A023444 -id:A023444 - cp's OEIS frontend

A023443 a(n) = n - 1.

-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 0

N. J. A. Sloane

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000027, A022958-A022996, A023444-A023482, A256958.

A023443:= func< n | n-1 >; // G. C. Greubel, Apr 18 2025

Range[0,80]-1 (* Harvey P. Dale, Apr 05 2012 *)
LinearRecurrence[{2,-1},{-1,0},75] (* Ray Chandler, Jul 30 2015 *)

a(n)=n-1 \\ Charles R Greathouse IV, Aug 26 2011

def A023443(n): return n-1 # G. C. Greubel, Apr 18 2025

From R. J. Mathar, Feb 11 2010: (Start)
a(n) = 2*a(n-1) -a(n-2).
G.f.: (-1+2*x)/(1-x)^2. (End)
E.g.f.: exp(x)*(x - 1). - Stefano Spezia, Feb 14 2025

A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)Q(m+1, 2k-1) - (2k-1)Q(m+2,2k-2), mQ(m, 2k+1) = (m - 2k)Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

Original entry on oeis.org

1, 1, 1, 1, 1, -2, 1, 1, -1, -5, 1, 1, 0, -4, 12, 1, 1, 1, -3, 3, 43, 1, 1, 2, -2, -4, 28, -120, 1, 1, 3, -1, -9, 15, -15, -531, 1, 1, 4, 0, -12, 4, 48, -288, 1680, 1, 1, 5, 1, -13, -5, 75, -105, 105, 8601, 1, 1, 6, 2, -12, -12, 72, 24, -624, 3984, -30240, 1, 1, 7, 3, -9, -17, 45, 105, -735, 945, -945, -172965

Offset: 0

N. J. A. Sloane, Mar 24 2007

Comment from N. J. A. Sloane, Jan 29 2020: (Start)
It looks like there was a missing 2 in the definition, which I have now corrected.  The old definition was:
(Wrong!) Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2, k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). (Wrong!) (End)

			Array begins:
     1,    1,    1,    1,    1,   1,   1,    1,    1,    1, ... (A000012)
     1,    1,    1,    1,    1,   1,   1,    1,    1,    1, ... (A000012)
    -2,   -1,    0,    1,    2,   3,   4,    5,    6,    7, ... (A023444)
    -5,   -4,   -3,   -2,   -1,   0,   1,    2,    3,    4, ... (A023447)
    12,    3,   -4,   -9,  -12, -13, -12,   -9,   -4,    3, ... (A127146)
    43,   28,   15,    4,   -5, -12, -17,  -20,  -21,  -20, ... (A127147)
  -120,  -15,   48,   75,   72,  45,   0,  -57, -120, -183, ... (A127148)
  -531, -288, -105,   24,  105, 144, 147,  120,   69,    0, ...
  1680,  105, -624, -735, -432, 105, 720, 1281, 1680, 1833, ...

V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.

G. C. Greubel, Antidiagonals n = 0..100, flattened

See A105937 for another version.
Columns give A127137, A127138, A127144, A127145;
Rows give A127146, A127147, A127148.

f:= proc(k) option remember;
      if `mod`(k,2)=0 then k!/(k/2)!
    else 2^(k-1)*((k-1)/2)!*add(binomial(2*j, j)/8^j, j=0..((k-1)/2))
      fi; end;
Q:= proc(n, k) option remember;
      if n=0 then (-1)^binomial(k, 2)*f(k)
    elif k<2 then 1
    elif `mod`(k,2)=0 then (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
      fi; end;
seq(seq(Q(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 30 2020

Q[0, k_]:= Q[0,k]= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^(k-1)*((k-1)/2)!* Sum[Binomial[2*j, j]/8^j, {j, 0, (k-1)/2}] ];
Q[n_, k_]:= Q[n,k]= If[k<2, 1, If[EvenQ[k], (n-k+1)*Q[n+1, k-1] - (k-1)*Q[n+2, k-2], ((n -k+1)*Q[n+1, k-1] - (k-1)*(n+1)*Q[n+2, k-2])/n]];
Table[Q[n-k,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 30 2020 *)

@CachedFunction
def f(k):
    if (mod(k, 2)==0): return factorial(k)/factorial(k/2)
    else: return 2^(k-1)*factorial((k-1)/2)*sum(binomial(2*j, j)/8^j for j in (0..(k-1)/2))
def Q(n,k):
    if (n==0): return (-1)^binomial(k, 2)*f(k)
    elif (k<2): return 1
    elif (mod(k,2)==0): return (n-k+1)*Q(n+1,k-1) - (k-1)*Q(n+2,k-2)
    else: return ( (n-k+1)*Q(n+1,k-1) - (k-1)*(n+1)*Q(n+2,k-2) )/n
[[Q(n-k,k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 30 2020

E.g.f.: Sum_{k >= 0} Q(m,2k) x^k/k! = (1+4x)^((m-1)/2)/(1+2x)^(m/2), Sum_{k >= 0} Q(m,2k+1) x^k/k! = (1+4x)^((m-2)/2)/(1+2x)^((m+1)/2).

More terms added by G. C. Greubel, Jan 30 2020

A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Mats Granvik, Jan 18 2009

Triangle can be used in matrix inverses. Signs in columns as in A153881.

Iff n is a triangular number, a(n)=1; otherwise, a(n)=-1. (This is explicitly implemented in the second Mathematica program below.) - Harvey P. Dale, Apr 27 2014

			Table begins:
   1;
  -1,  1;
  -1, -1,  1;
  -1, -1, -1,  1;
  -1, -1, -1, -1,  1;
  -1, -1, -1, -1, -1,  1;
  -1, -1, -1, -1, -1, -1, 1;

G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Cf. A023443, A023444.

[k eq n select 1 else -1: k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021

A154990 := proc(n,k)
    option remember;
    if k = n then
        1;
    elif k > n then
        0;
    else
        -1 ;
    end if;
end proc:
seq(seq(A154990(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Sep 16 2017

Flatten[Table[PadLeft[{1},n,-1],{n,15}]] (* or *) With[{tr=Accumulate[ Range[ 15]]}, Table[If[MemberQ[tr,n],1,-1],{n,Last[tr]}]] (* Harvey P. Dale, Apr 27 2014 *)

flatten([[1 if k==n else -1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 06 2021

From G. C. Greubel, Mar 06 2021: (Start)
T(n, k) = -1 with T(n, n) = 1.
Sum_{k=1..n} T(n, k) = 2-n = -A023443(n-1) = -A023444(n). (End)

A253639 Lesser of twin primes of the form (k^2 + 2, k^2 + 4).

Original entry on oeis.org

3, 11, 227, 1091, 2027, 3251, 13691, 21611, 59051, 65027, 91811, 140627, 178931, 199811, 205211, 227531, 328331, 567011, 700571, 804611, 815411, 1071227, 2241011, 3629027, 4223027, 4347227, 4809251, 5212091, 5919491, 6185171, 6426227, 6671891, 7601051, 7969331, 8661251, 8732027, 9018011, 10323371

Offset: 1

M. F. Hasler, Jan 16 2015

Companion sequence to A085554 (which yields the greater of the pair) and A086381 (which lists the x-values). Except for the first term, all values are a(n)=11 (mod 72). - M. F. Hasler, Jan 18 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001097, A001359, A006512, A085554, A086381, A253640.

Transpose[Select[Table[x^2+{2,4},{x,5000}],AllTrue[#,PrimeQ]&]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *)

forstep(x=1,9999,2,is_A086381(x)&&print1(x^2+2,",")) \\ M. F. Hasler, Jan 16 2015

Equals A059100 o A086381 = A023444 o A085554, i.e., a(n) = A086381(n)^2+2 = A085554(n)-2. - M. F. Hasler, Jan 18 2015

A129936 a(n) = (n-2)(n+3)(n+2)/6.

Original entry on oeis.org

-2, -2, 0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

Roger L. Bagula, Jun 09 2007

Essentially the same as A005586.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A002378, A005586, A023444, A034856, A214292.

seq(sum(i*(k-i+1), i=1..k+2), k=0..99); # Wesley Ivan Hurt, Sep 21 2013

f[n_] = Binomial[n + 3, 3] - (n + 3)*(n + 2)/2; Table[f[n], {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{-2,-2,0,5},50] (* Harvey P. Dale, Jul 03 2020 *)

a(n)=(n-2)*(n+3)*(n+2)/6 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(n+3,3) - (n + 3)*(n + 2)/2.
a(n) = A214292(n+2,2). - Reinhard Zumkeller, Jul 12 2012
G.f.: (x^3-4*x^2+6*x-2)/(x-1)^4. - Colin Barker, Sep 05 2012
From Wesley Ivan Hurt, Sep 21 2013: (Start)
a(n) = Sum_{i=1..n+2} i*(n-i+1).
a(n+2) = A000292(n+1) + A034856(n), n>0. (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=3} 1/a(n) = 77/200.
Sum_{n>=3} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
From Elmo R. Oliveira, Aug 04 2025: (Start)
E.g.f.: exp(x)*(x^3 + 6*x^2 - 12)/6.
a(n) = A023444(n)*A002378(n+2)/6.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)

More terms from Wesley Ivan Hurt, Sep 21 2013

A289207 a(n) = max(0, n-2).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 0

Jean-François Alcover and Paul Curtz, Jun 28 2017

This simple sequence is such that there is one and only one array of differences D(n,k) where the first and the second upper subdiagonal is a(n).
The rows of this array are existing sequences of the OEIS, prepended with zeros:
row 0 is A118425,
row 1 is A006478,
row 2 is A001629,
row 3 is A010049,
row 4 is A006367,
row 5 is not in the OEIS.
It can be observed that a(n) is an autosequence of the first kind whose second kind mate is A199969. In addition, the structure of the array D(n,k) shows that the first row is an autosequence.
For n = 1 to 8, rows with only one leading zero are also autosequences.

			Array of differences begin:
   0,   0,   0,   0,  0,   0,  0,  1,  4, 12, 30, 68, ...
   0,   0,   0,   0,  0,   0,  1,  3,  8, 18, 38, 76, ...
   0,   0,   0,   0,  0,   1,  2,  5, 10, 20, 38, 71, ...
   0,   0,   0,   0,  1,   1,  3,  5, 10, 18, 33, 59, ...
   0,   0,   0,   1,  0,   2,  2,  5,  8, 15, 26, 46, ...
   0,   0,   1,  -1,  2,   0,  3,  3,  7, 11, 20, 34, ...
   0,   1,  -2,   3, -2,   3,  0,  4,  4,  9, 14, 24, ...
   1,  -3,   5,  -5,  5,  -3,  4,  0,  5,  5, 10, 16, ...
  -4,   8, -10,  10, -8,   7, -4,  5,  0,  6,  6, 17, ...
  12, -18,  20, -18, 15, -11,  9, -5,  6,  0,  7,  7, ...
  ...

OEIS Wiki, Autosequence
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Essentially the same as A023444. Cf. A001477, A118425, A006478, A001629, A010049, A006367, A199969.

a[n_] := Max[0, n - 2];
D[n_, k_] /; k == n + 1 := a[n]; D[n_, k_] /; k == n + 2 := a[n]; D[n_, k_] /; k > n + 2 := D[n, k] = Sum[D[n + 1, j], {j, 0, k - 1}]; D[n_, k_] /; k <= n := D[n, k] = D[n - 1, k + 1] - D[n - 1, k];
Table[D[n, k], {n, 0, 11}, {k, 0, 11}]

G.f.: x^3 / (1-x)^2.

A023445 n-3.

Original entry on oeis.org

-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55

Offset: 0

N. J. A. Sloane

For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+3 n+5). Then a(n) = det(M(n)). [From K.V.Iyer, Apr 11 2009]

Tanya Khovanova, Recursive Sequences
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A256958, A001057, A000027, A001477, A023443, A023444, A023446, ..., A023482, A022958, ..., A022996.

Range[0,60]-3 (* Harvey P. Dale, Dec 09 2017 *)

PARI
```
a(n)=n-3
```

a(n) = 2 a(n-1) - a(n-2). a(n) = -A022959(n). G.f.: (-3 + 4*x)/(1 - x)^2. - M. F. Hasler, Apr 18 2015

A053565 a(n) = 2^(n-1)(3n-4).

Original entry on oeis.org

-2, -1, 4, 20, 64, 176, 448, 1088, 2560, 5888, 13312, 29696, 65536, 143360, 311296, 671744, 1441792, 3080192, 6553600, 13893632, 29360128, 61865984, 130023424, 272629760, 570425344, 1191182336, 2483027968, 5167382528, 10737418240

Offset: 0

Barry E. Williams, Jan 17 2000

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A023444.

Cf. A027992, A048496.

List([0..30], n-> 2^(n-1)*(3*n-4)) # G. C. Greubel, May 16 2019

[2^(n-1)*(3*n-4): n in [0..30]]; // Vincenzo Librandi, Sep 26 2011

Table[2^(n-1)*(3*n-4), {n,0,30}] (* G. C. Greubel, May 16 2019 *)

vector(30, n, n--; 2^(n-1)*(3*n-4)) \\ G. C. Greubel, May 16 2019

[2^(n-1)*(3*n-4) for n in (0..30)] # G. C. Greubel, May 16 2019

a(n) = 4*a(n-1) - 4*a(n-2), with a(0) = -2, a(1) = -1.
G.f.: -(2-7*x)/(1-2*x)^2. - Colin Barker, Apr 07 2012
E.g.f.: (3*x - 2)*exp(2*x). - G. C. Greubel, May 16 2019

A235774 Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.

Original entry on oeis.org

-1, -1, 1, 1, 59, 3, 169, 5, 179, 7, 533, 9, 26609, 11, 79, 13, 3523, 15, 56635, 17, -168671, 19, 857273, 21, -236304031, 23, 8553247, 25, -23749438409, 27, 8615841677021, 29, -7709321025917, 31, 2577687858559, 33, -26315271552988224913

Offset: 0

Paul Curtz, Jan 15 2014

(a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
Difference table for a(n)/A027642(n):
-1, -1/2,   1/6,      1,  59/30,     3, 169/42, ...
1/2, 2/3,   5/6,  29/30,  31/30, 43/42,  41/42, ... = A165161(n)/A051717(n+1)
1/6, 1/6,  2/15,   1/15, -1/105, -1/21, -1/105, ... not in the OEIS
0, -1/30, -1/15, -8/105, -4/105, 4/105,  8/105, ... etc.
Compare with the array in A190339.

Cf. A164558/A027642, A165161(n)/A051717(n+1).

b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jan 30 2014 *)

(a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
(A164558(n) - a(n))/A027642(n) = 2's = A007395.
(a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).

A023446 n-4.

Original entry on oeis.org

-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54

Offset: 0

N. J. A. Sloane

Tanya Khovanova, Recursive Sequences
Robert G. Wilson v, American English names for the numbers from 0 to 100999 without spaces or hyphens.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A256958, A001057, A000027, A001477, A023443, A023444, A023445, A023447, ..., A023482, A022958, ..., A022996.

Range[0,60]-4 (* Harvey P. Dale, Sep 17 2022 *)

PARI
```
a(n)=n-4
```

a(n) = 2 a(n-1) - a(n-2). a(n) = -A022960(n). G.f.: -(4-5*x)/(1-x)^2. - M. F. Hasler, Apr 18 2015

cp's OEIS Frontend

A023443 a(n) = n - 1.

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A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2,2k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

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A154990 Triangle read by rows. Main diagonal is positive. The rest of the terms are negative.

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A253639 Lesser of twin primes of the form (k^2 + 2, k^2 + 4).

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Mathematica

PARI

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A129936 a(n) = (n-2)*(n+3)*(n+2)/6.

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Mathematica

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A289207 a(n) = max(0, n-2).

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A023445 n-3.

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A127080 Infinite square array read by antidiagonals: Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)Q(m+1, 2k-1) - (2k-1)Q(m+2,2k-2), mQ(m, 2k+1) = (m - 2k)Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1).

A129936 a(n) = (n-2)(n+3)(n+2)/6.

A053565 a(n) = 2^(n-1)(3n-4).