022

			The even first column stolarsky array (EFC array), northwest corner:
  1......2.....3.....5.....8....13....21....34....55....89...144 ... A000045;
  4......7....11....18....29....47....76...123...199...322...521 ... A000032;
  6......9....15....24....39....63...102...165...267...432...699 ... A022086;
  10....17....27....44....71...115...186...301...487...788..1275 ... A022120;
  12....19....31....50....81...131...212...343...555...898..1453 ... A013655;
  14....23....37....60....97...157...254...411...665..1076..1741 ... A000285;
  16....25....41....66...107...173...280...453...733..1186..1919 ... A022113;
  20....33....53....86...139...225...364...589...953..1542..2495 ... A022096;
  22....35....57....92...149...241...390...631..1021..1652..2673 ... A022130;
Antidiagonal rows (T(n, k)):
   1;
   2,   4;
   3,   7,   6;
   5,  11,   9,  10;
   8,  18,  15,  17, 12;
  13,  29,  24,  27, 19, 14;
  21,  47,  39,  44, 31, 23, 16;
  34,  76,  63,  71, 50, 37, 25, 20;
  55, 123, 102, 115, 81, 60, 41, 33, 22;

Clark Kimberling, The first column of an interspersion, Fibonacci Quarterly 32 (1994), pp. 301-314.

Cf. A000032, A000045, A000285, A013655, A022086, A022088, A022095.
Cf. A022096, A022097, A022113, A022120, A022121, A022122, A022130.
Cf. A022388, A022390, A035506, A035513, A097135, A199536, A199537.

From G. C. Greubel, Jun 23 2022: (Start)
T(n, 1) = A000045(n+1).
T(n, 2) = A000032(n+1), n >= 2.
T(n, 3) = A022086(n) = A097135(n), n >= 3.
T(n, 4) = A022120(n-2), n >= 4.
T(n, 5) = A013655(n-1), n >= 5.
T(n, 6) = A000285(n-2), n >= 6.
T(n, 7) = A022113(n-4), n >= 7.
T(n, 8) = A022096(n-4), n >= 8.
T(n, 9) = A022130(n-6), n >= 9.
T(n, 10) = A022098(n-5), n >= 10.
T(n, 11) = A022095(n-7), n >= 11.
T(n, 12) = A022121(n-8), n >= 12.
T(n, 13) = A022388(n-10), n >= 13.
T(n, 14) = A022122(n-10), n >= 14.
T(n, 15) = A022097(n-10), n >= 15.
T(n, 16) = A022088(n-10), n >= 16.
T(n, 17) = A022390(n-14), n >= 17.
T(n, n) = A199536(n).
T(n, n-1) = A199537(n-1), n >= 2. (End)

More terms added by G. C. Greubel, Jun 23 2022

A017125 Table read by antidiagonals of Fibonacci-type sequences.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 5, 5, 4, 4, 1, 5, 8, 8, 7, 5, 5, 1, 6, 13, 13, 11, 9, 6, 6, 1, 7, 21, 21, 18, 14, 11, 7, 7, 1, 8, 34, 34, 29, 23, 17, 13, 8, 8, 1, 9, 55, 55, 47, 37, 28, 20, 15, 9, 9, 1, 10, 89, 89, 76, 60, 45, 33, 23, 17, 10, 10, 1, 11, 144, 144, 123, 97, 73

Offset: 0

Henry Bottomley, Jul 31 2000

Rows are (essentially) A000045, A000045, A000032, A000285, A022095, A022096, A022097, etc. Columns are (essentially) A001477, A000012, A000027, A005408, A016789, A016885, etc. One of the diagonals is A007502.

Antidiagonal sums are in A019274.

T(n, k) = T(n, k-1)+T(n, k-2) [with T(n, 0) = n and T(n, 1) = 1] = 2*T(n-1, k)-T(n-2, k) = Fib(k)+n*Fib(k-1) = (s^k*(1+2n/s)-t^k*(1+2n/t))/(2^k*sqrt(5)) where s = (1+sqrt(5))/2 and t = (1-sqrt(5))/2 = 1-s.

G.f. for n-th row: (n+x-nx)/(1-x-x^2).

A022311 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 6, 7, 14, 22, 37, 60, 98, 159, 258, 418, 677, 1096, 1774, 2871, 4646, 7518, 12165, 19684, 31850, 51535, 83386, 134922, 218309, 353232, 571542, 924775, 1496318, 2421094, 3917413, 6338508, 10255922, 16594431, 26850354, 43444786, 70295141, 113739928

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 (2,0,-1)

Cf. A000045.

LinearRecurrence[{2, 0, -1}, {0, 6, 7}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)

x='x+O('x^50); concat([0],Vec((6*x-5*x^2)/(1-2*x+x^3))) \\ G. C. Greubel, Aug 25 2017

Equals A022097(n) - 1.
G.f.: (6*x-5*x^2)/(1-2*x+x^3). - Franklin T. Adams-Watters, Oct 17 2006
a(n) = F(n+2) + 5*F(n) - 1, where F = A000045. - G. C. Greubel, Aug 25 2017

A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 2, 1, 2, 3, 1, 4, 3, 3, 2, 6, 5, 3, 4, 3, 10, 7, 5, 4, 5, 5, 16, 11, 7, 6, 5, 7, 8, 26, 17, 11, 8, 7, 7, 10, 13, 42, 27, 17, 12, 9, 9, 10, 15, 21, 68, 43, 27, 18, 13, 11, 12, 15, 23, 34, 110, 69, 43, 28, 19, 15, 14, 17, 23, 36, 55, 178, 111, 69, 44, 29, 21

Offset: 0

Philippe Deléham, Apr 07 2013

			Square array begins:
...0....2....2....4....6...10...16...26...42...
...1....3....3....5....7...11...17...27...43...
...1....3....3....5....7...11...17...27...43...
...2....4....4....6....8...12...18...28...44...
...3....5....5....7....9...13...19...29...45...
...5....7....7....9...11...15...21...31...47...
...8...10...10...12...14...18...24...34...50...
..13...15...15...17...19...23...29...39...55...
..21...23...23...25...27...31...37...47...63...
..34...36...36...38...40...44...50...60...76...
..55...57...57...59...61...65...71...81...97...
..89...91...91...93...95...99..105..115..131...
.144..146..146..148..150..154..160..170..186...
...

Cf. A000045

T(n,0) = A000045(n).
T(1,k) = A001588(k).
T(n,1) = T(n,2) = A157725(n).
T(n,3) = A157727(n).
T(n,n)= A022086(n) = 3*A000045(n).
T(n+1,n) = A000032(n+1) = A000204(n+1).
T(n+2,n) = A000285(n).
T(n+3,n) = A013655(n+1) = A001060(n+1).
T(n+4,n) = A021120(n).
T(n+5,n) = A022088(n+2) = 5*A000045(n+2).
T(n+6,n) = A022097(n+2).
T(n+7,n) = A022122(n+2).
T(n+8,n) = 3*A013655(n+2).
T(n+9,n) = A097657(n+2).
T(n+10,n) = A022118(n+4).
T(n,n+1) = A000045(n+3).
T(n,n+2) = A013655(n+1) = A001060(n+1).
T(n,n+3) = A000032(n+3).
T(n,n+4) = A022095(n+2).
T(n,n+5) = A022120(n+2).
T(n,n+6) = A022136(n+2).
T(n,n+7) = A022098(n+4).
T(n,n+8) = A022380(n+4).
T(n,n+9) = A206419(n+6).
Sum(T(n-k,k), 0<=k<=n) = 3*A000071(n+2).

cp's OEIS Frontend

A022369 Fibonacci sequence beginning 2, 14.

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A099255 Expansion of g.f. (7+6*x-6*x^2-3*x^3)/((x^2+x-1)*(x^2-x-1)).

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A099256 Expansion of g.f. (3-x)*(1+3*x+x^2)/((1-x-x^2)*(1+x-x^2)).

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A199535 Clark Kimberling's even first column Stolarsky array read by antidiagonals.

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A017125 Table read by antidiagonals of Fibonacci-type sequences.

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A022311 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=1.

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A217762 Square array T, read by antidiagonals: T(n,k) = F(n) + 2*F(k) where F(n) is the n-th Fibonacci number.

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A099255 Expansion of g.f. (7+6x-6x^2-3x^3)/((x^2+x-1)(x^2-x-1)).

A099256 Expansion of g.f. (3-x)(1+3x+x^2)/((1-x-x^2)*(1+x-x^2)).