A156093 -id:A156093 - cp's OEIS frontend

A214984 Array: T(m,n) = (F(m) + F(2m) + ... + F(nm))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

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

Offset: 1

Clark Kimberling, Oct 28 2012

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

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

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

Cf. A214978, A214985, A214986.

F[n_] := Fibonacci[n]; L[n_] := LucasL[n];
t[m_, n_] := (1/F[m])*Sum[F[m*k], {k, 1, n}]
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 rows (m odd):
T(m,n) = (F(m*n+m) + F(m*n) - F(m))/(F(m)*L(m)).
For even-numbered rows (m even):
T(m,n) = (F(m*n+m) - F(m*n) - F(m))/(F(m)*(L(m)-2)).

A156086 Sum of the squares of the first n Fibonacci numbers with index divisible by 4.

Original entry on oeis.org

0, 9, 450, 21186, 995355, 46760580, 2196752004, 103200583725, 4848230683206, 227763641527110, 10700042921091135, 502674253649756424, 23614989878617461000, 1109401850041370910801, 52118271962065815346890, 2448449380367051950393290

Offset: 0

Stuart Clary, Feb 04 2009

Natural bilateral extension (brackets mark index 0): ..., -21186, -450, -9, 0, [0], 9, 450, 21186, 995355, ... This is (-A156086)-reversed followed by A156086. That is, A156086(-n) = -A156086(n-1).

Cf. A156087, A156092, A156093.

a[n_Integer] := If[ n >= 0, Sum[ Fibonacci[4k]^2, {k, 1, n} ], -Sum[ Fibonacci[-4k]^2, {k, 1, -n - 1} ] ]

Let F(n) be the Fibonacci number A000045(n).
a(n) = sum_{k=1..n} F(4k)^2.
Closed form: a(n) = F(8n+4)/15 - (2n + 1)/5.
Recurrence: a(n) - 48 a(n-1) + 48 a(n-2) - a(n-3) = 18.
Recurrence: a(n) - 49 a(n-1) + 96 a(n-2) - 49 a(n-3) + a(n-4) = 0.
G.f.: A(x) = (9 x + 9 x^2)/(1 - 49 x + 96 x^2 - 49 x^3 + x^4) = 9 x (1 + x)/((1 - x)^2 (1 - 47 x + x^2)).

A156087 One ninth of the sum of the squares of the first n Fibonacci numbers with index divisible by 4.

Original entry on oeis.org

0, 1, 50, 2354, 110595, 5195620, 244083556, 11466731525, 538692298134, 25307071280790, 1188893657899015, 55852694849972936, 2623887764290829000, 123266872226818990089, 5790919106896201705210, 272049931151894661154810

Offset: 0

Stuart Clary, Feb 04 2009

Natural bilateral extension (brackets mark index 0): ..., -110595, -2354, -50, -1, 0, [0], 1, 50, 2354, 110595, 5195620, ... This is (-A156087)-reversed followed by A156087. That is, A156087(-n) = -A156087(n-1).

Cf. A156086, A156092, A156093.

a[n_Integer] := If[ n >= 0, Sum[ (1/9) Fibonacci[4k]^2, {k, 1, n} ], -Sum[ (1/9) Fibonacci[-4k]^2, {k, 1, -n - 1} ] ]
Accumulate[Fibonacci[4Range[0,20]]^2]/9 (* Harvey P. Dale, Sep 22 2011 *)

Let F(n) be the Fibonacci number A000045(n).
a(n) = sum_{k=1..n} F(4k)^2.
Closed form: a(n) = F(8n+4)/135 - (2n + 1)/45.
Recurrence: a(n) - 48 a(n-1) + 48 a(n-2) - a(n-3) = 2.
Recurrence: a(n) - 49 a(n-1) + 96 a(n-2) - 49 a(n-3) + a(n-4) = 0.
G.f.: A(x) = (x + x^2)/(1 - 49 x + 96 x^2 - 49 x^3 + x^4) = x (1 + x)/((1 - x)^2 (1 - 47 x + x^2)).

A156092 Alternating sum of the squares of the first n Fibonacci numbers with index divisible by 4.

Original entry on oeis.org

0, -9, 432, -20304, 953865, -44811360, 2105180064, -98898651657, 4646131447824, -218269279396080, 10254010000167945, -481720200728497344, 22630595424239207232, -1063156264738514242569, 49945713847285930193520, -2346385394557700204852880

Offset: 0

Stuart Clary, Feb 04 2009

Natural bilateral extension (brackets mark index 0): ..., -953865, 20304, -432, 9, 0, [0], -9, 432, -20304, 953865, -44811360, ... This is (-A156092)-reversed followed by A156092. That is, A156092(-n) = -A156092(n-1).

Cf. A156086, A156087, A156093.

a[n_Integer] := If[ n >= 0, Sum[ (-1)^k Fibonacci[4k]^2, {k, 1, n} ], Sum[ -(-1)^k Fibonacci[-4k]^2, {k, 1, -n - 1} ] ]

Let F(n) be the Fibonacci number A000045(n) and let L(n) be the Lucas number A000032(n).
a(n) = sum_{k=1..n} (-1)^k F(4k)^2.
Closed form: a(n) = (-1)^n (L(8n+4) - 7)/35.
Factored closed form: a(n) = (-1)^n F(4n) F(4n+4)/7.
Recurrence: a(n) + 47 a(n-1) + a(n-2) = (-1)^n 9.
Recurrence: a(n) + 48 a(n-1) + 48 a(n-2) + a(n-3) = 0.
G.f.: A(x) = -9 x/(1 + 48 x + 48 x^2 + x^3) = -9 x/((1 + x)(1 + 47 x + x^2)).

A161582 The list of the k values in the common solutions to the 2 equations 5k+1=A^2, 9k+1=B^2.

Original entry on oeis.org

0, 7, 336, 15792, 741895, 34853280, 1637362272, 76921173511, 3613657792752, 169764995085840, 7975341111241735, 374671267233275712, 17601574218852716736, 826899317018844410887, 38846666325666834594960, 1824966417989322381552240, 85734574979172485098360327

Offset: 1

Paul Weisenhorn, Jun 14 2009

The 2 equations are equivalent to the Pell equation x^2-45*y^2=1, with x=(45*k+7)/2 and y= A*B/2, case C=5 in A160682.

Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Cf. A160682, A049685 (sequence of A), A033890 (sequence of B).

t:=0: for n from 0 to 1000000 do a:=sqrt(5*n+1); b:=sqrt(9*n+1);
if (trunc(a)=a) and (trunc(b)=b) then t:=t+1; print(t,n,a,b): end if: end do:

LinearRecurrence[{48,-48,1},{0,7,336},30] (* or *) Rest[CoefficientList[ Series[ -7x^2/((x-1)(x^2-47x+1)),{x,0,30}],x]] (* Harvey P. Dale, Mar 21 2013 *)

k(t+3) = 48*(k(t+2)-k(t+1))+k(t).
With w = sqrt(5),
k(t) = ((7+3*w)*((47+21*w)/2)^(t-1)+(7-3*w)*((47-21*w)/2)^(t-1))/90.
k(t) = floor((7+3*w)*((47+21*w)/2)^(t-1)/90) = 7*|A156093(t-1)|.
G.f.: -7*x^2/((x-1)*(x^2-47*x+1)).
a(1)=0, a(2)=7, a(3)=336, a(n) = 48*a(n-1)-48*a(n-2)+a(n-3). - Harvey P. Dale, Mar 21 2013

Edited, extended by R. J. Mathar, Sep 02 2009

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

Original entry on oeis.org

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)).

A049654 a(n) = (F(8*n+1) - 1)/3 where F=A000045 (the Fibonacci sequence).

Original entry on oeis.org

0, 11, 532, 25008, 1174859, 55193380, 2592914016, 121811765387, 5722560059188, 268838511016464, 12629687457714635, 593326472001571396, 27873714496616140992, 1309471254868957055243, 61517275264344365455444

Offset: 0

Clark Kimberling

G. C. Greubel, Table of n, a(n) for n = 0..595
Index entries for linear recurrences with constant coefficients, signature (48,-48,1).

Cf. A000045, A156093.

[(Fibonacci(8*n+1) - 1)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017

LinearRecurrence[{48, -48, 1}, {0, 11, 532}, 50] (* or *) Table[( Fibonacci[8*n+1]-1)/3, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)
CoefficientList[Series[-x(11+4x)/((x-1)(x^2-47*x+1)),{x,0,14}],x] (* Stefano Spezia, Feb 18 2024 *)

for(n=0,30, print1((fibonacci(8*n+1) - 1)/3, ", ")) \\ G. C. Greubel, Dec 02 2017

From R. J. Mathar, Oct 26 2015: (Start)
G.f.: -x*(11+4*x) / ( (x-1)*(x^2-47*x+1) ).
a(n) = 11*|A156093(n)|+4*|A156093(n-1)|. (End)

cp's OEIS Frontend

A214984 Array: T(m,n) = (F(m) + F(2*m) + ... + F(n*m))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

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A156086 Sum of the squares of the first n Fibonacci numbers with index divisible by 4.

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A156087 One ninth of the sum of the squares of the first n Fibonacci numbers with index divisible by 4.

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A156092 Alternating sum of the squares of the first n Fibonacci numbers with index divisible by 4.

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A161582 The list of the k values in the common solutions to the 2 equations 5*k+1=A^2, 9*k+1=B^2.

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

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A049654 a(n) = (F(8*n+1) - 1)/3 where F=A000045 (the Fibonacci sequence).

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A214984 Array: T(m,n) = (F(m) + F(2m) + ... + F(nm))/F(m), by antidiagonals, where F = A000045 (Fibonacci numbers).

A161582 The list of the k values in the common solutions to the 2 equations 5k+1=A^2, 9k+1=B^2.

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