A028872 -id:A028872 - cp's OEIS frontend

A361950 Array read by antidiagonals: T(n,k) = n! * Sum_{s} 2^(Sum_{i=1..k-1} s(i)*s(i+1))/(Product_{i=1..k} s(i)!) where the sum is over all nonnegative compositions s of n into k parts.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 13, 26, 1, 0, 1, 5, 22, 81, 162, 1, 0, 1, 6, 33, 166, 721, 1442, 1, 0, 1, 7, 46, 287, 1726, 9153, 18306, 1, 0, 1, 8, 61, 450, 3309, 24814, 165313, 330626, 1, 0, 1, 9, 78, 661, 5650, 50975, 494902, 4244481, 8488962, 1, 0

Offset: 0

Andrew Howroyd, Mar 31 2023

T(n,k) corresponds to c(k,n) in the Klarner reference. This is an intermediate step in the computation of the number of labeled weakly graded (ranked) posets. The number of elements in the poset is n and the rank k.

			Array begins:
======================================================
n/k| 0 1      2       3        4        5        6 ...
---+--------------------------------------------------
0  | 1 1      1       1        1        1        1 ...
1  | 0 1      2       3        4        5        6 ...
2  | 0 1      6      13       22       33       46 ...
3  | 0 1     26      81      166      287      450 ...
4  | 0 1    162     721     1726     3309     5650 ...
5  | 0 1   1442    9153    24814    50975    91866 ...
6  | 0 1  18306  165313   494902  1058493  1957066 ...
7  | 0 1 330626 4244481 13729846 29885567 55363650 ...
  ...
T(3,2) = 26: the nonnegative integer compositions of 3 with 2 parts are (0,3), (1,2), (2,1), (3,0). These contribute, respectively 2^0*3!/(0!*3!) = 1, 2^2*3!/(1!*2!) = 12, 2^2*3!/(2!*1!) = 12, 2^0*3!/(0!*3!) = 1, so T(3,2) = 1 + 12 + 12 + 1 = 26.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.

Columns k=0..7 are A000007, A000012, A047863, A000684, A001827, A001828, A001829, A001830.
Rows 0..2 are A000012, A000027, A028872(n+1).
The unlabeled version is A361952.
Cf. A361951.

S(M)={matrix(#M, #M, i, j, sum(k=0, i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
C(n, m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
R(n)={Mat([Col(serlaplace(p)) | p<-C(n)])}
{ my(A=R(6)); for(i=1, #A, print(A[i,])) }

A028874 Primes of form k^2 - 3.

Original entry on oeis.org

13, 61, 97, 193, 397, 673, 1021, 1153, 1597, 1933, 2113, 3361, 4093, 4621, 6397, 7393, 7741, 8461, 9601, 12097, 12541, 13921, 15373, 16381, 18493, 19597, 20161, 21313, 26893, 29581, 36097, 37633, 40801, 42433, 43261, 47521, 48397

Offset: 1

Patrick De Geest

nonn

Also primes equal to the product of two consecutive odd numbers (A000466) minus 2. - Giovanni Teofilatto, Feb 11 2010

All terms are of the form 6m + 1. - Zak Seidov, May 01 2014

			61 is prime and equal to 8^2 - 3, so it is in the sequence.
67 is prime but it's 8^2 + 3 = 9^2 - 14, so it is not in the sequence.
9^2 - 3 = 78 but it's composite, so it's not in the sequence either.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
R. J. Mathar, Solutions to the exponential Diophantine 1 + p_1^x + p_2^y + p_3^z = w^2 for distinct primes p_1, p_2, p_3, 2022.
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A002476 (Primes of form 6m + 1), A028871, A028872, A028873.

Primes terms in A082109. Subsequence of A068228. - Klaus Purath, Jan 09 2023

[a: n in [2..300] | IsPrime(a) where a is n^2-3 ]; // Vincenzo Librandi, Nov 08 2014

Select[Range[2, 250]^2 - 3, PrimeQ] (* Harvey P. Dale, Aug 07 2013 *)
Select[Table[n^2 - 3, {n, 2, 300}], PrimeQ] (* Vincenzo Librandi, Nov 08 2014 *)

select(isprime, vector(100,n,n^2-3)) \\ Charles R Greathouse IV, Nov 19 2014

A028872 INTERSECT A000040. - Klaus Purath, Dec 07 2020

a(n) = A028873(n)^2 - 3. - Amiram Eldar, Mar 01 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)

A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 0, n >= 0.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 3, 13, 13, 3, 4, 22, 28, 22, 4, 5, 33, 49, 49, 33, 5, 6, 46, 74, 86, 74, 46, 6, 7, 61, 105, 131, 131, 105, 61, 7, 8, 78, 140, 188, 200, 188, 140, 78, 8, 9, 97, 181, 251, 289, 289, 251, 181, 97, 9, 10, 118, 226, 326, 386, 418, 386, 326, 226, 118, 10, 11, 141, 277

Offset: 0

N. J. A. Sloane, Feb 24 2006

This is the number of linear partitions of an m X n grid.

			The array begins:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
1, 6, 13, 22, 33, 46, 61, 78, 97, 118, ...
2, 13, 28, 49, 74, 105, 140, 181, 226, 277, ...
3, 22, 49, 86, 131, 188, 251, 326, 409, 502, ...
4, 33, 74, 131, 200, 289, 386, 503, 632, 777, ...
5, 46, 105, 188, 289, 418, 559, 730, 919, 1132, ...
6, 61, 140, 251, 386, 559, 748, 979, 1234, 1521, ...
7, 78, 181, 326, 503, 730, 979, 1282, 1617, 1994, ...
...

D. M. Acketa, J. D. Zunic: On the number of linear partitions of the (m,n)-grid. Inform. Process. Lett., 38 (3) (1991), 163-168. See Table A.1.
Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Max A. Alekseyev. On the number of two-dimensional threshold functions. SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184

The second and third rows are A028872 and A358296.
The main diagonal is A141255 = A114043 - 1.
The lower triangle is A332351.
Cf. A114999, A114043, A115004, A115005, A115006, A115007, A115010, A115011.

V:=proc(m,n) local t1,i,j; t1:=0; for i from 1 to m do for j from 1 to n do if gcd(i,j)=1 then t1:=t1+(m+1-i)*(n+1-j); fi; od; od; t1; end; T:=(m,n)->(2*m*n+m+n+2*V(m,n));

V[m_, n_] := Sum[If[GCD[i, j] == 1, (m-i+1)*(n-j+1), 0], {i, 1, m}, {j, 1, n}]; T[m_, n_] := 2*m*n+m+n+2*V[m, n]; Table[T[m-n, n], {m, 0, 11}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

A123968 a(n) = n^2 - 3, starting at n=1.

Original entry on oeis.org

-2, 1, 6, 13, 22, 33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622, 673, 726, 781, 838, 897, 958, 1021, 1086, 1153, 1222, 1293, 1366, 1441, 1518, 1597, 1678, 1761, 1846, 1933, 2022, 2113, 2206, 2301, 2398, 2497

Offset: 1

Gary W. Adamson and Roger L. Bagula, Oct 29 2006

Essentially the same as A028872 (n^2-3 with offset 2).
a(n) is the constant term of the quadratic factor of the characteristic polynomial of the 5 X 5 tridiagonal matrix M_n with M_n(i,j) = n for i = j, M_n(i,j) = -1 for i = j+1 and i = j-1, M_n(i,j) = 0 otherwise.
The characteristic polynomial of M_n is (x-(n-1))*(x-n)*(x-(n+1))*(x^2-2*n*x+c) with c = n^2-3.
The characteristic polynomials are related to chromatic polynomials, cf. links. They have roots n+sqrt(3).

			The quadratic factors of the characteristic polynomials of M_n for n = 1..6 are
  x^2 -  2*x -  2,
  x^2 -  4*x +  1,
  x^2 -  6*x +  6,
  x^2 -  8*x + 13,
  x^2 - 10*x + 22,
  x^2 - 12*x + 33.

Eric W. Weisstein, Chromatic Polynomial.
Wikipedia, Chromatic polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Essentially the same: A028872, A267874.

mat:=func< n | Matrix(IntegerRing(), 5, 5, [< i, j, i eq j select n else (i eq j+1 or i eq j-1) select -1 else 0 > : i, j in [1..5] ]) >; [ Coefficients(Factorization(CharacteristicPolynomial(mat(n)))[4][1])[1]:n in [1..50] ]; // Klaus Brockhaus, Nov 13 2010

with(combinat):seq(fibonacci(3, i)-4,i=1..55); # Zerinvary Lajos, Mar 20 2008

M[n_] := {{n, -1, 0, 0, 0}, {-1, n, -1, 0, 0}, {0, -1, n, -1, 0}, {0, 0, -1, n, -1}, {0, 0, 0, -1, n}}; p[n_, x_] = Factor[CharacteristicPolynomial[M[n], x]] Table[ -3 + n^2, {n, 1, 25}]

PARI
```
A123968(n) = n^2-3   /* or: */
```

a(n)=polcoeff(factor(charpoly(matrix(5,5,i,j,if(abs(i-j)>1,0,if(i==j,n,-1)))))[4,1], 0)

a(n) = 2*n + a(n-1) - 1. - Vincenzo Librandi, Nov 12 2010
G.f.: x*(-2+x)*(1-3*x)/(1-x)^3. - Colin Barker, Jan 29 2012
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(x^2 + x - 3) + 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited and extended by Klaus Brockhaus, Nov 13 2010

Definition simplified by M. F. Hasler, Nov 12 2010

A190576 a(n) = n^2 + 5*n - 5.

Original entry on oeis.org

1, 9, 19, 31, 45, 61, 79, 99, 121, 145, 171, 199, 229, 261, 295, 331, 369, 409, 451, 495, 541, 589, 639, 691, 745, 801, 859, 919, 981, 1045, 1111, 1179, 1249, 1321, 1395, 1471, 1549, 1629, 1711, 1795, 1881, 1969, 2059, 2151, 2245, 2341

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2011

Also a(n) = n^2 + 9*n + 9 if the offset is changed to -1. - R. J. Mathar, May 18 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. sequences of the form n^2 + k*n - k : A000290 (k=0), A028387 (k=1), A028872 (k=2), A082111 (k=3), A028884 (k=4).

[n^2+5*n-5: n in [1..50]]; // Vincenzo Librandi, Sep 30 2011

k = 5; Table[n^2 + k*n - k, {n, 100}]
LinearRecurrence[{3,-3,1},{1,9,19},50] (* Harvey P. Dale, May 28 2015 *)

a(n)=n^2+5*n-5 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(-1 - 6*x + 5*x^2) / (x-1)^3. - R. J. Mathar, May 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=9, a(3)=19. - Harvey P. Dale, May 28 2015
Sum_{n>=1} 1/a(n) = 199/495 + Pi*tan(3*sqrt(5)*Pi/2)/(3*sqrt(5)). - Amiram Eldar, Jan 18 2021

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).

A267874 Total number of ON (black) cells after n iterations of the "Rule 235" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 2, 6, 13, 22, 33, 46, 61, 78, 97, 118, 141, 166, 193, 222, 253, 286, 321, 358, 397, 438, 481, 526, 573, 622, 673, 726, 781, 838, 897, 958, 1021, 1086, 1153, 1222, 1293, 1366, 1441, 1518, 1597, 1678, 1761, 1846, 1933, 2022, 2113, 2206, 2301, 2398, 2497

Offset: 0

Robert Price, Jan 21 2016

With the exception of the first one or two entries the same as A123968 and A028872. - R. J. Mathar, Jan 24 2016

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
Stephen Wolfram, A New Kind of Science, 2002; p. 55.
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A267869.

Cf. A028872, A123968.

rule=235; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) nbc=Table[Total[catri[[k]]],{k,1,rows}]; (* Number of Black cells in stage n *) Table[Total[Take[nbc,k]],{k,1,rows}] (* Number of Black cells through stage n *)

From Colin Barker, Jan 22 2016 and Apr 20 2019: (Start)
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
G.f.: (1-x+3*x^2-x^4) / (1-x)^3.
(End)

A349947 Triangular array: row n gives the positions of n+1 in A349946.

Original entry on oeis.org

1, 2, 4, 3, 5, 9, 6, 7, 10, 16, 8, 11, 12, 17, 25, 13, 14, 18, 19, 26, 36, 15, 20, 21, 27, 28, 37, 49, 22, 23, 29, 30, 38, 39, 50, 64, 24, 31, 32, 40, 41, 51, 52, 65, 81, 33, 34, 42, 43, 53, 54, 66, 67, 82, 100, 35, 44, 45, 55, 56, 68, 69, 83, 84, 101, 121

Offset: 1

Clark Kimberling, Dec 07 2021

Every positive integer occurs exactly once, so as a sequence, this is a permutation of the positive integers.
Row n ends in n^2. The first term in row n is (1 + n/1)^2 - 3 if n >= 4 and n is even; as in A028872(n) for n >= 3.
The first term in row n is ((n+1)/2)^2 - 1 if n >= 3 and n is odd, as in A132411(n) for n >= 3.

			First 7 rows:
   1
   2   4
   3   5   9
   6   7  10  16
   8  11  12  17  25
  13  14  18  19  26  36
  14  20  21  27  28  37  49

Cf. A349526, A349946.

t = {1, 1}; Do[t = Join[t, Riffle[Range[n], n], {n}], {n, 2, 100}];
u = Flatten[Partition[t, 2]];
v = Table[n (n + 1), {n, 1, 80}];
d = Delete[u, Map[{#} &, v]]; (* A349526 *)
p = Table[{d[[n]], d[[n + 1]]}, {n, 1, 150}];
q = Map[Total, p]  (* A349946 *)
r = Table[Flatten[Position[q, n]], {n, 2, 12}]  (* A349947 array *)
Flatten[r]  (* A349947 sequence *)

cp's OEIS Frontend

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A123968 a(n) = n^2 - 3, starting at n=1.

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A115009 Array read by antidiagonals: let V(m,n) = Sum_{i=1..m, j=1..n, gcd(i,j)=1} (m+1-i)(n+1-j), then T(m,n) = 2mn+m+n+2V(m,n), for m >= 0, n >= 0.