A028875 -id:A028875 - cp's OEIS frontend

A028876 Numbers k such that k^2 - 5 is prime.

4, 6, 8, 12, 14, 16, 22, 24, 32, 34, 36, 38, 42, 44, 46, 52, 58, 64, 72, 74, 78, 82, 94, 102, 112, 116, 122, 132, 144, 152, 164, 166, 168, 174, 176, 182, 184, 186, 188, 198, 204, 212, 222, 226, 232, 234, 236, 252, 262, 264, 278, 284, 288, 292, 298, 302, 318, 324

Offset: 1

Patrick De Geest

G. C. Greubel, Table of n, a(n) for n = 1..1000
P. De Geest, Palindromic Quasipronics of the form n(n+x)
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A028875, A028877.

[n: n in [2..1000] |IsPrime(n^2 - 5 )]; // Vincenzo Librandi, Nov 18 2010

Select[Range[400], PrimeQ[#^2 - 5] &] (* Alonso del Arte, Aug 27 2013 *)

is(n)=isprime(n^2-5) \\ Charles R Greathouse IV, Jun 06 2017

A028877 Primes of form k^2 - 5.

Original entry on oeis.org

11, 31, 59, 139, 191, 251, 479, 571, 1019, 1151, 1291, 1439, 1759, 1931, 2111, 2699, 3359, 4091, 5179, 5471, 6079, 6719, 8831, 10399, 12539, 13451, 14879, 17419, 20731, 23099, 26891, 27551, 28219, 30271, 30971, 33119, 33851, 34591, 35339, 39199, 41611, 44939, 49279

Offset: 1

Patrick De Geest

These numbers are prime in Z but not in Z[sqrt(5)] nor in Z[phi] (where phi is the golden ratio), since (k - sqrt(5))(k + sqrt(5)) = ((k + 1) - 2*phi)((k - 1) + 2*phi) = k^2 - 5. - Alonso del Arte, Aug 27 2013

			31 is in the sequence as it is equal to 6^2 - 5.
59 is in the sequence since it is equal to 8^2 - 5.
95 is not in the sequence though it does equal 10^2 - 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A028875 (superset), A028876.

[a: n in [1..300] | IsPrime(a) where a is n^2-5]; // Vincenzo Librandi, Dec 01 2011

Select[Table[n^2 - 5, {n, 200}], PrimeQ] (* Harvey P. Dale, Jan 17 2011 *)

a(n) = A028875(A028876(n)). - Elmo R. Oliveira, Feb 22 2025

A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

Original entry on oeis.org

-1, 0, 1, 3, 2, 7, 5, 8, 3, 13, 10, 17, 7, 18, 11, 15, 4, 21, 17, 30, 13, 35, 22, 31, 9, 32, 23, 37, 14, 33, 19, 24, 5, 31, 26, 47, 21, 58, 37, 53, 16, 59, 43, 70, 27, 65, 38, 49, 11, 50, 39, 67, 28, 73, 45, 62, 17, 57, 40, 63, 23, 52, 29, 35, 6, 43, 37, 68, 31, 87, 56, 81, 25, 94, 69

Offset: 0

Arie Werksma (Werksma(AT)Tiscali.nl), Feb 04 2009

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A002487, A007814.
From Yosu Yurramendi, Mar 09 2018: (Start)
a(2^m +  0) = A000027(m),   m >= 0.
a(2^m +  1) = A002061(m+2), m >= 1.
a(2^m +  2) = A002522(m),   m >= 2.
a(2^m +  3) = A033816(m-1), m >= 2.
a(2^m +  4) = A002061(m),   m >= 2.
a(2^m +  5) = A141631(m),   m >= 3.
a(2^m +  6) = A084849(m-1), m >= 3.
a(2^m +  7) = A056108(m-1), m >= 3.
a(2^m +  8) = A000290(m-1), m >= 3.
a(2^m +  9) = A185950(m-1), m >= 4.
a(2^m + 10) = A144390(m-1), m >= 4.
a(2^m + 12) = A014106(m-2), m >= 4.
a(2^m + 16) = A028387(m-3), m >= 4.
a(2^m + 18) = A250657(m-4), m >= 5.
a(2^m + 20) = A140677(m-3), m >= 5.
a(2^m + 32) = A028872(m-2), m >= 5.
a(2^m -  1) = A005563(m-1), m >= 0.
a(2^m -  2) = A028387(m-2), m >= 2.
a(2^m -  3) = A033537(m-2), m >= 2.
a(2^m -  4) = A008865(m-1), m >= 3.
a(2^m -  7) = A140678(m-3), m >= 3.
a(2^m -  8) = A014209(m-3), m >= 4.
a(2^m - 16) = A028875(m-2), m >= 5.
a(2^m - 32) = A108195(m-5), m >= 6.
(End)

A156140 := proc(n)
    option remember ;
    if n <= 1 then
        n-1 ;
    else
        (2*A007814(n-1)+1)*procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A156140(n),n=0..80) ; # R. J. Mathar, Mar 14 2009

Fold[Append[#1, (2 IntegerExponent[#2, 2] + 1) #1[[-1]] - #1[[-2]] ] &, {-1, 0}, Range[73]] (* Michael De Vlieger, Mar 09 2018 *)

first(n)=my(v=vector(n+1)); v[1]=-1; v[2]=0; for(k=1,n-1,v[k+2]=(2*valuation(k,2)+1)*v[k+1] - v[k]); v \\ Charles R Greathouse IV, Apr 05 2016

fusc(n)=my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); b
a(n)=my(m=1,s,t); if(n==0, return(-1)); while(n%2==0, s+=fusc(n>>=1)); while(n>1, t=logint(n,2); n-=2^t; s+=m*fusc(n)*(t^2+t+1); m*=-t); m*(n-1) + s \\ Charles R Greathouse IV, Dec 13 2016

a <- c(0,1)
maxlevel <- 6 # by choice
for(m in 1:maxlevel) {
  a[2^(m+1)] <- m + 1
  for(k in 1:(2^m-1)) {
    r <- m - floor(log2(k)) - 1
    a[2^r*(2*k+1)] <- a[2^r*(2*k)] + a[2^r*(2*k+2)]
}}
a
# Yosu Yurramendi, May 08 2018

Let b(n) = A002487(n), Stern's diatomic series.
a(n+1)*b(n) - a(n)*b(n+1) = 1 for n >= 0.
a(2n+1) = a(n) + a(n+1) + b(n) + b(n+1) for n >= 0.
a(2n) = a(n) + b(n) for n >= 0.
a(2^n + k) = -n*a(k) + (n^2 + n + 1)*b(k) for 0 <= k <= 2^n.
b(2^n + k) = -a(k) + (n + 1)*b(k) for 0 <= k <= 2^n.
a(2^m + k) = b(2^m+k)*m + b(k), m >= 0, 0 <= k < 2^m. - Yosu Yurramendi, Mar 09 2018
a(2^(m+1)+2^m+1) = 2*m+1, m >= 0. - Yosu Yurramendi, Mar 09 2018
From Yosu Yurramendi, May 08 2018: (Start)
a(2^m) = m, m >= 0.
a(2^r*(2*k+1)) = a(2^r*(2*k)) + a(2^r*(2*k+2)), r = m - floor(log_2(k)) - 1, m > 0, 1 <= k < 2^m.
(End)

A179770 The fourth central column of triangle A122242, i.e., A179761(4), A179761(11), A179761(20), A179761(31), ...

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024

Cf. A179771-A179774, and also A179775, A179830.

a(n) = A179761(A028875(n+2)).

A179775 The fourth central column of triangle A122245, i.e., A179762(4), A179762(11), A179762(20), A179762(31), ...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

A. Karttunen, Table of n, a(n) for n = 1..1024

Cf. A179776-A179779, and also A179770, A179830.

a(n) = A179762(A028875(n+2)).

A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 8, 9, 13, 17, 14, 6, 16, 21, 26, 22, 11, 12, 25, 31, 37, 32, 18, 15, 20, 36, 43, 50, 44, 27, 23, 24, 30, 49, 57, 65, 58, 38, 33, 19, 35, 42, 64, 73, 82, 74, 51, 45, 28, 29, 48, 56, 81, 91, 101, 92, 66, 59, 39, 34, 41, 63, 72, 100, 111

Offset: 1

Boris Putievskiy, Mar 05 2013

A permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(1,2), T(2,1), T(2,2);
. . .
T(1,n), T(3,n), ... T(n,3), T(n,1), T(2,n), T(4,n), ... T(n,4), T(n,2);
...

			The start of the sequence as table:
   1   2   5  10  17  26 ...
   3   4   8  14  22  32 ...
   7   9   6  11  18  27 ...
  13  16  12  15  23  33 ...
  21  25  20  24  19  28 ...
  31  36  30  35  29  34 ...
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   5,  4,  7;
  10,  8,  9, 13;
  17, 14,  6, 16, 21;
  26, 22, 11, 12, 25, 31;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i > j:
   result=i*i-(j%2)*i+2-int((j+2)/2)
else:
   result=j*j-((i%2)+1)*j + int((i+3)/2)

As a table:
T(n,k) = n*n - (k mod 2)*n + 2 - floor((k+2)/2), if n>k;
T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.
As a linear sequence:
a(n) = i*i - (j mod 2)*i + 2 - floor((j+2)/2), if i>j;
a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/2), if i<=j; where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

Original entry on oeis.org

1, 3, 4, 8, 2, 9, 15, 6, 7, 16, 24, 13, 5, 14, 25, 35, 22, 11, 12, 23, 36, 48, 33, 20, 10, 21, 34, 49, 63, 46, 31, 18, 19, 32, 47, 64, 80, 61, 44, 29, 17, 30, 45, 62, 81, 99, 78, 59, 42, 27, 28, 43, 60, 79, 100, 120, 97, 76, 57, 40, 26, 41, 58, 77, 98, 121

Offset: 1

Boris Putievskiy, Mar 05 2013

Permutation of the natural numbers.
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.
Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1). Enumeration table T(n,k) is layer by layer. The order of the list:
T(1,1)=1;
T(2,2), T(1,2), T(2,1);
...
T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1);
...

			The start of the sequence as a table:
   1,  3,  8, 15, 24, 35, ...
   4,  2,  6, 13, 22, 33, ...
   9,  7,  5, 11, 20, 31, ...
  16, 14, 12, 10, 18, 29, ...
  25, 23, 21, 19, 17, 27, ...
  36, 34, 32, 30, 28, 26, ...
...
The start of the sequence as triangular array read by rows:
   1;
   3,  4;
   8,  2,  9;
  15,  6,  7, 16;
  24, 13,  5, 14, 25;
  35, 22, 11, 12, 23, 36;
  ...

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A060734, A060736; table T(n,k) contains: in rows A005563, A028872, A028875, A028881, A028560, A116711; in columns A000290, A008865, A028347, A028878, A028884.

f[n_, k_] := n^2 - 2*k + 2 /; n >= k; f[n_, k_] := k^2 - 2*n + 1 /; n < k; TableForm[Table[f[n, k], {n, 1, 5}, {k, 1, 10}]]; Table[f[n - k + 1, k], {n, 5}, {k, n, 1, -1}] // Flatten (* G. C. Greubel, Aug 19 2017 *)

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
if i >= j:
   result=i*i-2*j+2
else:
   result=j*j-2*i+1

As a table,
  T(n,k) = n*n - 2*k + 2, if n >= k;
  T(n,k) = k*k - 2*n + 1, if n < k.
As a linear sequence,
  a(n) = i*i - 2*j + 2, if i >= j;
  a(n) = j*j - 2*i + 1, if i < j
  where
    i = n - t*(t+1)/2,
    j = (t*t + 3*t + 4)/2 - n,
    t = floor((-1 + sqrt(8*n-7))/2).

A281442 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 8, 0, 6, 0, 1, 0, 0, 22, 0, 8, 0, 1, 0, 42, 0, 40, 0, 10, 0, 1, 0, 0, 140, 0, 62, 0, 12, 0, 1, 0, 262, 0, 288, 0, 88, 0, 14, 0, 1, 0, 0, 992, 0, 492, 0, 118, 0, 16, 0, 1

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

T(n,r) is also the number of idempotent basis elements of rank r in the Temperley-Lieb algebra of degree n in the generic case (when the twisting parameter is not an m-th root of unity for any m <= n).

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A281438 (row sums), A281441, A289620.

T(2n-1,1) = A005315(n). Empirical: T(2n,2) = A077056(n); T(n+2,n-2) = 2*A028875(n) for n>2. - Andrey Zabolotskiy, Oct 19 2017

A358519 Decimal expansion of Sum_{k >= 1} (-1)^(k+1)/(k^2 + 4*k - 1).

Original entry on oeis.org

1, 8, 9, 9, 5, 7, 9, 0, 7, 7, 1, 8, 0, 6, 2, 7, 2, 5, 2, 7, 1, 9, 0, 8, 4, 0, 9, 0, 6, 3, 6, 3, 6, 6, 5, 5, 2, 7, 9, 5, 6, 6, 4, 9, 1, 4, 5, 4, 0, 6, 7, 9, 5, 6, 3, 1, 3, 2, 8, 9, 3, 1, 3, 6, 2, 2, 1, 3, 1, 4, 6, 7, 1, 4, 9, 3, 6, 1, 9, 8, 5, 9, 8, 5, 3, 2, 5, 6, 4, 7, 7, 5, 6, 1, 3

Offset: 0

Claude H. R. Dequatre, Nov 20 2022

			0.189957907718062725271908409063636655279566491...

Michael I. Shamos, A catalog of the real numbers (2011), p. 253.

Cf. A028875, A242023, A242024, A358516, A358517.

Cf. A002162.

RealDigits[(2*Sqrt[5]*Pi*Csc[Sqrt[5]*Pi] - 17)/20, 10, 120][[1]] (* Amiram Eldar, Nov 21 2022 *)

(Pi/(2*sqrt(5)))*1/sin(Pi*sqrt(5)) - 17/20

Equals Sum_{k >=1} (-1)^(k+1)/(k^2 + 4*k - 1).
Equals (Pi/(2*sqrt(5)))*csc(Pi*sqrt(5)) - 17/20 = Sum_{k>=3} (-1)^(k+1)/(k^2-5). [from the Shamos reference]
Equals Sum_{k>=3} (-1)^(k+1)/A028875(k). - Amiram Eldar, Nov 21 2022

A144204 Array A(k,n) = (n+k-2)*(n-1) - 1 (k >= 1, n >= 1) read by antidiagonals.

Original entry on oeis.org

-1, -1, 0, -1, 1, 3, -1, 2, 5, 8, -1, 3, 7, 11, 15, -1, 4, 9, 14, 19, 24, -1, 5, 11, 17, 23, 29, 35, -1, 6, 13, 20, 27, 34, 41, 48, -1, 7, 15, 23, 31, 39, 47, 55, 63, -1, 8, 17, 26, 35, 44, 53, 62, 71, 80, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, 99, -1, 10, 21, 32, 43, 54, 65, 76, 87

Offset: 1

Jonathan Vos Post, Sep 13 2008

Arises in complete intersection threefolds,
Also can be produced as a triangle read by rows: a(n, k) = nk - (n + k). - Alonso del Arte, Jul 09 2009
Kosta: Let X be a complete intersection of two hypersurfaces F_n and F_k in the projective space P^5 of degree n and k respectively. with n=>k, such that the singularities of X are nodal and F_k is smooth. We prove that if the threefold X has at most (n+k-2)*(n-1) - 1 singular points, then it is factorial.

			From _R. J. Mathar_, Jul 10 2009: (Start)
The rows A(n,1), A(n,2), A(n,3), etc., are :
.-1...0...3...8..15..24..35..48..63..80..99.120.143.168 A067998
.-1...1...5..11..19..29..41..55..71..89.109.131.155.181 A028387
.-1...2...7..14..23..34..47..62..79..98.119.142.167.194 A008865
.-1...3...9..17..27..39..53..69..87.107.129.153.179.207 A014209
.-1...4..11..20..31..44..59..76..95.116.139.164.191.220 A028875
.-1...5..13..23..35..49..65..83.103.125.149.175.203.233 A108195
.-1...6..15..26..39..54..71..90.111.134.159.186.215.246
.-1...7..17..29..43..59..77..97.119.143.169.197.227.259
.-1...8..19..32..47..64..83.104.127.152.179.208.239.272
.-1...9..21..35..51..69..89.111.135.161.189.219.251.285
.-1..10..23..38..55..74..95.118.143.170.199.230.263.298
.-1..11..25..41..59..79.101.125.151.179.209.241.275.311
.-1..12..27..44..63..84.107.132.159.188.219.252.287.324
.-1..13..29..47..67..89.113.139.167.197.229.263.299.337 Cf. A126719.
(End)
As a triangle:
. 0
. 1, 3
. 2, 5, 8
. 3, 7, 11, 15
. 4, 9, 14, 19, 24
. 5, 11, 17, 23, 29, 35
. 6, 13, 20, 27, 34, 41, 48
. 7, 15, 23, 31, 39, 47, 55, 63
. 8, 17, 26, 35, 44, 53, 62, 71, 80

Dimitra Kosta, Factoriality of complete intersection threefolds arXiv:0808.4071 [math.AG]

Row 1 = A067998(n) for n>0. Row 2 = A028387(n) for n>0.Column 1 = -A000012(n). Column 2 = A001477. Column 3 = A005408(k). Column 4 = A016789(k+1). Column 5 = A004767(k+2). Column 6 = A016897(k+3). Column 7 = A016969(k+4). Column 8 = A017053(k+5). Column 9 = A004771(k+6). Column 10 = A017257(k+7).

Cf. A000012, A001477, A004767, A004771, A005408, A016789, A016897, A016969, A017053, A028387, A067998, A126719.

A := proc(k,n) (n+k-2)*(n-1)-1 ; end: for d from 1 to 13 do for n from 1 to d do printf("%d,",A(d-n+1,n)) ; od: od: # R. J. Mathar, Jul 10 2009

a[n_, k_] := a[n, k] = n*k - (n + k); ColumnForm[Table[a[n, k], {n, 10}, {k, n}], Center] (* Alonso del Arte, Jul 09 2009 *)

A[k,n] = (n+k-2)*(n-1) - 1.

Antidiagonal sum: Sum_{n=1..d} A(d-n+1,n) = d*(d^2-2d-1)/2 = -A110427(d). - R. J. Mathar, Jul 10 2009

Duplicate of 6th antidiagonal removed by R. J. Mathar, Jul 10 2009
Keyword:tabl added by R. J. Mathar, Jul 23 2009
Edited by N. J. A. Sloane, Sep 14 2009. There was a comment that the defining formula was wrong, but it is perfectly correct.

cp's OEIS Frontend

A028876 Numbers k such that k^2 - 5 is prime.

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A028877 Primes of form k^2 - 5.

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A156140 Accumulation of Stern's diatomic series: a(0)=-1, a(1)=0, and a(n+1) = (2e(n)+1)*a(n) - a(n-1) for n > 1, where e(n) is the highest power of 2 dividing n.

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PARI

PARI

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A179770 The fourth central column of triangle A122242, i.e., A179761(4), A179761(11), A179761(20), A179761(31), ...

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A179775 The fourth central column of triangle A122245, i.e., A179762(4), A179762(11), A179762(20), A179762(31), ...

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A213921 Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places clockwise. Table T(n,k) read by antidiagonals.

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A213922 Natural numbers placed in table T(n,k) layer by layer. The order of placement: T(n,n), T(n-1,n), T(n,n-1), ... T(1,n), T(n,1). Table T(n,k) read by antidiagonals.

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A281442 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.

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