A022098 -id:A022098 - cp's OEIS frontend

A022312 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=7.

0, 7, 8, 16, 25, 42, 68, 111, 180, 292, 473, 766, 1240, 2007, 3248, 5256, 8505, 13762, 22268, 36031, 58300, 94332, 152633, 246966, 399600, 646567, 1046168, 1692736, 2738905, 4431642, 7170548, 11602191, 18772740, 30374932, 49147673, 79522606, 128670280

Offset: 0

N. J. A. Sloane

G. C. Greubel, 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, 7, 8}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

x='x+O('x^50); concat([0],Vec(x*(7-6*x)/( (1-x)*(1-x-x^2) ))) \\ G. C. Greubel, Aug 25 2017

Equals A022098(n) - 1.
G.f.: x*(7-6*x)/( (1-x)*(1-x-x^2) ). - R. J. Mathar, Apr 07 2011
a(n) = F(n+2) + 6*F(n) - 1, where F = A000045. - G. C. Greubel, Aug 25 2017

A295678 a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 1, a(3) = 3.

Original entry on oeis.org

1, 2, 1, 3, 6, 9, 13, 22, 37, 59, 94, 153, 249, 402, 649, 1051, 1702, 2753, 4453, 7206, 11661, 18867, 30526, 49393, 79921, 129314, 209233, 338547, 547782, 886329, 1434109, 2320438, 3754549, 6074987, 9829534, 15904521, 25734057, 41638578, 67372633, 109011211

Offset: 0

Clark Kimberling, Nov 27 2017

Lim_{n->inf} a(n)/a(n-1) = (1 + sqrt(5))/2 = golden ratio (A001622), so that a( ) has the growth rate of the Fibonacci numbers (A000045).

Clark Kimberling, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, 1)

Cf. A001622, A000045.

LinearRecurrence[{1, 0, 1, 1}, {1, 2, 1, 3}, 100]

a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 1, a(3) = 3.
G.f.: (-1 - x + x^2 - x^3)/( (x^2+x-1)*(1+x^2)).
5*a(n) = A022098(n)+2*( A000034(n+1)*(-1)^floor(n/2)). - R. J. Mathar, Apr 26 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

A022312 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=7.

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A295678 a(n) = a(n-1) + a(n-3) + a(n-4), where a(0) = 1, a(1) = 2, a(2) = 1, a(3) = 3.

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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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