A033887 -id:A033887 - cp's OEIS frontend

A134502 a(n) = Fibonacci(7n + 4).

3, 89, 2584, 75025, 2178309, 63245986, 1836311903, 53316291173, 1548008755920, 44945570212853, 1304969544928657, 37889062373143906, 1100087778366101931, 31940434634990099905, 927372692193078999176, 26925748508234281076009

Offset: 0

Artur Jasinski, Oct 28 2007

Index entries for linear recurrences with constant coefficients, signature (29,1)

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888, A033889, A033890, A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

[Fibonacci(7*n +4): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

Mathematica
```
Table[Fibonacci[7n+4], {n, 0, 30}]
```

a(n)=fibonacci(7*n+4) \\ Charles R Greathouse IV, Jun 11 2015

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-3-2*x) / (-1 + 29*x + x^2).
a(n) = 2*A049667(n) + 3*A049667(n+1). (End)
a(n) = A000045(A017029(n)). - Michel Marcus, Nov 07 2013

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 17 2011

A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

Original entry on oeis.org

3, 11, 47, 199, 843, 3571, 15127, 64079, 271443, 1149851, 4870847, 20633239, 87403803, 370248451, 1568397607, 6643838879, 28143753123, 119218851371, 505019158607, 2139295485799, 9062201101803, 38388099893011

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

Binomial transform of A163062. Second binomial transform of A163114. Inverse binomial transform of A098648 without initial 1.

Nathaniel Johnston, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A163062, A163114, A098648, A001077 (L(3*n)/L(2)), A048876 (L(3*n+1)).

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((3+r)*(2+r)^n+(3-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

[Lucas(3*n+2): n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

with(combinat):A163063:=proc(n)return fibonacci(3*n+1) + fibonacci(3*n+3): end:seq(A163063(n), n=0..21); # Nathaniel Johnston, Apr 18 2011

Table[Fibonacci[3n + 1] + Fibonacci[3n + 3], {n, 0, 21}] (* Alonso del Arte, Nov 29 2010 *)
LinearRecurrence[{4,1},{3,11},30] (* Harvey P. Dale, Apr 14 2021 *)

Vec((3-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

a(n) = 4*a(n-1)+a(n-2) for n > 1; a(0) = 3, a(1) = 11.
G.f.: (3-x)/(1-4*x-x^2).
a(n) = A033887(n) + A014445(n+1).
a(n) = ((3+sqrt(5))*(2+sqrt(5))^n+(3-sqrt(5))*(2-sqrt(5))^n)/2.
a(n) = A000032(3*n+2), n>=0, (Lucas trisection). - Wolfdieter Lang, Mar 09 2011.
a(n) = 5*F(n)*F(n+1)*L(n+1) + L(n+2)*(-1)^n with F(n)=A000045(n) and L(n)=A000032(n). - J. M. Bergot, Dec 10 2015

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 28, 45, 117, 189, 494, 799, 2091, 3383, 8856, 14329, 37513, 60697, 158906, 257115, 673135, 1089155, 2851444, 4613733, 12078909, 19544085, 51167078, 82790071, 216747219, 350704367, 918155952, 1485607537, 3889371025

Offset: 1

Roger L. Bagula, Jun 12 2005

A switched sequence with alternating limits of the golden mean and its square. The sequence uses only one initial term. Note that lim_{n->oo} a(n)/a(n-1) does not exist.

The consecutive pairs (2,3), (7,11), (28,45) occur as pairs in columns 2 and 3 of the Wythoff array, A035513. Suppose (l(n)) and (u(n)) are the lower and upper Beatty sequences of positive irrational numbers rClark Kimberling, Nov 24 2010

Index entries for linear recurrences with constant coefficients, signature (1,4,-4,1,-1).

Cf. A000045, A000201, A001950, A015448, A033887.

[ n eq 1 select 1 else Floor(((Sqrt(5)+1)/2+(n mod 2))*Self(n-1)): n in [1..35] ];

Phi = N[(Sqrt[5] + 1)/2] F[1] = 1; F[n__] := F[n] = If[Mod[n, 2] == 0, Floor[Phi*F[n - 1]], Floor[(Phi + 1)*F[n -1]]] a = Table[F[n], {n, 1, 50}]
LinearRecurrence[{1,4,-4,1,-1},{1,1,2,3,7},40] (* Harvey P. Dale, Mar 31 2023 *)

PARI
```
a(n)=if(n<2,1,floor((phi+n%2)*a(n-1)))
```

G.f.: -x*(-1+3*x^2-x^3+x^4) / ( (x-1)*(x^4+4*x^2-1) ). - R. J. Mathar, Sep 11 2011
a(2n+2) = (1/2)*(Fib(3n+2) + 1), a(2n+1) = (1/2)*(Fib(3n+1) + 1).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) + a(n-4) - a(n-5). - Wesley Ivan Hurt, May 04 2025

Edited and better name by Ralf Stephan, Nov 24 2010

A140413 a(2n) = A000045(6n) + 1, a(2n+1) = A000045(6n+3) - 1.

Original entry on oeis.org

1, 1, 9, 33, 145, 609, 2585, 10945, 46369, 196417, 832041, 3524577, 14930353, 63245985, 267914297, 1134903169, 4807526977, 20365011073, 86267571273, 365435296161, 1548008755921, 6557470319841, 27777890035289, 117669030460993, 498454011879265, 2111485077978049

Offset: 0

Paul Curtz, Jun 17 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A000045, A014445, A141325.

a:=[1,1,9];; for n in [4..30] do a[n]:=3*a[n-1]+5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Jun 08 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)^2/((1+x)*(1-4*x-x^2)) )); // G. C. Greubel, Jun 08 2019

LinearRecurrence[{3,5,1},{1,1,9},30] (* or *) CoefficientList[Series[ (1-x)^2/((1+x)(1-4*x-x^2)),{x,0,30}],x] (* Harvey P. Dale, Jun 20 2011 *)

Vec((1-x)^2/((1+x)*(1-4*x-x^2)) + O(x^30)) \\ Colin Barker, Jun 06 2017

((1-x)^2/((1+x)*(1-4*x-x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 08 2019

a(n) = A141325(3*n) = (-1)^n + A014445(n).
a(n) = +3*a(n-1) +5*a(n-2) +a(n-3). - R. J. Mathar, Dec 17 2010
G.f.: (1-x)^2 / ( (1+x)*(1-4*x-x^2) ). - R. J. Mathar, Dec 17 2010
a(n) = ((-1)^n + (-(2-sqrt(5))^n + (2+sqrt(5))^n) / sqrt(5)). - Colin Barker, Jun 06 2017
a(n) = -A033887(n) + 2*Sum_{k=0..n} A033887(k)*(-1)^(n-k). - Yomna Bakr and Greg Dresden, Jun 03 2024

A048878 Generalized Pellian with second term of 9.

Original entry on oeis.org

1, 9, 37, 157, 665, 2817, 11933, 50549, 214129, 907065, 3842389, 16276621, 68948873, 292072113, 1237237325, 5241021413, 22201322977, 94046313321, 398386576261, 1687592618365, 7148757049721, 30282620817249, 128279240318717, 543399582092117, 2301877568687185

Offset: 0

Barry E. Williams

			a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=9.

Harry J. Smith, Table of n, a(n) for n = 0..1584
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000045, A015448, A001077, A001076, A033887.

with(combinat): a:=n->5*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4,1},{1,9},31] (* or *) CoefficientList[ Series[ (1+5x)/(1-4x-x^2),{x,0,30}],x] (* Harvey P. Dale, Jul 12 2011 *)

{ default(realprecision, 2000); for (n=0, 2000, a=round(((7+sqrt(5))*(2+sqrt(5))^n - (7-sqrt(5))*(2-sqrt(5))^n )/10*sqrt(5)); if (a > 10^(10^3 - 6), break); write("b048878.txt", n, " ", a); ); } \\ Harry J. Smith, May 31 2009

a(n) = ( (7+sqrt(5))(2+sqrt(5))^n - (7-sqrt(5))(2-sqrt(5))^n )/2*sqrt(5).
G.f.: (1+5*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = F(3*n+3) + F(3*n-2); F = A000045. - Yomna Bakr and Greg Dresden, May 25 2024

A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

Original entry on oeis.org

0, 1, 3, 14, 58, 247, 1045, 4428, 18756, 79453, 336567, 1425722, 6039454, 25583539, 108373609, 459077976, 1944685512, 8237820025, 34895965611, 147821682470, 626182695490, 2652552464431, 11236392553213, 47598122677284

Offset: 0

Creighton Dement, Jul 24 2005

A001076(n) = a(n) + a(n+1). Program "Superseeker" finds: A033887(n+1) = a(n+2) - a(n); Elements of even index in the sequence: A049661(n) = (F(6n+1)-1)/4; A015448(n+2) = a(n+2) + 2*a(n+1) + a(n)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3, 5, 1).

Cf. A110526, A110527, A033887, A001076, A049661, A033887, A000045.

seriestolist(series(-x/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1jbaseseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')]

Table[(Fibonacci[3n+1]-(-1)^n)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

concat(0, Vec(x/((1+x)*(1-x^2-4*x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

G.f.: -x/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n/2 * Sum_{k=0..n} (-1)^k*Fibonacci(3*k). - Gary Detlefs, Jan 03 2013
a(n) = (Fibonacci(3*n+1)-(-1)^n)/4, where Fibonacci(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015

A122070 Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.

Original entry on oeis.org

1, 1, 2, 2, 6, 5, 3, 15, 24, 13, 5, 32, 78, 84, 34, 8, 65, 210, 340, 275, 89, 13, 126, 510, 1100, 1335, 864, 233, 21, 238, 1155, 3115, 5040, 4893, 2639, 610, 34, 440, 2492, 8064, 16310, 21112, 17080, 7896, 1597, 55, 801, 5184, 19572, 47502, 76860, 82908, 57492, 23256, 4181

Offset: 0

Philippe Deléham, Oct 15 2006, Mar 13 2012

Subtriangle of (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Mirror image of the triangle in A185384.

			Triangle begins:
   1;
   1,   2;
   2,   6,   5;
   3,  15,  24,   13;
   5,  32,  78,   84,   34;
   8,  65, 210,  340,  275,  89;
  13, 126, 510, 1100, 1335, 864, 233;
(0, 1, 1, -1, 0, 0, ...) DELTA (1, 1, 1, 0, 0, ...) begins :
  1;
  0,  1;
  0,  1,   2;
  0,  2,   6,   5;
  0,  3,  15,  24,   13;
  0,  5,  32,  78,   84,   34;
  0,  8,  65, 210,  340,  275,  89;
  0, 13, 126, 510, 1100, 1335, 864, 233;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000045, A001519, A033887, A033889, A185384.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n+ k+1) ))); # G. C. Greubel, Oct 02 2019

[Binomial(n,k)*Fibonacci(n+k+1): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 02 2019

with(combinat): seq(seq(binomial(n,k)*fibonacci(n+k+1), k=0..n), n=0..10); # G. C. Greubel, Oct 02 2019

Table[Fibonacci[n+k+1]*Binomial[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 02 2019 *)

T(n,k) = binomial(n,k)*fibonacci(n+k+1);
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 02 2019

[[binomial(n,k)*fibonacci(n+k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Oct 02 2019

T(n,k) = A000045(n+k+1)*A007318(n,k) .
T(n,n) = Fibonacci(2*n+1) = A001519(n+1) .
Sum_{k=0..n} T(n,k) = Fibonacci(3*n+1) = A033887(n) .
Sum_{k=0..n}(-1)^k*T(n,k) = (-1)^n = A033999(n) .
Sum_{k=0..floor(n/2)} T(n-k,k) = (Fibonacci(n+1))^2 = A007598(n+1).
Sum_{k=0..n} T(n,k)*2^k = Fibonacci(4*n+1) = A033889(n).
Sum_{k=0..n} T(n,k)^2 = A208588(n).
G.f.: (1-y*x)/(1-(1+3y)*x-(1+y-y^2)*x^2).
T(n,k) = T(n-1,k) + 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185384(n,n-k).
T(2n,n) = binomial(2n,n)*Fibonacci(3*n+1) = A208473(n).

Corrected and extended by Philippe Deléham, Mar 13 2012

Term a(50) corrected by G. C. Greubel, Oct 02 2019

A363753 a(n) = Sum_{k=0..n} (-1)^kF(k-1)F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

Original entry on oeis.org

0, 0, 1, -2, 13, -47, 213, -879, 3762, -15873, 67342, -285098, 1207966, -5116586, 21674919, -91815276, 388937619, -1647563169, 6979194475, -29564334305, 125236542640, -530510487155, 2247278519916, -9519624520452, 40325776676748, -170822731106052, 723616701297373

Offset: 0

Hans J. H. Tuenter, Jun 19 2023

Alternating sum of the product of three consecutive Fibonacci numbers, divided by two.
Can also be seen as the alternating sum of the Fibonomial coefficients (n+1,3), A001655.
This sequence is part of a suite of sums over triple products of Fibonacci numbers. Subba Rao (1953) gives closed-form expressions for several Fibonacci sums of this type.

K. Subba Rao, Some properties of Fibonacci numbers, The American Mathematical Monthly, 60(10):680-684, December 1953.
Index entries for linear recurrences with constant coefficients, signature (-2,9,-3,-4,1).

Cf. A000045, A001655.
Other sequences with the product of three Fibonacci numbers as a summand (the sequence may have a shifted [and scaled] version of the summand given here).
A005968: F(k)^3,  A119284: (-1)^k*F(k)^3,  A215037: F(k-1)*F(k)*F(k+1),
A363753: (-1)^k*F(k-1)*F(k)*F(k+1),  A163198: F(2k)^3,  A163200: F(2k+1)^3,
A256178: F(2k)*F(2k+1)*F(2k+2),  this sequence: (-1)^k*F(k-1)*F(k)*F(k+1),
A363754: F(2k-1)*F(2k)*F(2k+1).

LinearRecurrence[{-2, 9, -3, -4, 1}, {0, 0, 1, -2, 13}, 27]

a(n) = ((-1)^n*(F(n+1)^3 - F(n)^3) + F(n+2) - 2)/8.
a(n) = ((-1)^n*F(3*n+1) + 4*F(n+2) - 5)/20.
a(n) = -2*a(n-1) + 9*a(n-2) - 3*a(n-3) - 4*a(n-4) + a(n-5).
a(-n) = A215037(n-3).
G.f.: x^2/((1 - x)*(1 + 4*x - x^2)*(1 - x - x^2)).
20*a(n) = (-1)^n*A033887(n) + 4*A000045(n+2) - 5. - R. J. Mathar, Jun 27 2023

A100232 Triangle, read by rows, of the coefficients of [x^k] in G100231(x)^n such that the row sums are 5^n-1 for n>0, where G100231(x) is the g.f. of A100231.

Original entry on oeis.org

1, 1, 3, 1, 6, 17, 1, 9, 39, 75, 1, 12, 70, 220, 321, 1, 15, 110, 470, 1165, 1363, 1, 18, 159, 852, 2895, 5922, 5777, 1, 21, 217, 1393, 5943, 16807, 29267, 24475, 1, 24, 284, 2120, 10822, 38536, 93468, 141688, 103681, 1, 27, 360, 3060, 18126, 77274, 236748

Offset: 0

Paul D. Hanna, Nov 29 2004

The main diagonal forms A100233. Secondary diagonal is: T(n+1,n) = (n+1)*A033887(n) = (n+1)*Fibonacci(3*n+1). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

			Rows begin:
[1],
[1,3],
[1,6,17],
[1,9,39,75],
[1,12,70,220,321],
[1,15,110,470,1165,1363],
[1,18,159,852,2895,5922,5777],
[1,21,217,1393,5943,16807,29267,24475],
[1,24,284,2120,10822,38536,93468,141688,103681],...
where row sums form 5^n-1 for n>0:
5^1-1 = 1+3 = 4
5^2-1 = 1+6+17 = 24
5^3-1 = 1+9+39+75 = 124
5^4-1 = 1+12+70+220+321 = 624
5^5-1 = 1+15+110+470+1165+1363 = 3124.
The main diagonal forms A100233 = [1,3,17,75,321,1363,5777,...],
where Sum_{n>=1} A100233(n)/n*x^n = log((1-x)/(1-4*x-x^2)).

Cf. A100231, A100233, A033887, A100229.

PARI
```
T(n,k,m=5)=if(n
				
```

G.f.: A(x, y)=(1-2*x*y+5*x^2*y^2)/((1-x*y)*(1-4*x*y-x^2*y^2-x*(1-x*y))).

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

Original entry on oeis.org

0, 1, 8, 29, 128, 537, 2280, 9653, 40896, 173233, 733832, 3108557, 13168064, 55780809, 236291304, 1000946021, 4240075392, 17961247585, 76085065736, 322301510525, 1365291107840, 5783465941881, 24499154875368

Offset: 0

Creighton Dement, Jul 24 2005

A048878(n) = a(n) + a(n+1). Compare with A110526.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A110526, A110528, A033887, A001076, A049661, A033887.

seriestolist(series(-x*(1+5*x)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')], apart from initial term.

LinearRecurrence[{3,5,1},{0,1,8},30] (* Harvey P. Dale, Feb 12 2015 *)

x='x+O('x^50); concat(0, Vec(-x*(1+5*x)/((1+x)*(x^2+4*x-1)))) \\ G. C. Greubel, Aug 30 2017

G.f.: -x*(1+5*x)/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n + 3*A001076(n) - A015448(n). - Ehren Metcalfe, Nov 18 2017
a(n) = (-1)^n + 2*A110526(n) + A110679(n-2) for n >= 2. - Yomna Bakr and Greg Dresden, May 25 2024

cp's OEIS Frontend

A134502 a(n) = Fibonacci(7n + 4).

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A163063 Lucas(3n+2) = Fibonacci(3n+1) + Fibonacci(3n+3).

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A107857 a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.

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A140413 a(2n) = A000045(6n) + 1, a(2n+1) = A000045(6n+3) - 1.

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A048878 Generalized Pellian with second term of 9.

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A110526 a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

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PARI

Formula

A122070 Triangle given by T(n,k) = Fibonacci(n+k+1)*binomial(n,k) for 0<=k<=n.

Views

A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

A363753 a(n) = Sum_{k=0..n} (-1)^kF(k-1)F(k)*F(k+1)/2, where F(n) is the Fibonacci number A000045(n).

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.