A054977 -id:A054977 - cp's OEIS frontend

A306821 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Denominators.

2, 2, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, 1365, 1, 3, 1, 255, 1, 399, 1, 165, 1, 69, 1, 1365, 1, 3, 1, 435, 1, 7161, 1, 255, 1, 3, 1, 959595, 1, 3, 1, 6765, 1, 903, 1, 345, 1, 141, 1, 23205, 1, 33, 1, 795, 1, 399, 1

Offset: 0

Paul Curtz, Jun 04 2019

Fractions: 1/2, 1/2, -4/3, 2, -38/15, 3, -73/21, 4, -68/15, 5, -179/33, 6, -9218/1365, 7, ... .
Numerators are A307974(n).
a(2n) same as denominators of cosecant numbers A001897 for n>0 (conjectured).

Essentially the same as A141459.

Cf. A001897, A027642, A054977, A141044, A164555, A307974.

b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1;
a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Denominator;
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 04 2019 *)

a(n) = A141459(n) * A141044(n).
a(n) = A141459(n) for n>2.
a(2n+1) = A054977(n).
a(2n) = A001897(n) * A054977(n).

A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

Original entry on oeis.org

2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 15, 16, 9, 2, 1, 7, 25, 37, 25, 11, 2, 1, 8, 43, 94, 77, 36, 13, 2, 1, 9, 77, 259, 273, 141, 49, 15, 2, 1, 10, 143, 748, 1045, 646, 235, 64, 17, 2, 1, 11, 273, 2209, 4121, 3151, 1321, 365, 81, 19, 2

Offset: 0

Stefano Spezia, Aug 19 2024

			Array begins:
  2, 2,  2,   2,    2,     2, ...
  1, 3,  5,   7,    9,    11, ...
  1, 4,  9,  16,   25,    36, ...
  1, 5, 15,  37,   77,   141, ...
  1, 6, 25,  94,  273,   646, ...
  1, 7, 43, 259, 1045,  3151, ...
  1, 8, 77, 748, 4121, 15656, ...
  ...

Cf. A000290, A004247, A004248, A005408 (n=1), A005491 (n=3), A007395 (n=0), A054977 (k=0), A176691 (k=2), A176805 (k=3), A176916 (k=5), A176972 (k=7), A214647.

Cf. A375578 (antidiagonal sums).

A[0,0]=2; A[n_,k_]:=k^n+k*n+1;Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

G.f. for the k-th column: (2*x^2 - 3*x - k^2 + k + 1)/((x - 1)^2*(x - k)).
E.g.f. for the k-th column: exp(x)*(1 + exp((k-1)*x) + k*x).
A(n,1) = n + 2.
A(2,n) = A000290(n+1).
A(n,n) = 2*A214647(n) + 1.

A080209 Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e., the diagonal of leading successive absolute differences of A005384.

Original entry on oeis.org

2, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 1, 3

Offset: 1

John W. Layman, Mar 20 2003

nonn

Conjecture: The diagonal of leading successive absolute differences of the Sophie Germain primes consists, except for the initial 2, only of 1's and 3s.

			The difference table begins:
   2;
   3,  1;
   5,  2,  1;
  11,  6,  4,  3;
  23, 12,  6,  2,  1;
  29,  6,  6,  0,  2,  1;

Cristian Cobeli, Mihai Prunescu, Alexandru Zaharescu, A growth model based on the arithmetic Z-game, arXiv:1511.04315 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Sophie Germain Prime.
Eric Weisstein's World of Mathematics, Gilbreath's Conjecture

Cf. A005384, A036262, A054977.

sgp[1] = Select[Prime[Range[1000]], PrimeQ[2 # + 1]&];
sgp[n_] := Differences[sgp[n - 1]] // Abs;
Table[sgp[n], {n, 1, 105}][[All, 1]] (* Jean-François Alcover, Feb 04 2019 *)

A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

46, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1, 45, 1

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

More generally, the ordinary generating function for the continued fraction expansion of phi^(2*k + 1), where phi = (1 + sqrt(5))/2, k = 1, 2, 3,... is floor(phi^(2*k + 1))/(1 - x), and for the continued fraction expansion of phi^(2*k) is (floor(phi^(2*k)) + x - x^2)/(1 - x^2).

			phi^8 = (47 + 21*sqrt(5))/2 = 46 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/(1 + 1/(45 + 1/...)))))).

Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, Golden ratio
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A001622.

Cf. continued fraction expansion of phi^k: A000012 (k = 1), A054977 (k = 2), A010709 (k = 3), A176260 (k = 4, for n>0), A010850 (k = 5), A040071 (k = 6, for n>0), A010868 (k = 7), this sequence (k = 8).

[46] cat &cat [[1, 45]^^50]; // Vincenzo Librandi, Jan 13 2016

ContinuedFraction[(47 + 21 Sqrt[5])/2, 82]

G.f.: (46 + x - x^2)/(1 - x^2).

a(n) = 23 + 22*(-1)^n for n>0. - Bruno Berselli, Jan 18 2016

A273153 a(n) = Numerator of (0 followed by 1's) - n/2^n.

Original entry on oeis.org

0, 1, 1, 5, 3, 27, 29, 121, 31, 503, 507, 2037, 1021, 8179, 8185, 32753, 4095, 131055, 131063, 524269, 262139, 2097131, 2097141, 8388585, 2097149, 33554407, 33554419, 134217701, 67108857, 536870883, 536870897, 2147483617, 134217727, 8589934559, 8589934575, 34359738333

Offset: 0

Paul Curtz, May 16 2016

A060576(n+1) = 0, 1, 1, 1, 1, 1, 1, ... - (0(n) = Oresme(n) = 0, 1/2, 1/2, 3/8, 1/4, 5/32, 3/32, ...). Both sequences are autosequences of the first kind. f(n) = 0, 1/2, 1/2, 5/8, 3/4, 27/32, 29/32, 121/128, ... is an autosequence of the first kind. Without one 1/2, f(n) is an increasing sequence.

The numerators (1 followed by A075101(n)) are the same as in n/2^n.

			Array of differences of fractions (characteristic aspect of an autosequence of the first kind):
0,     1/2,   1/2,   5/8,   3/4, ...
1/2,     0,   1/8,   1/8,  3/32, ...
-1/2,  1/8,     0, -1/32, -1/32, ...
5/8,  -1/8, -1/32,     0, 1/128, ...
-3/4, 3/32,  1/32, 1/128,     0, ...
...

OEIS Wiki, Autosequence

Cf. A060576, A054977, A075101, A209308.

{0}~Join~Array[Numerator@ Abs[1 - Binomial[0, # - 1] - #/2^#] &, 30] (* Michael De Vlieger, May 17 2016 *)

A305499 Square array A(n,k), n > 0 and k > 0, read by antidiagonals, with initial values A(1,k) = k and recurrence equations A(n+1,k) = A(n,k) for 0 < k <= n and A(n+1,k) = A(n,k) - A000035(n+k) for 0 < n < k.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 3, 4, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 6, 1, 1, 2, 2, 4, 5, 7, 1, 1, 2, 2, 4, 5, 7, 8, 1, 1, 2, 2, 3, 4, 6, 7, 9, 1, 1, 2, 2, 3, 4, 6, 7, 9, 10, 1, 1, 2, 2, 3, 3, 5, 6, 8, 9, 11, 1, 1, 2, 2, 3, 3, 5, 6, 8, 9, 11, 12, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 11, 13

Offset: 1

Werner Schulte, Jun 03 2018

			The square array begins:
  n\k |  1  2  3  4  5  6  7  8  9  10  11  12
  ====+=======================================
    1 |  1  2  3  4  5  6  7  8  9  10  11  12
    2 |  1  1  3  3  5  5  7  7  9   9  11  11
    3 |  1  1  2  3  4  5  6  7  8   9  10  11
    4 |  1  1  2  2  4  4  6  6  8   8  10  10
    5 |  1  1  2  2  3  4  5  6  7   8   9  10
    6 |  1  1  2  2  3  3  5  5  7   7   9   9
    7 |  1  1  2  2  3  3  4  5  6   7   8   9
    8 |  1  1  2  2  3  3  4  4  6   6   8   8
    9 |  1  1  2  2  3  3  4  4  5   6   7   8
   10 |  1  1  2  2  3  3  4  4  5   5   7   7
   11 |  1  1  2  2  3  3  4  4  5   5   6   7
etc.

Cf. A000012 (col 1), A054977 (col 2), A000027 (row 1), A109613 (row 2), A028310 (row 3), A008619 (main diagonal and subdiagonals).

Cf. A000035, A001651, A209229, A211538.

A(n,k) = floor((k+1)/2) for 1 <= k <= n and A(n,k) = floor((k+1)/2) + floor((k+1-n)/2) for 1 <= n < k.
A(n+m,n) = floor((n+1)/2) for n > 0 and some fixed m >= 0.
A(n,n+m) = floor((m+1)/2) + floor((n+1+m)/2) for n>0 and some fixed m >= 0.
A(n+1,k+1) = A(n,k+1) + A(n,k) - A(n-1,k) for k > 0 and n > 1.
A(n,k) = A(n,k-1) + 2*A(n,k-2) - 2*A(n,k-3) - A(n,k-4) + A(n,k-5) for n > 0 and k > 5.
A(n,n) = A008619(n-1) for n > 0.
A(n+1,2*n-1) = A001651(n) for n > 0.
Sum_{i=1..n} A(i,i)*A209229(i) = 2^floor(log_2(n)) for n > 0.
P(n,x) = Sum_{k>0} A(n,k)*x^(k-1) = (1-x^(2*n))/((1-x^n)*(1-x^2)*(1-x)) = (1+x^n)/((1-x^2)*(1-x)) for n > 0.
P(n+1,x) = P(n,x) - x^n/(1-x^2) for n > 0 and P(1,x) = 1/(1-x)^2.
G.f.: Sum_{n>0, k>0} A(n,k)*x^(k-1)*y^(n-1) = (1+x-2*x*y)/((1-x)*(1-x^2) * (1-y)*(1-x*y)).
Conjecture: Sum_{i=1..n} A(n+1-i,i) = A211538(n+3) for n > 0.

A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 0, 0, 3, 1, 1, 3, 0, 6, 4, 7, 5, 2, 9, 9, 8, 10, 8, 9, 1, 13, 18, 1, 2, 8, 15, 26, 10, 22, 1, 18, 9, 20, 10, 2, 6, 13, 19, 16, 38, 38, 3, 32, 5, 39, 24, 7, 27, 14, 41, 20, 39, 32, 7, 20, 35, 44, 50, 24, 34, 51, 14, 39, 47, 49, 15, 61, 54, 60, 52, 34, 60, 32, 72, 48, 12, 67, 52, 22, 48

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/exp(2) = 0 + 0/2! + 0/3! + 3/4! + 1/5! + 1/6! + 3/7! + 0/8! + 6/9! +...

Index entries for factorial base representation

Cf. A092553 (decimal expansion), 0 U A001204 (continued fraction).

Cf. A054977 (e), A067840 (e^2), A068453 (sqrt(e)), A237420 (1/e).

SetDefaultRealField(RealField(250));  [Floor(Exp(-2))] cat [Floor(Factorial(n)*Exp(-2)) - n*Floor(Factorial((n-1))*Exp(-2)) : n in [2..80]];

With[{b = 1/E^2}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(-2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=exp(-2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

A383134 Array read by ascending antidiagonals: A(n,k) is the length of the arithmetic progression of only primes having difference n and first term prime(k).

Original entry on oeis.org

2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Stefano Spezia, Apr 17 2025

			The array begins as:
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 2, 1, 1, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 5, 3, 4, 2, 3, 1, 2, 1, ...
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 2, 1, 2, 1, 1, 1, 2, 2, ...
  2, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 3, 1, 2, 1, 2, 1, 2, 1, 1, ...
  ...
A(2,2) = 3 since 3 primes are in arithmetic progression with a difference of 2 and the first term equal to the 2nd prime: 3, 5, and 7.
A(6,3) = 5 since 5 primes are in arithmetic progression with a difference of 6 and the first term equal to the 3rd prime: 5, 11, 17, 23, and 29.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 139.

Cf. A000012, A006512, A040976, A054977, A175191 (n=2).

Cf. A088430, A206045, A237453.

A[n_,k_]:=Module[{count=1,sum=Prime[k]},While[PrimeQ[sum+=n], count++]; count]; Table[A[n-k+1,k],{n,13},{k,n}]//Flatten

A(A006512(n),k) = 1 for n > 1.

A(A040976(n),k) = A054977(k+1).

cp's OEIS Frontend

A306821 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Denominators.

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A375577 Array read by ascending antidiagonals: A(n,k) = k^n + k*n + 1.

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A080209 Gilbreath transform of the sequence of Sophie Germain primes (A005384), i.e., the diagonal of leading successive absolute differences of A005384.

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A267319 Continued fraction expansion of phi^8, where phi = (1 + sqrt(5))/2.

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A273153 a(n) = Numerator of (0 followed by 1's) - n/2^n.

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A305499 Square array A(n,k), n > 0 and k > 0, read by antidiagonals, with initial values A(1,k) = k and recurrence equations A(n+1,k) = A(n,k) for 0 < k <= n and A(n+1,k) = A(n,k) - A000035(n+k) for 0 < n < k.

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A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

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A383134 Array read by ascending antidiagonals: A(n,k) is the length of the arithmetic progression of only primes having difference n and first term prime(k).

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