A127647 -id:A127647 - cp's OEIS frontend

A128540 Triangle A127647 * A097806, read by rows.

1, 1, 1, 0, 2, 2, 0, 0, 3, 3, 0, 0, 0, 5, 5, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 13, 13, 0, 0, 0, 0, 0, 0, 21, 21, 0, 0, 0, 0, 0, 0, 0, 34, 34, 0, 0, 0, 0, 0, 0, 0, 0, 55, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 89, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 144

Offset: 1

Gary W. Adamson, Mar 10 2007

Row sums = A094895 starting (1, 2, 4, 6, 10, 16, 26, ...). A128541 = A097806 * A127647.

			First few rows of the triangle:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 3, 3;
  0, 0, 0, 5, 5;
  0, 0, 0, 0, 8, 8;
  ...

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A127647, A097806, A128541.

[k eq n select Fibonacci(n) else k eq n-1 select Fibonacci(n) else 0: k in [1..n], n in [1..15]]; // G. C. Greubel, Jul 11 2019

Table[If[k==n || k==n-1, Fibonacci[n], 0], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)

T(n,k) = if(k==n || k==n-1, fibonacci(n), 0); \\ G. C. Greubel, Jul 11 2019

def T(n, k):
    if (k==n): return fibonacci(n)
    elif (k==n-1): return fibonacci(n)
    else: return 0
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jul 11 2019

A127646 * A097806 as infinite lower triangular matrices.

A131410 A127647 * A000012.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 5, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 21, 21, 21, 21, 21, 21, 21, 21, 34, 34, 34, 34, 34, 34, 34, 34, 34, 55, 55, 55, 55, 55, 55, 55, 55, 55, 55, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 89, 144, 144, 144

Offset: 1

Gary W. Adamson, Jul 08 2007

Row sums = A045925, n*Fib(n): (1, 2, 6, 12, 25, 48,...).

A104763 = A000012 * A127647.

			First few rows of the triangle are:
1;
1, 1;
2, 2, 2;
3, 3, 3, 3;
5, 5, 5, 5, 5;
8, 8, 8, 8, 8, 8;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A127647, A045925, A104763, A000045.

a131410 n k = a131410_tabl !! (n-1) !! (n-1)
a131410_row n = a131410_tabl !! (n-1)
a131410_tabl = zipWith replicate [1..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Table[Fibonacci[n], {n, 15}, {n}] // Flatten (* Vincenzo Librandi, Jan 28 2017 *)

A127647 * A000012 as infinite lower triangular matrices.
Partial sums of A127647 starting from the right, read by rows.
By rows, F(n) occurs n times.

A143061 Triangle read by rows, A000012 * A127647 * A000012.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 5, 6, 7, 5, 8, 10, 11, 12, 8, 13, 16, 18, 19, 20, 13, 21, 26, 29, 31, 32, 33, 21, 34, 42, 47, 50, 52, 53, 54, 34, 55, 68, 76, 81, 84, 86, 87, 88, 55, 89, 110, 123, 131, 136, 139, 141, 142, 143, 89, 144, 178, 199, 212, 220, 225, 228, 230, 231

Offset: 1

Gary W. Adamson, Jul 20 2008

Row sums = A014286 (1, 3, 9, 21, 46, 94, ...); left border = Fibonacci numbers.

			First few rows of the triangle are:
  1;
  1,  2;
  2,  3,  4;
  3,  5,  6,  7;
  5,  8, 10, 11, 12;
  8, 13, 16, 18, 19, 20;
  ...

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Cf. A000012, A000045, A014286, A127647.

seq(seq(combinat:-fibonacci(i+2)-combinat:-fibonacci(i+2-j),j=1..i),i=1..20); # Robert Israel, Nov 06 2016

From Robert Israel, Nov 06 2016: (Start)
T(n,k) = A000045(n+2) - A000045(n+2-k) for 1 <= k <= n.
G.f. as triangle: x*y*(1+x^2*y)/((1-x*y)*(1-x-x^2)*(1-x*y-x^2*y^2)). (End)

Corrected by Dintle N Kagiso, Nov 06 2016

A128541 Triangle, A097806 * A127647, read by rows.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 0, 3, 5, 0, 0, 0, 0, 5, 8, 0, 0, 0, 0, 0, 8, 13, 0, 0, 0, 0, 0, 0, 13, 21, 0, 0, 0, 0, 0, 0, 0, 21, 34, 0, 0, 0, 0, 0, 0, 0, 0, 34, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 89, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 89, 144, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 233

Offset: 0

Gary W. Adamson, Mar 10 2007

Row sums = A000045 starting (1, 2, 3, 5, 8, 13, ...). A128540 = A127647 * A097806.

			First few rows of the triangle:
  1;
  1, 1;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 0, 3, 5;
  0, 0, 0, 0, 5, 8;
  ...

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

Cf. A000045, A097806, A127647, A128540.

[k eq n select Fibonacci(n+1) else k eq n-1 select Fibonacci(n) else 0: k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 11 2019

Table[If[k==n, Fibonacci[n+1], If[k==n-1, Fibonacci[n], 0]], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 11 2019 *)

T(n,k) = if(k==n, fibonacci(n+1), if(k==n-1, fibonacci(n), 0)); \\ G. C. Greubel, Jul 11 2019

def T(n, k):
    if (k==n): return fibonacci(n+1)
    elif (k==n-1): return fibonacci(n)
    else: return 0
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jul 11 2019

A097806 * A127647 as infinite lower triangular matrices.

More terms added by G. C. Greubel, Jul 11 2019

A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 3, 0, 3, 5, 0, 5, 0, 5, 0, 8, 0, 8, 0, 8, 13, 0, 13, 0, 13, 0, 13, 0, 21, 0, 21, 0, 21, 0, 21, 34, 0, 34, 0, 34, 0, 34, 0, 34, 0, 55, 0, 55, 0, 55, 0, 55, 0, 55

Offset: 1

Gary W. Adamson, Mar 14 2007

This triangle is different from A128618, which is equal to A128174 * A127647.

			First few rows of the triangle are:
   1;
   0,  1;
   2,  0,  2;
   0,  3,  0,  3;
   5,  0,  5,  0,  5;
   0,  8,  0,  8,  0,  8;
  13,  0, 13,  0, 13,  0, 13;
   0, 21,  0, 21,  0, 21,  0, 21,
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000045, A096140, A127646, A128174, A128610, A128618, A128620.

Cf. A128620 (row sums).

[((n+k+1) mod 2)*Fibonacci(n): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Table[Fibonacci[n]*Mod[n+k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 16 2024 *)

flatten([[((n+k+1)%2)*fibonacci(n) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

T(n, k) = A127647 * A128174, an infinite lower triangular matrix. In odd rows, n terms of F(n), 0, F(n),...; in the n-th row. In even rows, n terms of 0, F(n), 0,...; in the n-th row.
Sum_{k=1..n} T(n, k) = A128620(n-1).
From G. C. Greubel, Mar 16 2024: (Start)
T(n, k) = Fibonacci(n)*(1 + (-1)^(n+k))/2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^n*A128620(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A096140(floor((n + 1)/2)).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*( Fibonacci(n-1) + (-1)^floor((n-1)/2) * Fibonacci(floor((n-3)/2)) ). (End)

A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 5, 0, 1, 0, 3, 0, 8, 1, 0, 2, 0, 5, 0, 13, 0, 1, 0, 3, 0, 8, 0, 21, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233

Offset: 1

Gary W. Adamson, Mar 14 2007

This triangle is different from A128619, which is A128619 = A127647 * A128174.

			First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 3;
  1, 0, 2, 0, 5;
  0, 1, 0, 3, 0, 8;
  1, 0, 2, 0, 5, 0, 13;
  0, 1, 0, 3, 0, 8,  0, 21;
  1, 0, 2, 0, 5, 0, 13,  0, 34;
  0, 1, 0, 3, 0, 8,  0, 21,  0, 55;
  1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000045, A052952, A127647, A128174, A128619, A355020.

[((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)

flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
From G. C. Greubel, Mar 17 2024: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
T(n, n) = A000045(n).
T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) *  A355020(floor((n-1)/2)). (End)

a(6) corrected and more terms from Georg Fischer, May 30 2023

A153281 Triangle read by rows, A130321 * A127647. Also, number of subsets of [n+2] with consecutive integers that start at k.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 8, 4, 4, 3, 16, 8, 8, 6, 5, 32, 16, 16, 12, 10, 8, 64, 32, 32, 24, 20, 16, 13, 128, 64, 64, 48, 40, 32, 26, 21, 256, 128, 128, 96, 80, 64, 52, 42, 34, 512, 256, 256, 192, 160, 128, 104, 84, 68, 55

Offset: 0

Gary W. Adamson, Dec 23 2008

Row sums = A008466(k-2): (1, 3, 8, 19, 43, 94, ...).

T(n,k) is the number of subsets of {1,...,n+2} that contain consecutive integers and that have k as the first integer in the first consecutive string. (See the example below.) Hence rows sums of T(n,k) give the number of subsets of {1,...,n+2} that contain consecutive integers. Also, T(n,k) = F(k)*2^(n+1-k), where F(k) is the k-th Fibonacci number, since there are F(k) subsets of {1,...,k-2} that contain no consecutive integers and there are 2^(n+1-k) subsets of {k+2,...,n+2}. [Dennis P. Walsh, Dec 21 2011]

			First few rows of the triangle:
    1;
    2,   1;
    4,   2,   2;
    8,   4,   4,   3;
   16,   8,   8,   6,   5;
   32,  16,  16,  12,  10,   8;
   64,  32,  32,  24,  20,  16,  13;
  128,  64,  64,  48,  40,  32,  26,  21;
  256, 128, 128,  96,  80,  64,  52,  42,  34;
  512, 256, 256, 192, 160, 128, 104,  84,  68,  55;
  ...
Row 4 = (16, 8, 8, 6, 5) = termwise products of (16, 8, 4, 2, 1) and (1, 1, 2, 3, 5).
For n=5 and k=3, T(5,3)=16 since there are 16 subsets of {1,2,3,4,5,6,7} containing consecutive integers with 3 as the first integer in the first consecutive string, namely,
  {1,3,4}, {1,3,4,5}, {1,3,4,6}, {1,3,4,7}, {1,3,4,5,6}, {1,3,4,5,7}, {1,3,4,6,7}, {1,3,4,5,6,7}, {3,4}, {3,4,5}, {3,4,6}, {3,4,7}, {3,4,5,6}, {3,4,5,7}, {3,4,6,7}, and {3,4,5,6,7}. [_Dennis P. Walsh_, Dec 21 2011]

Cf. A008466, A127647, A130321.

with(combinat, fibonacci):
seq(seq(2^(n+1-k)*fibonacci(k),k=1..(n+1)),n=0..10);

Table[2^(n+1-k) Fibonacci[k],{n,0,10},{k,n+1}]//Flatten (* Harvey P. Dale, Apr 26 2020 *)

Triangle read by rows, A130321 * A127647. A130321 = an infinite lower triangular matrix with powers of 2: (A000079) in every column: (1, 2, 4, 8, ...).
A127647 = an infinite lower triangular matrix with the Fibonacci numbers, A000045 as the main diagonal and the rest zeros.
T(n,k)=2^(n+1-k)*F(k) where F(k) is the k-th Fibonacci number. [Dennis Walsh, Dec 21 2011]

A128589 A051731 * A127647.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 5, 1, 1, 2, 0, 0, 8, 1, 0, 0, 0, 0, 0, 13, 1, 1, 0, 3, 0, 0, 0, 21, 1, 0, 2, 0, 0, 0, 0, 0, 34, 1, 1, 0, 0, 5, 0, 0, 0, 0, 55

Offset: 1

Gary W. Adamson, Mar 11 2007

Row sums = A007435: (1, 2, 3, 5, 6, 12, 14, ...), inverse Moebius transform of the Fibonacci numbers.

			First few rows of the triangle:
  1;
  1, 1;
  1, 0, 2;
  1, 1, 0, 3;
  1, 0, 0, 0, 5;
  1, 1, 2, 0, 0, 8;
  ...

Cf. A051731, A127647, A007435.

Inverse Moebius transform of A127647, as infinite lower triangular matrices.

A007435 Inverse Moebius transform of Fibonacci numbers 1,1,2,3,5,8,...

Original entry on oeis.org

1, 2, 3, 5, 6, 12, 14, 26, 37, 62, 90, 159, 234, 392, 618, 1013, 1598, 2630, 4182, 6830, 10962, 17802, 28658, 46548, 75031, 121628, 196455, 318206, 514230, 832722, 1346270, 2179322, 3524670, 5704486, 9227484, 14933129, 24157818, 39092352, 63246222, 102341006

Offset: 1

N. J. A. Sloane

For p prime, a(p) == k (mod p) where k = 0 if p == 2, 3 (mod 5), k = 2 if p == 1, 4 (mod 5) and k = 1 if p = 5. - Michael Somos, Apr 15 2012

			G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 12*x^6 + 14*x^7 + 26*x^8 + 37*x^9 + 62*x^10 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms

Cf. A000045, A051731, A127647, A245282, A108046, A034772, A100107, A108031, A008683.

with(numtheory): seq(add(combinat[fibonacci](d), d in divisors(n)), n=1..60); # Ridouane Oudra, Apr 12 2025

Table[Plus @@ Map[Function[d, Fibonacci[d]], Divisors[n]], {n, 100}] (* T. D. Noe, Aug 14 2012 *)
a[n_] := DivisorSum[n, Fibonacci]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, k, fibonacci(k)))} /* Michael Somos, Apr 15 2012 */

Row sums of A051731 * A127647. - Gary W. Adamson, Jan 22 2007
G.f.: Sum_{k>0} Fibonacci(k)*x^k/(1-x^k) = Sum_{k>0} x^k/(1-x^k-x^(2*k)). - Vladeta Jovovic, Dec 17 2002
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(Fibonacci(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 20 2018
a(n) ~ 5^(-1/2) * phi^n, where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2018
From Ridouane Oudra, Apr 12 2025 : (Start)
a(n) = Sum_{d|n} Fibonacci(d).
a(n) = Sum_{d|n} mu(d)*A034772(n/d).
a(n) = A245282(n) - A108046(n).
a(n) = 2*A245282(n) - A100107(n).
a(n) = (A108031(n) + A108046(n))/2. (End)

More terms from Joerg Arndt, Aug 14 2012

A007436 Moebius transform of Fibonacci numbers.

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 12, 18, 32, 50, 88, 134, 232, 364, 604, 966, 1596, 2544, 4180, 6708, 10932, 17622, 28656, 46206, 75020, 121160, 196384, 317432, 514228, 831374, 1346268, 2177322, 3524488, 5701290, 9227448, 14927632, 24157816, 39083988

Offset: 1

N. J. A. Sloane

After a(4) = 2, there are no primes in this sequence. Every element thereafter has at least two prime factors, the semiprimes (intersection of A007436 and A001358) starting 4, 6, 134, 831374, ... - Jonathan Vos Post, Dec 15 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms

Cf. A001358.

mt[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*Fibonacci /@ d)]; Table[ mt[n], {n, 38}] (* Robert G. Wilson v Dec 10 2004 *)
a[n_] := DivisorSum[n, Fibonacci[#] MoebiusMu[n/#]&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)

a(n)=sumdiv(n,d,fibonacci(d)*moebius(n/d))

Row sums of the triangle generated by A054525 * A127647. - Gary W. Adamson, Jan 22 2007

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/(1 - x - x^2). - Ilya Gutkovskiy, Apr 25 2017

More terms from Robert G. Wilson v, Dec 10 2004

cp's OEIS Frontend

A128540 Triangle A127647 * A097806, read by rows.

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A131410 A127647 * A000012.

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A143061 Triangle read by rows, A000012 * A127647 * A000012.

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A128541 Triangle, A097806 * A127647, read by rows.

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A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.

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A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

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