A022120 -id:A022120 - cp's OEIS frontend

A187891 a(0)=0, a(1)=5, a(n)=a(n-1)+a(n-2)-1.

0, 5, 4, 8, 11, 18, 28, 45, 72, 116, 187, 302, 488, 789, 1276, 2064, 3339, 5402, 8740, 14141, 22880, 37020, 59899, 96918, 156816, 253733, 410548, 664280, 1074827, 1739106, 2813932, 4553037, 7366968, 11920004, 19286971, 31206974, 50493944, 81700917

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 15 2011

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Cf. A000071, A001611, A001612, A022120, A187890.

Join[{a=0,b=5},Table[c=a+b-1;a=b;b=c,{n,100}]]
nxt[{a_,b_}]:={b,a+b-1}; NestList[nxt,{0,5},40][[All,1]] (* Harvey P. Dale, Nov 03 2022 *)

a(n) = 1+A022120(n-2), n>2. - R. J. Mathar, Mar 15 2011

G.f.: -x^2*(-5+6*x) / ( (x-1)*(x^2+x-1) ). - R. J. Mathar, Mar 15 2011

A356807 Tetranacci sequence beginning with 3, 7, 12, 24.

Original entry on oeis.org

3, 7, 12, 24, 46, 89, 171, 330, 636, 1226, 2363, 4555, 8780, 16924, 32622, 62881, 121207, 233634, 450344, 868066, 1673251, 3225295, 6216956, 11983568, 23099070, 44524889, 85824483, 165432010, 318880452, 614661834, 1184798779, 2283773075, 4402114140, 8485347828

Offset: 1

Greg Dresden and Hangyu Liang, Aug 29 2022

By "Tetranacci sequence" we mean a sequence in which each term is the sum of the four previous terms.
For n>1, a(n) is the number of ways to tile this figure of length n with squares, dominoes, trominoes, and tetraminoes:
   _
  |||_________       _
  |||_|||_|| ... ||

			Here is one of the a(6) = 89 ways to tile this figure of length 6 with tiles of length <= 4, this one using three squares, one domino, and one tromino:
   ___
  | |_|_______
  |_|_____|_|_|

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Cf. A022120, A100683.

LinearRecurrence[{1, 1, 1, 1}, {3, 7, 12, 24}, 50] (* Paolo Xausa, Aug 30 2024 *)

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4).
a(n) = 5*b(n+2) + 2*b(n+1) - 2*b(n-2) for b(n) = A000078(n) the tetranacci numbers.
a(n) = L(n+2) - F(n-2) + Sum_{k=0..n-3} a(k)*F(n-k-1), for L(n) and F(n) the Lucas and Fibonacci numbers.
G.f.: x*(-2*x^3 - 2*x^2 - 4*x - 3)/(x^4 + x^3 + x^2 + x - 1). - Chai Wah Wu, Aug 30 2022

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

A187891 a(0)=0, a(1)=5, a(n)=a(n-1)+a(n-2)-1.

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A356807 Tetranacci sequence beginning with 3, 7, 12, 24.

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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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