A165900 -id:A165900 - cp's OEIS frontend

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

1, 1, 0, 1, -1, -1, 1, -2, 1, 1, 1, -3, 5, -1, 0, 1, -4, 11, -13, 1, -1, 1, -5, 19, -41, 34, -1, 1, 1, -6, 29, -91, 153, -89, 1, 0, 1, -7, 41, -169, 436, -571, 233, -1, -1, 1, -8, 55, -281, 985, -2089, 2131, -610, 1, 1, 1, -9, 71, -433, 1926, -5741, 10009, -7953, 1597, -1, 0

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Feb 01 2018

This array is used to compute A269252: A269252(n) = least k such that |A(n,k)| is a prime, or -1 if no such k exists.
For detailed theory, see [Hone].
The array can be extended to k<0 with A(n, k) = -A(n, -k-1) for all k in Z. - Michael Somos, Jun 19 2023

			Array begins:
1   0  -1     1     0      -1       1         0        -1           1
1  -1   1    -1     1      -1       1        -1         1          -1
1  -2   5   -13    34     -89     233      -610      1597       -4181
1  -3  11   -41   153    -571    2131     -7953     29681     -110771
1  -4  19   -91   436   -2089   10009    -47956    229771    -1100899
1  -5  29  -169   985   -5741   33461   -195025   1136689    -6625109
1  -6  41  -281  1926  -13201   90481   -620166   4250681   -29134601
1  -7  55  -433  3409  -26839  211303  -1663585  13097377  -103115431
1  -8  71  -631  5608  -49841  442961  -3936808  34988311  -310957991
1  -9  89  -881  8721  -86329  854569  -8459361  83739041  -828931049

Robert Price, Table of n, a(n) for n = 1..5050
Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269251, A269252, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.
Cf. A094954 (unsigned version of this array, but missing the first row).
Cf. Rows: A057078, A033999, A099496, A079935 (or A001835), A004253, A001653, A049685, A070997, A070998, A138288 (or A072256), ...
Cf. Columns: A000012, A001477 (A000027), A110331 (A165900), A123972, A192398, ...

(* Array: *)
Grid[Table[LinearRecurrence[{-n, -1}, {1, 1 - n}, 10], {n, 10}]]
(*Array antidiagonals flattened (gives this sequence):*)
A299045[n_, k_] := Sum[(-1)^(Floor[j/2]) Binomial[k - Floor[(j + 1)/2], Floor[j/2]] (-n)^(k - j), {j, 0, k}]; Flatten[Table[A299045[n - k, k], {n, 11}, {k, 0, n - 1}]]

{A(n, k) = sum(j=0, k, (-1)^(j\2)*binomial(k-(j+1)\2, j\2)*(-n)^(k-j))}; /* Michael Somos, Jun 19 2023 */

G.f. for row n: (1 + x)/(1 + n*x + x^2), n >= 1.

A(n, k) = B(-n, k) where B = A294099. - Michael Somos, Jun 19 2023

A070313 a(n) = 2^n - (2*n+1).

Original entry on oeis.org

0, -1, -1, 1, 7, 21, 51, 113, 239, 493, 1003, 2025, 4071, 8165, 16355, 32737, 65503, 131037, 262107, 524249, 1048535, 2097109, 4194259, 8388561, 16777167, 33554381, 67108811, 134217673, 268435399, 536870853, 1073741763, 2147483585

Offset: 0

N. J. A. Sloane, May 16 2002

Binomial transform of (-1)^n! + !n. - Paul Barry, May 13 2004
This appears as the exponent in Krotov, who writes on p. 2: "in general, two extended Hamming codes can intersect in 2^(2^m - 2m - 1) elements." - Jonathan Vos Post, Jan 13 2013
Primes appear at positions n = 4, 7, 8, 28, 32, 81, 669, 1108, ... (A344781). - R. J. Mathar, Jan 22 2013
a(n) is the total number of dollars lost when using the Martingale method (bet $1, if win then continue to bet $1, if lose then double next bet) for n trials of a wager with exactly one win, n-1 losses. For the case with exactly one loss, n-1 wins, see A165900. - Max Winnick, Jun 28 2022

Denis Krotov, A partition of the hypercube into cosets of maximally nonparallel Hamming codes, arXiv:1210.0010v1 [cs.IT], Sep 28, 2012.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Second diagonal of A046739.

Cf. A344781.

lst={};s=-1;Do[s+=s+n;AppendTo[lst, s], {n, 1, 5!, 2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 18 2008 *)
Table[2^n-(2n+1),{n,0,40}] (* Harvey P. Dale, Feb 13 2024 *)

makelist(2^n - (2*n+1),n,0,20); /* Martin Ettl, Jan 25 2013 */

a(n)=2^n-(2*n+1) \\ Charles R Greathouse IV, Oct 07 2015

E.g.f.: (exp(x))^2 - exp(x) - 2*x*exp(x). - Paul Barry, May 13 2004
From Colin Barker, Mar 21 2012: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: -x*(1-3*x)/((1-x)^2*(1-2*x)). (End)

A144139 Chebyshev polynomial of the second kind U(4,n).

Original entry on oeis.org

1, 5, 209, 1189, 3905, 9701, 20305, 37829, 64769, 104005, 158801, 232805, 330049, 454949, 612305, 807301, 1045505, 1332869, 1675729, 2080805, 2555201, 3106405, 3742289, 4471109, 5301505, 6242501, 7303505, 8494309, 9825089, 11306405

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A057722, A079414, A165900, A193250.

[16*n^4-12*n^2+1: n in [0..40]]; // Vincenzo Librandi, May 29 2014

lst={}; Do[AppendTo[lst, ChebyshevU[4, n]], {n, 0, 9^2}]; lst
CoefficientList[Series[(1 + 194 x^2 + 184 x^3 + 5 x^4)/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

G.f.: (1 + 194*x^2 + 184*x^3 + 5*x^4)/(1 - x)^5. - Vincenzo Librandi, May 29 2014
a(n) = 16*n^4-12*n^2+1 = (4*n^2-2*n-1)*(4*n^2+2*n-1). - Vincenzo Librandi, May 29 2014
From Klaus Purath, Sep 08 2022: (Start)
a(n) = A165900(2*n)*A165900(2*n+1).
a(n) = A057722(2*n).
a(n) = 4*(Sum_{i=1..n} A193250(i)) + 1 = 4*A079414(n) + 1.
(End)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Wesley Ivan Hurt, Aug 04 2025

Changed offset from 1 to 0 by Vincenzo Librandi, May 29 2014

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 2, -1, 0, 1, 2, 3, -1, 1, 0, 1, 3, 4, 1, 1, -1, 0, 1, 4, 5, 5, 3, 1, 1, 0, 1, 5, 6, 11, 7, 5, 3, -1, 0, 1, 6, 7, 19, 13, 11, 7, -2, 1, 0, 1, 7, 8, 29, 21, 19, 13, 1, 0, -1, 0, 1, 8, 9, 41, 31, 29, 21, 14, 3, 0, 1, 0, 1, 9, 10, 55, 43, 41, 31, 43, 16, 5, 2, -1

Offset: 0

Alois P. Heinz, Jan 25 2023

The empty bit string is used as binary expansion of 0, so A(0,k) = 0.

			Square array A(n,k) begins:
   0,  0, 0,  0,  0,   0,   0,   0,   0,   0,   0, ...
   1,  1, 1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  -1,  0, 1,  2,  3,   4,   5,   6,   7,   8,   9, ...
   1,  2, 3,  4,  5,   6,   7,   8,   9,  10,  11, ...
  -1, -1, 1,  5, 11,  19,  29,  41,  55,  71,  89, ...
   1,  1, 3,  7, 13,  21,  31,  43,  57,  73,  91, ...
  -1,  1, 5, 11, 19,  29,  41,  55,  71,  89, 109, ...
   1,  3, 7, 13, 21,  31,  43,  57,  73,  91, 111, ...
  -1, -2, 1, 14, 43,  94, 173, 286, 439, 638, 889, ...
   1,  0, 3, 16, 45,  96, 175, 288, 441, 640, 891, ...
  -1,  0, 5, 20, 51, 104, 185, 300, 455, 656, 909, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-6, 10 give: A062157, A145037, A006257, A147991, A147992, A153777, A147993, A359925.
Rows n=0-10 give: A000004, A000012, A023443, A000027(k+1), A165900, A002061, A165900(k+1), A002061(k+1), A083074, A152618, A062158.
Main diagonal gives A360096.
Cf. A030300, A057427.

A:= proc(n, k) option remember; local m;
     `if`(n=0, 0, k*A(iquo(n, 2, 'm'), k)+2*m-1)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= (n, k)-> (l-> add((2*l[i]-1)*k^(i-1), i=1..nops(l)))(Bits[Split](n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

G.f. for column k satisfies g_k(x) = k*(x+1)*g_k(x^2) + x/(1+x).
A(n,k) = k*A(floor(n/2),k)+2*(n mod 2)-1 for n>0, A(0,k)=0.
A(n,k) mod 2 = A057427(n) if k is even.
A(n,k) mod 2 = A030300(n) if k is odd and n>=1.
A(2^(n+1),1) + n = 0.

A110331 Row sums of a number triangle related to the Pell numbers.

Original entry on oeis.org

1, -1, -5, -11, -19, -29, -41, -55, -71, -89, -109, -131, -155, -181, -209, -239, -271, -305, -341, -379, -419, -461, -505, -551, -599, -649, -701, -755, -811, -869, -929, -991, -1055, -1121, -1189, -1259, -1331, -1405, -1481, -1559, -1639, -1721, -1805, -1891, -1979, -2069, -2161, -2255, -2351, -2449

Offset: 0

Paul Barry, Jul 20 2005

Row sums of A110330. Results from a general construction: the row sums of the inverse of the number triangle whose columns have e.g.f. (x^k/k!)/(1-a*x-b*x^2) have g.f. (1-(a+2)x-(2b-a-1)x^2)/(1-x)^3 and general term 1+(b-a)*n-b*n^2. This is the binomial transform of (1,-a,-2b,0,0,0,...).

Hankel transform of A007054(n)-2*0^n. - Paul Barry, Jul 20 2008

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

Cf. A028387 (absolute values). A165900 is another version.

[1-n-n^2: n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[(1-4x+x^2)/(1-x)^3,{x,0,50}],x]  (* Vincenzo Librandi, Jul 08 2012 *)
LinearRecurrence[{3,-3,1},{1,-1,-5},60] (* Harvey P. Dale, Mar 22 2022 *)

a(n)=1-n-n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 1-n-n^2.
G.f.: (1-4*x+x^2)/(1-x)^3.
a(n) = binomial(n+2, 2) - 4*binomial(n+1, 2) + binomial(n, 2).
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - Vincenzo Librandi, Jul 08 2012
E.g.f.: exp(x)*(1-2*x-x^2). - Tom Copeland, Dec 02 2013
a(n) = -A165900(n+1) (= -A028387(n-1) for n > 0). - M. F. Hasler, Mar 01 2014

A188652 First differences of A000463.

Original entry on oeis.org

0, 1, 2, -1, 6, -5, 12, -11, 20, -19, 30, -29, 42, -41, 56, -55, 72, -71, 90, -89, 110, -109, 132, -131, 156, -155, 182, -181, 210, -209, 240, -239, 272, -271, 306, -305, 342, -341, 380, -379, 420, -419, 462, -461, 506, -505, 552, -551, 600, -599, 650, -649, 702, -701, 756, -755, 812, -811, 870, -869, 930, -929, 992, -991, 1056, -1055, 1122, -1121, 1190, -1189, 1260, -1259, 1332, -1331, 1406

Offset: 1

Reinhard Zumkeller, Apr 13 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-1,2,2,-1,-1).

Cf. A188653 (first differences).

Cf. A000463, A002378, A165900.

a188652 n = a188652_list !! (n-1)
a188652_list = zipWith (-) (tail a000463_list) a000463_list

Differences[Flatten[Array[{#,#^2}&,40]]] (* Harvey P. Dale, Aug 04 2012 *)

a(2n) = 1 - a(2n-1), a(2n+1) = 2*n + 1 - a(2n).
a(n) = A000463(n+1) - A000463(n).
a(2n-1) = A002378(n-1), a(2n) = - A165900(n).
G.f.: -x^2*(-1-3*x+x^2+x^3) / ( (x-1)^2*(1+x)^3 ). - R. J. Mathar, Apr 14 2011
a(n) = (2*n+3-(2*n^2-2*n-5)*(-1)^n)/8. - Luce ETIENNE, Dec 18 2014
E.g.f.: ((4 + x - x^2)*cosh(x) - (1 - x - x^2)*sinh(x) - 4)/4. - Stefano Spezia, Jul 08 2023
Sum_{n>=2} 1/a(n) = 1 + (sqrt(5)-5)*Pi*tan(sqrt(5)*Pi/2)/(5*(sqrt(5)-1)). - Amiram Eldar, May 11 2025

A356247 Denominator of the continued fraction 1/(2 - 3/(3 - 4/(4 - 5/(...(n-1) - n/(-1))))).

Original entry on oeis.org

1, 5, 11, 19, 29, 41, 11, 71, 89, 109, 131, 31, 181, 19, 239, 271, 61, 31, 379, 419, 461, 101, 29, 599, 59, 701, 151, 811, 79, 929, 991, 211, 59, 41, 1259, 1, 281, 1481, 1559, 149, 1721, 1, 61, 1979, 2069, 2161, 1, 2351, 79, 2549, 241, 1, 2861, 2969, 3079, 3191

Offset: 2

Mohammed Bouras, Jul 30 2022

Conjecture 1: Every term of this sequence is either a prime number or 1.
Conjecture 2: The sequence contains all prime numbers which ends with a 1 or 9.
Conjecture 3: Except for 5, the primes all appear exactly twice.
a(n) divides n^2 - n - 1, which is the unreduced denominator.
Conjecture: The ordered sequence of prime values is A038872. - Bill McEachen, Jul 28 2025
For a proof of Conjectures 1-3, see Cloitre (2025). - Sean A. Irvine, Aug 25 2025

			For n=2, 1/(2 - 3) = -1, so a(2)=1.
For n=3, 1/(2 - 3/(3 - 4)) = 1/5, so a(3)=5.
For n=4, 1/(2 - 3/(3 - 4/(4 - 5))) = 7/11, so a(4)=11.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6)))) = 23/19, so a(5)=19.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 - 7))))) = 73/29, so a(6)=29.
a(23) = a(79) = 23 + 79 - 1 = 101.
a(26) = a(34) = gcd(26^2 - 26 -1, 34^2 - 34 - 1) = gcd(649, 1121) = 59.

Michael De Vlieger, Table of n, a(n) for n = 2..10001
Mohammed Bouras, A new sequence of prime numbers, Romanian Math. Mag. (15 August 2022). See Table 2 p. 2.
Mohammed Bouras, A New Sequence of Prime Numbers, EasyChair Preprint 8686, 2022.
Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022).
Benoit Cloitre, On the sequence A356247, 2025.

Cf. A002327 (primes of the form k^2-k-1), A028387, A051403, A165900, A356684.

a[n_] := ContinuedFractionK[-i-1, If[i == n, 1, i+1], {i, 1, n}] //
   Denominator;
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Aug 11 2022 *)

a(n) = if (n==1, 1, n--; my(v = vector(2*n, k, (k+4)\2)); my(q = 1/(v[2*n-1] - v[2*n])); forstep(k=2*n-3, 1, -2, q = v[k] - v[k+1]/q; ); denominator(1/q)); \\ Michel Marcus, Aug 07 2022

from fractions import Fraction
def A356247(n):
    k = -1
    for i in range(n-1,1,-1):
        k = i-Fraction(i+1,k)
    return abs(k.numerator) # Chai Wah Wu, Aug 23 2022

a(n) = (n^2 - n - 1)/gcd(n^2 - n - 1, A356684(n)).
If conjecture 3 is true, then we have:
   a(n) = a(m) = n + m - 1.
   a(n) = a(m) = gcd(n^2 - n - 1, m^2 - m - 1).
   a(n) = a(a(n) - n + 1).

A214803 Frobenius numbers of numerators and denominators of rational numbers in order of their canonical enumeration.

Original entry on oeis.org

-1, -1, 1, -1, 5, -1, 3, 7, 11, -1, 19, -1, 5, 11, 17, 23, 29, -1, 13, 27, 41, -1, 7, 23, 31, 47, 55, -1, 17, 53, 71, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, -1, 43, 65, 109, -1, 11, 23, 35, 47, 59, 71, 83, 95, 107, 119, 131, -1, 25, 51, 103, 129, 155, -1, 13

Offset: 1

Reinhard Zumkeller, Jul 29 2012

sign

a(n) = A020652(n) * A038567(n) - A020652(n) - A038567(n);
for n > 1: a(A015614(n)) = A165900(n-1);
a(A002088(n)) = -1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Frobenius Number.
Wikipedia, Coin problem

a214803 n = a214803_list !! (n-1)
a214803_list = [x * y - x - y | y <- [1..], x <- [1..y-1], gcd x y == 1]

A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

Original entry on oeis.org

1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289

Offset: 1

Kritsada Moomuang, Jan 28 2019

nonn

Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).

			a(3) = 19 because 5^2 - 5 - 1 = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A091568.
Subsequence of A028387 or A165900.
Second column of A378979.
Cf. A000040, A001248, A030431, A033879, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.

map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019

Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)

a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
a(n) = A033879(A001248(n)). [Deficiency of squares of primes] - Antti Karttunen, Dec 13 2024

A243436 Numbers n such that n^2-n-1 is semiprime.

Original entry on oeis.org

8, 13, 15, 18, 19, 23, 24, 26, 28, 30, 33, 34, 35, 38, 41, 44, 50, 52, 58, 59, 62, 64, 68, 70, 72, 73, 74, 75, 76, 78, 79, 80, 82, 83, 88, 89, 91, 92, 96, 98, 99, 100, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 117, 119, 120, 122, 123, 124, 125, 128, 130

Offset: 1

K. D. Bajpai, Jun 06 2014

nonn

			13 is in the sequence because 13^2 - 13 - 1 = 155 = 5 * 31 is semiprime.
18 is in the sequence because 18^2 - 18 - 1 = 305 = 5 * 61 is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A001358, A165900.

with(numtheory):A243436 := proc() if bigomega(n^2-n-1)=2 then RETURN (n); fi; end: seq(A243436 (), n=1..200);

c = 0; Do[If[PrimeOmega[n^2-n-1] == 2, c++; Print[c," ",n]], {n,1,30000}];
Select[Range[200],PrimeOmega[#^2-#-1]==2&] (* Harvey P. Dale, Sep 21 2016 *)

cp's OEIS Frontend

A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)*binomial(k-floor((j+1)/2), floor(j/2))*(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.

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Mathematica

PARI

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A070313 a(n) = 2^n - (2*n+1).

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Mathematica

Maxima

PARI

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A144139 Chebyshev polynomial of the second kind U(4,n).

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Magma

Mathematica

Formula

Extensions

A360099 To get A(n,k), replace 0's in the binary expansion of n with (-1) and interpret the result in base k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

Formula

A110331 Row sums of a number triangle related to the Pell numbers.

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Magma

Mathematica

PARI

Formula

A188652 First differences of A000463.

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Haskell

Mathematica

Formula

A356247 Denominator of the continued fraction 1/(2 - 3/(3 - 4/(4 - 5/(...(n-1) - n/(-1))))).

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A299045 Rectangular array: A(n,k) = Sum_{j=0..k} (-1)^floor(j/2)binomial(k-floor((j+1)/2), floor(j/2))(-n)^(k-j), n >= 1, k >= 0, read by antidiagonals.