A027941 -id:A027941 - cp's OEIS frontend

A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.

1, 1, 2, 1, 4, 4, 1, 5, 12, 7, 1, 8, 22, 33, 12, 1, 12, 56, 94, 88, 20, 1, 19, 134, 385, 399, 232, 33, 1, 30, 342, 1487, 2640, 1691, 609, 54, 1, 48, 872, 6138, 16492, 18096, 7164, 1596, 88, 1, 77, 2256, 25319, 110143, 182900, 124033, 30348, 4180, 143

Offset: 1

Clark Kimberling, Oct 28 2012

row 1: A001612 (except for initial term)
col 1: A000071
col 2: A027941
col 3: A049652
col 4: A092521
col 6: A049664
col 8: A156093 without minus signs

			Northwest corner:
1....1.....1......1.......1
2....4.....5......8.......12
4....12....22.....56......134
7....33....94.....385.....1487
12...88....399....2640....16492
20...232...1691...18096...182900

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A214978, A214984, A214986.

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[n])*Sum[F[k*n], {k, 1, m}]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
Flatten[Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}]]

For odd-numbered columns (m odd):
T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
For even-numbered columns (m even):
T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-1)).

A219020 Sum of the cubes of the first n even-indexed Fibonacci numbers divided by the sum of the first n terms.

Original entry on oeis.org

1, 7, 45, 297, 2002, 13630, 93177, 638001, 4371235, 29956465, 205313076, 1407206412, 9645056785, 66107994667, 453110391657, 3105663400665, 21286529888422, 145900036590826, 1000013702089545, 6854195814790005, 46979356835860351, 322001301602738017, 2207029753248402600, 15127206968164865112

Offset: 1

Max Alekseyev, Nov 09 2012

For a Lucas sequence U(k,1), the sum of the cubes of the first n terms is divisible by the sum of the first n terms. This sequence corresponds to the case of k=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A001906, A027941, A163198, A219021.

Table[Fibonacci[2*n+1]/4 + LucasL[4*n+2]/20 - 2/5, {n, 1, 20}] (* Vaclav Kotesovec, May 23 2013 *)
With[{f=Fibonacci[Range[2,50,2]]},Accumulate[f^3]/Accumulate[f]] (* Harvey P. Dale, Feb 17 2020 *)

Vec(x*(1-4*x+x^2)/((1-x)*(1-7*x+x^2)*(1-3*x+x^2)) + O(x^100)) \\ Altug Alkan, Dec 09 2015

a(n) = Sum_{k=1..n} A001906(k)^3 / Sum_{k=1..n} A001906(k).
a(n) = A163198(n) / A027941(n).
a(n) = 11*a(n-1) - 33*a(n-2) + 33*a(n-3) - 11*a(n-4) + a(n-5). - Vaclav Kotesovec, May 23 2013
G.f.: x*(1-4*x+x^2)/((1-x)*(1-7*x+x^2)*(1-3*x+x^2)). [Bruno Berselli, Jun 07 2013]

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 10, 32, 101, 319, 1006, 3172, 10001, 31531, 99410, 313416, 988125, 3115319, 9821846, 30965900, 97627977, 307797347, 970410426, 3059468848, 9645763669, 30410754735, 95877738174, 302279267892, 953013259777, 3004619799579, 9472837914274, 29865561746840

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-2).

Partial sums of S(n, x), for x=1...10, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784.

Partial sums of A077833.

LinearRecurrence[{3,1,-1,-2},{1,3,10,32},30] (* Harvey P. Dale, May 12 2024 *)

Vec((1-x)^(-1)/(1-2*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

From Wesley Ivan Hurt, Jun 26 2022: (Start)
G.f.: (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).
a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4). (End)

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

Original entry on oeis.org

1, 3, 11, 42, 157, 588, 2204, 8259, 30949, 115977, 434606, 1628619, 6103000, 22870056, 85702025, 321155187, 1203479779, 4509855786, 16899992477, 63330128340, 237319937332, 889320046107, 3332590398005, 12488371098225, 46798254229006, 175369276077483

Offset: 0

N. J. A. Sloane, Nov 17 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,3).

Partial sums of S(n, x), for x=1...15, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829.

CoefficientList[Series[1/(1-3x-2x^2-3x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,3},{1,3,11},40] (* Harvey P. Dale, Nov 05 2021 *)

A101919 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at even heights.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 12, 8, 1, 1, 33, 42, 13, 1, 1, 88, 183, 102, 19, 1, 1, 232, 717, 624, 205, 26, 1, 1, 609, 2622, 3275, 1650, 366, 34, 1, 1, 1596, 9134, 15473, 11020, 3716, 602, 43, 1, 1, 4180, 30691, 67684, 64553, 30520, 7483, 932, 53, 1, 1, 10945, 100284, 279106

Offset: 0

Emeric Deutsch, Dec 20 2004

A Schroeder path of length 2n is a lattice path starting from (0,0), ending at (2n,0), consisting only of steps U=(1,1) (up steps), D=(1,-1) (down steps) and H=(2,0) (level steps) and never going below the x-axis (Schroeder paths are counted by the large Schroeder numbers, A006318). Also number of Schroeder paths of length 2n and having k humps. A hump is an up step U followed by 0 or more level steps H followed by a down step D. The T(3,2)=8 Schroeder paths of length 6 and having 2 humps are: H(UD)(UD), (UD)H(UD), (UD)(UD)H, (UD)(UHD), (UD)(UUDD), (UHD)(UD), (UUDD)(UD) and U(UD)(UD)D, the humps being shown between parentheses. Row sums are the large Schroeder numbers (A006318). Column 1 yields the odd-indexed Fibonacci numbers minus 1 (A027941). T(n,n-1)=A034856(n)=binomial(n + 1, 2) + n - 1.

Product A085478*A090181 (Morgan-Voyce times Narayana). [From Paul Barry, Jan 29 2009]

			T(3,2)=8 because we have HU'DU'D, U'DHU'D, U'DU'DH, U'DU'HD, U'DU'UDD, U'HDU'D, U'UDDU'D and U'UU'DDD, the up steps starting at an even height being shown with a prime sign.
Triangle begins:
1;
1,1;
1,4,1;
1,12,8,1;
1,33,42,13,1;

Cf. A006318, A027941, A034856, A101920.

G:=1/2/(-z+z^2)*(-1+z+t*z-z^2+sqrt(1-6*z-2*t*z+11*z^2+2*t*z^2-6*z^3+t^2*z^2-2*t*z^3+z^4)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields the sequence in triangular form

G.f.=G=G(t, z) satisfies z(1-z)G^2-(1-z-tz+z^2)G+1-z=0.

G.f.: 1/(1-x-xy/(1-x-x/(1-x-xy/(1-x-xy/(1-x-x/(1-x-xy/(1-.... (continued fraction). [From Paul Barry, Jan 29 2009]

A105073 Define a(1)=0, a(2)=2 then a(n) = 3a(n-1) - a(n-2), a(n+1) = 3a(n)-a(n-1) and a(n+2) = 3*a(n+1) - a(n) + 2.

Original entry on oeis.org

0, 2, 6, 16, 44, 116, 304, 798, 2090, 5472, 14328, 37512, 98208, 257114, 673134, 1762288, 4613732, 12078908, 31622992, 82790070, 216747218, 567451584, 1485607536, 3889371024, 10182505536, 26658145586, 69791931222, 182717648080, 478361013020, 1252365390980

Offset: 1

Pierre CAMI, Apr 06 2005

From Jon E. Schoenfield, Jan 18 2019: (Start)
Previously, the Name had included the comment, "This sequence is such that 20*(a(n)^2) + 20*a(n) + 1 = j^2 = a square."
However, Anthony Hernandez observed that this statement is not true for all terms; e.g., at a(4)=16, 20*16^2 + 20*16 + 1 = 5441, a nonsquare.
It is true that 20*a(n)^2 + 20*a(n) + 1 = A305315(n/3)^2 when n == 0 (mod 3) and A305316((n-2)/3)^2 when n == 2 (mod 3); however, for n == 1 (mod 3) with n > 1, sqrt(20*a(n)^2 + 20*a(n) + 1) is a noninteger number whose fractional part apparently approaches 3 - sqrt(5) as n increases, and Andrey Zabolotskiy observes that round(sqrt(20*a(n)^2 + 20*a(n) + 1) + sqrt(5)) appears to be equal to A002878(n). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,1,-3,1).

Cf. A001519, A061347, A027941.

Cf. A305315, A305316.

I:=[0,2,6,16,44]; [n le 5 select I[n] else 3*Self(n-1) - Self(n-2) + Self(n-3) - 3*Self(n-4) + Self(n-5): n in [1..35]]; // Vincenzo Librandi, Jan 13 2019

a[n_]:=(1/6)*(Fibonacci[2*n+4] - 2*Fibonacci[2*n] - 2*Cos[(n+2)*(2*Pi/3)] - 4 ); Array[a,50] (* Stefano Spezia, Jan 11 2019 *)
RecurrenceTable[{a[1]==0, a[2]==2, a[3]==6, a[4]==16, a[5]==44, a[n]== 3 a[n-1] - a[n-2] + a[n-3] - 3 a[n-4] + a[n-5]}, a, {n, 35}] (* Vincenzo Librandi, Jan 13 2019 *)

a(n) = (1/6)*(Fibonacci(2n+4) - 2*Fibonacci(2n) - 2*cos((n+2)(2*Pi/3)) - 4). - Ralf Stephan, May 20 2007
From R. J. Mathar, Nov 13 2009: (Start)
a(n) = 3*a(n-1) - a(n-2) + a(n-3) - 3*a(n-4) + a(n-5).
G.f.: 2*x^2/((1-x) * (1+x+x^2) * (1-3*x+x^2)).
a(n) = A061347(n+2)/6 + A001519(n+2)/2 - 2/3. (End)
a(n) = floor(A027941(n)/2). - Anthony Hernandez, Jan 03 2019

Extended by R. J. Mathar, Nov 13 2009

A134561 Array T by antidiagonals: T(n,k) = k-th number whose Zeckendorf representation has exactly n terms.

Original entry on oeis.org

1, 2, 4, 3, 6, 12, 5, 7, 17, 33, 8, 9, 19, 46, 88, 13, 10, 20, 51, 122, 232, 21, 11, 25, 53, 135, 321, 609, 34, 14, 27, 54, 140, 355, 842, 1596, 55, 15, 28, 67, 142, 368, 931, 2206, 4180, 89, 16, 30, 72, 143, 373, 965

Offset: 1

Clark Kimberling, Nov 01 2007

A permutation of the natural numbers.

Except for initial terms in some cases, (Row 1) = A000045 (Row 2) = A095096 (Row 3) = A059390 (Row 4) = A111458 (Col 1) = A027941 (Col 2) = A005592.

			19 = 13 + 5 + 1 is the 3rd-largest number (after 12 and 17) that has a 3-term Zeckendorf representation; i.e., the (unique) sum of distinct non-neighboring Fibonacci numbers.
Northwest corner:
1 2 3 5 8 13
4 6 7 9 10 11
12 17 19 20 25 27
33 46 51 53 54 67

C. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
Index entries for sequences that are permutations of the natural numbers

Cf. A035513.

A214729 Member m=6 of the m-family of sums b(m,n) = Sum_{k=0..n} F(k+m)*F(k), m >= 0, n >= 0, with the Fibonacci numbers F.

Original entry on oeis.org

0, 13, 34, 102, 267, 712, 1864, 4893, 12810, 33550, 87835, 229968, 602064, 1576237, 4126642, 10803702, 28284459, 74049688, 193864600, 507544125, 1328767770, 3478759198, 9107509819, 23843770272, 62423800992, 163427632717, 427859097154, 1120149658758

Offset: 0

Wolfdieter Lang, Jul 27 2012

See the comment section on A080144 for the general formula and the o.g.f. for b(m,n).

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

Cf. A001654, A027941, A059840(n+2), A064831, A080097, A080143 and A080144 for the m=0,1,...,5 members.

Cf. A027941.

[(9*(-1)^(n+1)-20+Lucas(2*n+7))/5: n in [0..40]]; // Vincenzo Librandi, Aug 26 2017

With[{m = 6}, Table[Sum[Fibonacci[k + m]*Fibonacci[k], {k, 0, n}], {n, 0, 25}]] (* or *)
Table[(9 (-1)^(n + 1) - 20 + LucasL[2 n + 7])/5, {n, 0, 25}] (* Michael De Vlieger, Aug 23 2017 *)
LinearRecurrence[{3,0,-3,1},{0,13,34,102},40] (* Harvey P. Dale, Jun 13 2022 *)

concat(0, Vec(x*(13 - 5*x) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)) + O(x^30))) \\ Colin Barker, Aug 25 2017

[fibonacci(n+3)*fibonacci(n+4) - 2*(2+(-1)^n) for n in range(41)] # G. C. Greubel, Dec 31 2023

a(n) = b(6,n) = 4*A027941(n) + 9*A001654(n), with A027941(n) = Fibonacci(2*n+1) - 1 and A001654(n) = Fibonacci(n+1)*Fibonacci(n), n >= 0. 4 = Fibonacci(6)/2 and 9 = LucasL(6)/2.
O.g.f.: x*(13-5*x)/((1-x^2)*(1-3*x+x^2)) (see a comment above). - Wolfdieter Lang, Jul 30 2012
a(n) = (9*(-1)^(n+1) - 20 + Lucas(2*n + 7))/5. - Ehren Metcalfe, Aug 21 2017
From Colin Barker, Aug 25 2017: (Start)
a(n) = (1/10)*((29 - 13*sqrt(5))*((3 - sqrt(5))/2)^n + (29 + 13*sqrt(5))*((3 + sqrt(5))/2)^n - 2*(20 + 9*(-1)^n) ).
a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = A001654(n+3) - 2*(2 + (-1)^n). - G. C. Greubel, Dec 31 2023

A293064 Number of vertices of type B at level n of the hyperbolic Pascal pyramid PP_(4,5).

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 33, 88, 232, 609, 1596, 4180, 10945, 28656, 75024, 196417, 514228, 1346268, 3524577, 9227464, 24157816, 63245985, 165580140, 433494436, 1134903169, 2971215072, 7778742048, 20365011073, 53316291172, 139583862444, 365435296161, 956722026040

Offset: 0

Eric M. Schmidt, Sep 30 2017

a(n+2) = A027941(n) = Fibonacci(2n+1) - 1 for n >= 0. - Georg Fischer, Oct 09 2018

Colin Barker, Table of n, a(n) for n = 0..1000
László Németh, Pascal pyramid in the space H^2 x R, arXiv:1701.06022 [math.CO], 2017 (2nd line of Table 1).
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A293066, A027941, A293070.

Join[{0}, LinearRecurrence[{4, -4, 1}, {0, 0, 1}, 31]] (* Jean-François Alcover, Oct 07 2017 *)

concat(vector(3), Vec(x^3 / ((1 - x)*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Oct 07 2017

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 4.
From Colin Barker, Oct 07 2017: (Start)
G.f.: x^3 / ((1 - x)*(1 - 3*x + x^2)).
a(n) = -1 + (2^(-n)*((3-sqrt(5))^n*(2+sqrt(5)) + (-2+sqrt(5))*(3+sqrt(5))^n)) / sqrt(5) for n>0.
(End)

cp's OEIS Frontend

A214985 Array: T(m,n) = (F(n) + F(2*n) + ... + F(n*m))/F(n), by antidiagonals; transpose of A214984.

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A219020 Sum of the cubes of the first n even-indexed Fibonacci numbers divided by the sum of the first n terms.

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A077826 Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

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A077827 Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).

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A077830 Expansion of 1/(1-3*x-2*x^2-3*x^3).

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A101919 Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k up steps starting at even heights.

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A105073 Define a(1)=0, a(2)=2 then a(n) = 3*a(n-1) - a(n-2), a(n+1) = 3*a(n)-a(n-1) and a(n+2) = 3*a(n+1) - a(n) + 2.

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A134561 Array T by antidiagonals: T(n,k) = k-th number whose Zeckendorf representation has exactly n terms.

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A214985 Array: T(m,n) = (F(n) + F(2n) + ... + F(nm))/F(n), by antidiagonals; transpose of A214984.

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

A105073 Define a(1)=0, a(2)=2 then a(n) = 3a(n-1) - a(n-2), a(n+1) = 3a(n)-a(n-1) and a(n+2) = 3*a(n+1) - a(n) + 2.