A034856 -id:A034856 - cp's OEIS frontend

A224038 T(n,k)=Number of nXk 0..1 arrays with antidiagonals unimodal and rows and diagonals nondecreasing.

2, 3, 4, 4, 8, 8, 5, 13, 21, 16, 6, 19, 37, 55, 32, 7, 26, 58, 105, 144, 64, 8, 34, 85, 168, 298, 377, 128, 9, 43, 119, 252, 488, 848, 987, 256, 10, 53, 161, 363, 734, 1422, 2419, 2584, 512, 11, 64, 212, 508, 1064, 2149, 4160, 6908, 6765, 1024, 12, 76, 273, 695, 1505, 3107

Offset: 1

R. H. Hardin Mar 30 2013

Table starts
....2.....3.....4......5......6......7......8......9.....10.....11......12
....4.....8....13.....19.....26.....34.....43.....53.....64.....76......89
....8....21....37.....58.....85....119....161....212....273....345.....429
...16....55...105....168....252....363....508....695....933...1232....1603
...32...144...298....488....734...1064...1505...2091...2864...3875....5185
...64...377...848...1422...2149...3107...4395...6124...8439..11527...15626
..128...987..2419...4160...6321...9125..12856..17875..24623..33686...45837
..256..2584..6908..12214..18673..26928..37808..52356..71923..98238..133616
..512..6765.19737..35960..55373..79800.111683.154076.210890.287271..389967
.1024.17711.56401.106072.164729.237348.331170.455195.620860.843084.1141689

			Some solutions for n=3 k=4
..0..0..0..1....1..1..1..1....0..0..0..1....0..0..0..1....0..0..0..1
..0..1..1..1....1..1..1..1....1..1..1..1....0..0..0..0....0..1..1..1
..0..0..1..1....0..1..1..1....1..1..1..1....0..0..0..0....1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..1012

Column 1 is A000079
Column 2 is A001906(n+1)
Row 1 is A000027(n+1)
Row 2 is A034856(n+1)

Empirical: columns k=1..7 have recurrences of order 1,2,5,7,9,11,13 for n>0,0,0,10,13,16

Empirical: rows n=1..7 are polynomials of degree n for k>0,0,1,2,3,4,5

A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1

Offset: 0

Alois P. Heinz, Nov 29 2014

A(n,n) = A034856(n+2) for n>=2.

			Square array A(n,k) begins:
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,     13,    1,    1,   1,   1,   1,  1, ...
  1, 1,     41,   19,    1,   1,   1,   1,  1, ...
  1, 1,    281,   57,   26,   1,   1,   1,  1, ...
  1, 1,   1183,  121,   75,  34,   1,   1,  1, ...
  1, 1,   6728,  783,  154,  95,  43,   1,  1, ...
  1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...
  1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened
Wikipedia, Polyomino

Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.

Cf. A034856, A250662.

b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
      if n=0 then 1
    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
    else for k while l[k]>0 do od;
         `if`(n2*d+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1,  Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A027379 Expansion of (1+x^2-x^3)/(1-x)^3.

Original entry on oeis.org

1, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078

Offset: 0

N. J. A. Sloane

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

Essentially the triangular numbers (A000217) minus 3.
Also essentially the same as A055998.
Cf. A000217, A034856, A046691.

[n eq 0 select 1 else n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Jul 31 2022

a(n)=if(n,n*(n+5)/2,1) \\ Charles R Greathouse IV, Jun 11 2015

[n*(n+5)/2 + bool(n==0) for n in (0..50)] # G. C. Greubel, Jul 31 2022

a(n) = n*(n+5)/2 = A000217(n+2) - 3 for n>=1. - Emeric Deutsch, Mar 01 2004
a(n) = n + a(n-1) + 2, n>1. - Vincenzo Librandi, Dec 06 2009
E.g.f.: 1 + x*(x + 6)*exp(x)/2. - G. C. Greubel, May 14 2017

A209268 Inverse permutation A054582.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 10, 7, 15, 9, 21, 8, 28, 14, 36, 11, 45, 20, 55, 13, 66, 27, 78, 12, 91, 35, 105, 19, 120, 44, 136, 16, 153, 54, 171, 26, 190, 65, 210, 18, 231, 77, 253, 34, 276, 90, 300, 17, 325, 104, 351, 43, 378, 119, 406, 25, 435, 135, 465, 53, 496, 152

Offset: 1

Boris Putievskiy, Jan 15 2013

nonn

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.

			The start of the sequence for n = 1..32 as table, distributed by exponent of highest power of 2 dividing n:
   |   Exponent of highest power of 2 dividing n
n  |--------------------------------------------------
   |    0      1      2       3      4         5    ...
------------------------------------------------------
1  |....1
2  |...........2
3  |....3
4  |..................4
5  |....6
6  |...........5
7  |...10
8  |..........................7
9  |...15
10 |...........9
11 |...21
12 |..................8
13 |...28
14 |..........14
15 |...36
16 |................................11
17 |...45
18 |..........20
19 |...55
20 |.................13
21 |...66
22 |..........27
23 |...78
24 |................................12
25 |...91
26 |..........35
27 |..105
28 |.................19
29 |..120
30 |..........44
31 |..136
32 |.........................................16
. . .
Let r_c be number row inside the column number c.
r_c = (n+2^c)/2^(c+1).
The column number 0 contains numbers r_0*(r_0+1)/2,     A000217,
The column number 1 contains numbers r_1*(r_1+3)/2,     A000096,
The column number 2 contains numbers r_2*(r_2+5)/2 + 1, A034856,
The column number 3 contains numbers r_3*(r_3+7)/2 + 3, A055998,
The column number 4 contains numbers r_4*(r_4+9)/2 + 6, A046691.

Boris Putievskiy, Table of n, a(n) for n = 1..10000
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
R. J. Mathar, oeisPy
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A054582, A003602, A007814, A000217, A000096, A034856, A055998, A046691, A014132.

a[n_] := (v = IntegerExponent[n, 2]; (1/2)*(((1/2)*(n/2^v + 1) + v)^2 + (1/2)*(n/2^v + 1) - v)); Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Jan 15 2013, from 1st formula *)

f = open("result.csv", "w")
def A007814(n):
### author        Richard J. Mathar 2010-09-06 (Start)
### http://oeis.org/wiki/User:R._J._Mathar/oeisPy/oeisPy/oeis_bulk.py
        a = 0
        nshft = n
        while (nshft %2 == 0):
                a += 1
                nshft >>= 1
        return a
###(End)
for  n in range(1,10001):
     x = A007814(n)
     y = (n+2**x)/2**(x+1)
     m = ((x+y)**2-x+y)/2
     f.write('%d;%d;%d;%d;\n' % (n, x, y, m))
f.close()

a(n) = (((A003602)+A007814(n))^2 - A007814(n) + A003602(n))/2.

a(n) = ((x+y)^2-x+y)/2, where x = max {k: 2^k | n}, y = (n+2^x)/2^(x+1).

A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

Original entry on oeis.org

1, -4, 1, -4, 1, -5, 4, 1, -6, 8, 1, -7, 13, -4, 1, -8, 19, -12, 1, -9, 26, -25, 4, 1, -10, 34, -44, 16, 1, -11, 43, -70, 41, -4, 1, -12, 53, -104, 85, -20, 1, -13, 64, -147, 155, -61, 4, 1, -14, 76, -200, 259, -146, 24

Offset: 0

Tom Copeland, Jan 18 2016

			Row polynomials:
P(0,t) = 1 - 4t
P(1,t) = 1 - 4t = [-t(0) + (1-4t)] = -t(0) + P(0,t)
P(2,t) = 1 - 5t + 4t^2 = [-t(1-4t) + (1-4t)] = -t P(0,t) + P(1,t)
P(3,t) = 1 - 6t + 8t^2 = [-t(1-4t) + (1-5t+4t^2)] = -t P(1,t) + P(2,t)
P(4,t) = 1 - 7t + 13t^2 - 4t^3 = [-t(1-5t+4t^2) + (1-6t+8t^2)]
P(5,t) = 1 - 8t + 19t^2 - 12t^3 = [-t(1-6t+8t^2) + (1-7t+13t^2)]
P(6,t) = 1 - 9t + 26t^2 - 25t^3 + 4t^4
P(7,t) = 1 - 10t + 34t^2 - 44t^3 + 16t^4
P(8,t) = 1 - 11t + 43t^2 - 70t^3 + 41t^4 - 4t^5
P(9,t) = 1 - 12t + 53t^2 - 104t^3 + 85t^4 - 20t^5
P(10,t) = 1 - 13t + 64t^2 - 147t^3 + 155t^4 - 61t^5 + 4t^6
P(11,t) = 1 - 14t + 76t^2 - 200t^3 + 259t^4 - 146t^5 + 24t^6
...
Apparently: The odd rows for n>1 are reversed rows of A140882 (mod signs), and the even rows for n>0, the 9th, 15th, 21st, 27th, etc. rows of A228785 (mod signs). The diagonals are reverse rows of A202241.

Tom Copeland, Addendum to Elliptic Lie Triad

Cf. A000108, A000297, A008574, A008586, A011973, A022088 A034856, A084103, A098474, A124644, A140882, A202241, A228785.

p = (1 - 4 t) / (1 - x + t x^2) + O[x]^12 // CoefficientList[#, x] &;
CoefficientList[#, t] & /@ p // Flatten (* Andrey Zabolotskiy, Mar 07 2024 *)

O.g.f. G(x,t) = (1 - 4t)/(1 - x + t x^2) = a /  [t (x - (1+sqrt(a))/(2t))(x - (1-sqrt(a))/(2t))] with a = 1-4t.
Recursion P(n,t) = -t P(n-2,t) + P(n-1,t) with P(-1,t)=0 and P(0,t) = 1-4t.
Convolution of the Fibonacci polynomials of signed A011973 Fb(n,-t) with coefficients of (1-4t), e.g., (1-4t)Fb(5,-t) = (1-4t)(1-3t+t^2) = 1-7t+13t^2-4t^3, so, for n>=1 and k<=(n-1), T(n,k) = (-1)^k [-4*binomial(n-(k-1),k-1) - binomial(n-k,k)] with the convention that 1/(-m)! = 0 for m>=1, i.e., let binomial(n,k) = nint[n!/((k+c)!(n-k+c)!)] for c sufficiently small in magnitude.
Third column is A034856, and the fourth, A000297. Embedded in the coefficients of the highest order term of the polynomials is A008586 (cf. also A008574).
With P(0,t)=0, the o.g.f. is H(x,t) = (1-4t) x(1-tx)/[1-x(1-tx)] = (1-4t) Linv(Cinv(tx)/t), where Linv(x) = x/(1-x) with inverse L(x) = x/(1+x) and Cinv(x) = x (1-x) is the inverse of the o.g.f. of the shifted Catalan numbers A000108, C(x) = [1-sqrt(1-4x)]/2. Then Hinv(x,t) = C[t L(x/(1-4t))]/t = {1 - sqrt[1-4t(x/(1-4t))/[1+x/(1-4t)]]}/2t = {1-sqrt[1-4tx/(1-4t+x)]}/2t = 1/(1-4t) + (-1+t)/(1-4t)^2 x + (1-2t+2t^2)/(1-4t)^3 x^ + (-1+3t-6t^2+5t^3)/(1-4t)^4 + ..., where the numerators are the signed polynomials of A098474, reverse of A124644.
Row sums (t=1) are periodic -3,-3,0,3,3,0, repeat the six terms ... with o.g.f. -3 - 3x (1-x) / [1-x(1-x)]. Cf. A084103.
Unsigned row sums (t=-1) are shifted A022088 with o.g.f. 5 + 5x(1+x) / [x(1+x)].

Data corrected by Andrey Zabolotskiy, Mar 07 2024

A101494 Triangle, read by rows, where T(n,k) = Sum_{j=0..n-k-1} C(j+k,j)*T(n-1,j+k) for n>k>=0 with T(n,n)=1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 8, 4, 1, 1, 23, 23, 13, 5, 1, 1, 66, 73, 44, 19, 6, 1, 1, 210, 253, 162, 73, 26, 7, 1, 1, 733, 948, 643, 302, 111, 34, 8, 1, 1, 2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1, 11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1, 49864, 75295

Offset: 0

Paul D. Hanna, Jan 21 2005

Column 0 equals row sums (A026898) shift right.

T(n,k) is the number of m-tuples of nonnegative integers satisfying these two criteria: (i) there are exactly k 0’s, and (ii) the remaining m-k elements are positive integers less than or equal to n-m. - Mathew Englander, Feb 25 2021

			4th row sum = 23 = (5-0)^0+(5-1)^1+(5-2)^2+(5-3)^3+(5-4)^4.
5th row sum = 66 = (6-0)^0+(6-1)^1+(6-2)^2+(6-3)^3+(6-4)^4+(6-5)^5.
T(6,0) = 66 = 1*23 + 1*23 + 1*13 + 1*5 + 1*1 + 1*1.
T(6,1) = 73 = 1*23 + 2*13 + 3*5 + 4*1 + 5*1.
T(6,2) = 44 = 1*13 + 3*5 + 6*1 + 10*1.
Rows begin:
1;
1, 1;
2, 1, 1;
4, 3, 1, 1;
9, 8, 4, 1, 1;
23, 23, 13, 5, 1, 1;
66, 73, 44, 19, 6, 1, 1;
210, 253, 162, 73, 26, 7, 1, 1;
733, 948, 643, 302, 111, 34, 8, 1, 1;
2781, 3817, 2724, 1337, 506, 159, 43, 9, 1, 1;
11378, 16433, 12259, 6266, 2457, 788, 218, 53, 10, 1, 1;
49864, 75295, 58423, 30953, 12558, 4147, 1163, 289, 64, 11, 1, 1;
232769, 365600, 293902, 160823, 67259, 22878, 6574, 1647, 373, 76, 12, 1, 1; ...

Muniru A Asiru, Rows n=0..150 of triangle, flattened

Cf. A101495, A026898, A089246 (first differences by column), A304357 (antidiagonal sums, empirically), A034856 (fourth diagonal).

Flat(List([0..10],n->List([0..n],k->Sum([0..n-k],j->Binomial(j+k,j)*(n-k-j)^j)))); # Muniru A Asiru, Mar 07 2019

PARI
```
T(n,k)=if(n
				
```

T(n,k)=polcoeff(sum(m=0,n-k, x^m/(1-m*x +x*O(x^(n-k)))^(k+1)),n-k)
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print()) \\ Paul D. Hanna, Mar 06 2013

T(n,0) = A026898(n-1).
T(n,k) = Sum_{j=0..n-k} binomial(j+k,j)*(n-k-j)^j. - Vladeta Jovovic, Sep 07 2006
G.f.: A(x,y) = Sum_{n>=0} Sum_{k>=0} x^(n+k)*y^k / (1 - n*x)^(k+1). - Paul D. Hanna, Mar 06 2013
From Mathew Englander, Feb 25 2021: (Start)
G.f. of row n: Sum_{i=0..n} (x+n-i)^i.
T(n,k) = Sum_{j=k..n} A089246(j,k).
Antidiagonal sums: Sum_{j = 0..n} Sum_{i = j..floor((n+j)/2)} binomial(i,j)*(n+j-2*i)^j. (End)

A101881 Write two numbers, skip one, write two, skip two, write two, skip three ... and so on.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 13, 14, 19, 20, 26, 27, 34, 35, 43, 44, 53, 54, 64, 65, 76, 77, 89, 90, 103, 104, 118, 119, 134, 135, 151, 152, 169, 170, 188, 189, 208, 209, 229, 230, 251, 252, 274, 275, 298, 299, 323, 324, 349, 350, 376, 377, 404, 405, 433, 434, 463, 464, 494

Offset: 0

Candace Mills (scorpiocand(AT)yahoo.com), Dec 19 2004

Equals row sums of triangle A177994. - Gary W. Adamson, May 16 2010
From Ralf Stephan, Mar 09 2014: (Start)
Write the positive integers in a skewed triangle:
  1,  2;
  0,  3,  4,  5;
  0,  0,  6,  7,  8,  9;
  0,  0,  0, 10, 11, 12, 13, 14;
  ...
Sequence consists of the first number in each column. (End)
In a regular k-polygon draw lines connecting all the vertices. Select a triangle that tiles the polygon into k pieces. This triangle contains two adjacent polygon vertices. The third vertex is for even k the center of the polygon and for odd k one of the vertices of the central k-polygon (which is not included in the tiling). Count all lines connecting vertices in the original k-polygon that passes through the interior of the tiling triangle. That count is a(k-5). (See illustrations below.) - Lars Blomberg, Feb 20 2020
a(n) is the smallest number which has n+1 as a part in any of its maximally refined strict partitions. The first such are:(1),(2),(1,3),(1,4),(1,2,5),(1,2,6),(1,2,3,7),(1,2,3,8),(1,2,3,4,9) etc. - Sigurd Kittilsen, Oct 18 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lars Blomberg, Illustration for 14-polygon
Lars Blomberg, Illustration for 15-polygon
Rene Marczinzik, Finitistic Auslander algebras, arXiv:1701.00972 [math.RT], 2017. [Page 9, Conjecture]
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A101882, A101883, A177994.

Cf. A000096, A034856.

import Data.List (intersperse)
a101881 n = a101881_list !! n
a101881_list = scanl1 (+) $ intersperse 1 [1..]
-- Reinhard Zumkeller, Feb 20 2015

[(1/16)*(2*n^2+18*n+15+(2*n+1)*(-1)^n): n in [0..60]]; // Vincenzo Librandi, Mar 11 2014

CoefficientList[Series[(-1 + x^3 - x)/((x + 1)^2 (x - 1)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Mar 11 2014 *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,4,5,8},60] (* Harvey P. Dale, Dec 07 2016 *)
With[{nn=60},Take[#,2]&/@TakeList[Range[(nn^2+nn-6)/2],Range[3,nn]]]// Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2019 *)

Vec((-1+x^3-x)/((x+1)^2*(x-1)^3) + O(x^60)) \\ Iain Fox, Nov 17 2017

G.f.: (-1+x^3-x)/((x+1)^2*(x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
a(n) = (1/16)*(2*n^2 + 18*n + 15 + (2*n+1)*(-1)^n). - Ralf Stephan, Mar 09 2014
a(2*n) = A034856(n+1); a(2*n+1) = A000096(n+1). - Reinhard Zumkeller, Feb 20 2015
a(n) = n + 1 + A008805(n-2). - Wesley Ivan Hurt, Nov 17 2017
E.g.f.: (cosh(x) - sinh(x))*(1 - 2*x + (15 + 20*x + 2*x^2)*(cosh(2*x) + sinh(2*x)))/16. - Stefano Spezia, Feb 20 2020

A131818 A130296 + A002260 - A000012. Triangle read by rows: row n consists of n, 2, 3, 4, ..., n.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Jul 18 2007

Row sums = A034856; (1, 4, 8, 13, 19, 26, 34, ...).

			First few rows of the triangle:
  1;
  2, 2;
  3, 2, 3;
  4, 2, 3, 4;
  5, 2, 3, 4, 5;
  6, 2, 3, 4, 5, 6;
  7, 2, 3, 4, 5, 6, 7;
  ...

Cf. A130296, A002260, A034856.

Table[Join[{n},Range[2,n]],{n,15}]//Flatten (* Harvey P. Dale, Feb 24 2021 *)

t(n, k) = if (k==1, n, k); \\ Michel Marcus, Feb 12 2014

from math import isqrt, comb
def A131818(n):
    y = (m:=isqrt(k:=n-1<<1))+(k>m*(m+1))
    return n-comb(y,2) # Chai Wah Wu, Jul 07 2025

A130296 + A002260 - A000012 as infinite lower triangular matrices.

T(n, 1) = n, T(n, k) = k for k > 1. - Michel Marcus, Feb 12 2014

More terms from Michel Marcus, Feb 12 2014

A262221 a(n) = 25n(n + 1)/2 + 1.

Original entry on oeis.org

1, 26, 76, 151, 251, 376, 526, 701, 901, 1126, 1376, 1651, 1951, 2276, 2626, 3001, 3401, 3826, 4276, 4751, 5251, 5776, 6326, 6901, 7501, 8126, 8776, 9451, 10151, 10876, 11626, 12401, 13201, 14026, 14876, 15751, 16651, 17576, 18526, 19501, 20501, 21526, 22576, 23651

Offset: 0

Bruno Berselli, Sep 15 2015

Also centered 25-gonal (or icosipentagonal) numbers.
This is the case k=25 of the formula (k*n*(n+1) - (-1)^k + 1)/2. See table in Links section for similar sequences.
For k=2*n, the formula shown above gives A011379.
Primes in sequence: 151, 251, 701, 1951, 3001, 4751, 10151, 12401, ...

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 51 (23rd row of the table).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of numbers of the form (k*n*(n+1)-(-1)^k+1)/2.
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A008607 (first differences), A011379, A034856, A054254, A080956, A123296, A186349, A255184.
Cf. centered polygonal numbers listed in A069190.
Similar sequences of the form (k*n*(n+1) - (-1)^k + 1)/2 with -1 <= k <= 26: A000004, A000124, A002378, A005448, A005891, A028896, A033996, A035008, A046092, A049598, A060544, A064200, A069099, A069125, A069126, A069128, A069130, A069132, A069174, A069178, A080956, A124080, A163756, A163758, A163761, A164136, A173307.

Magma
```
[25*n*(n+1)/2+1: n in [0..50]];
```

Table[25 n (n + 1)/2 + 1, {n, 0, 50}]
25*Accumulate[Range[0,50]]+1 (* or *) LinearRecurrence[{3,-3,1},{1,26,76},50] (* Harvey P. Dale, Jan 29 2023 *)

PARI
```
vector(50, n, n--; 25*n*(n+1)/2+1)
```
Sage
```
[25*n*(n+1)/2+1 for n in (0..50)]
```

G.f.: (1 + 23*x + x^2)/(1 - x)^3.
a(n) = a(-n-1) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = A123296(n) + 1.
a(n) = A000217(5*n+2) - 2.
a(n) = A034856(5*n+1).
a(n) = A186349(10*n+1).
a(n) = A054254(5*n+2) with n>0, a(0)=1.
a(n) = A000217(n+1) + 23*A000217(n) + A000217(n-1) with A000217(-1)=0.
Sum_{i>=0} 1/a(i) = 1.078209111... = 2*Pi*tan(Pi*sqrt(17)/10)/(5*sqrt(17)).
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=0} a(n)/n! = 77*e/2.
Sum_{n>=0} (-1)^(n+1) * a(n)/n! = 23/(2*e). (End)
E.g.f.: exp(x)*(2 + 50*x + 25*x^2)/2. - Elmo R. Oliveira, Dec 24 2024

A188843 T(n,k) is the number of n X k binary arrays without the pattern 0 1 diagonally or vertically.

Original entry on oeis.org

2, 4, 3, 8, 8, 4, 16, 21, 13, 5, 32, 55, 40, 19, 6, 64, 144, 121, 66, 26, 7, 128, 377, 364, 221, 100, 34, 8, 256, 987, 1093, 728, 364, 143, 43, 9, 512, 2584, 3280, 2380, 1288, 560, 196, 53, 10, 1024, 6765, 9841, 7753, 4488, 2108, 820, 260, 64, 11, 2048, 17711, 29524

Offset: 1

R. H. Hardin, Apr 12 2011

Table starts
4   8   16   32    64    128    256     512     1024     2048      4096
8  21   55  144   377    987   2584    6765    17711    46368    121393
13  40  121  364  1093   3280   9841   29524    88573   265720    797161
19  66  221  728  2380   7753  25213   81927   266110   864201   2806272
26 100  364 1288  4488  15504  53296  182688   625184  2137408   7303360
34 143  560 2108  7752  28101 100947  360526  1282735  4552624  16131656
43 196  820 3264 12597  47652 177859  657800  2417416  8844448  32256553
53 260 1156 4845 19551  76912 297275 1134705  4292145 16128061  60304951
64 336 1581 6954 29260 119416 476905 1874730  7283640 28048800 107286661
76 425 2109 9709 42504 179630 740025 2991495 11920740 46981740 183579396

			Some solutions for 5 X 3:
  0 0 1    1 1 0    1 1 1    0 1 0    1 1 0    1 1 0    1 1 1
  0 0 0    1 0 0    1 1 0    0 0 0    1 1 0    1 1 0    1 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 0    1 1 0    0 1 1
  0 0 0    0 0 0    1 1 0    0 0 0    1 0 0    1 0 0    0 0 0
  0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0    0 0 0

R. H. Hardin, Table of n, a(n) for n = 1..1741

Diagonal is A143388.
Column 2 is A034856(n+1).
Column 3 is A137742(n+1).
Row 2 is A001906(n+1).
Row 3 is A003462(n+1).
Row 4 is A005021.
Row 5 is A005022.
Row 6 is A005023.
Row 7 is A005024.
Row 8 is A005025.

Row recurrence
Empirical: T(n,k) = Sum_{i=1..floor((n+2)/2)} binomial(n+2-i,i)*T(n,k-i)*(-1)^(i-1).
E.g.,
empirical: T(1,k) = 2*T(1,k-1),
empirical: T(2,k) = 3*T(2,k-1) -    T(2,k-2),
empirical: T(3,k) = 4*T(3,k-1) -  3*T(3,k-2),
empirical: T(4,k) = 5*T(4,k-1) -  6*T(4,k-2) +    T(4,k-3),
empirical: T(5,k) = 6*T(5,k-1) - 10*T(5,k-2) +  4*T(5,k-3),
empirical: T(6,k) = 7*T(6,k-1) - 15*T(6,k-2) + 10*T(6,k-3) -    T(6,k-4),
empirical: T(7,k) = 8*T(7,k-1) - 21*T(7,k-2) + 20*T(7,k-3) -  5*T(7,k-4),
empirical: T(8,k) = 9*T(8,k-1) - 28*T(8,k-2) + 35*T(8,k-3) - 15*T(8,k-4) + T(8,k-5).
Columns are polynomials for n > k-3.
Empirical: T(n,1) = n + 1.
Empirical: T(n,2) = (1/2)*n^2 + (5/2)*n + 1.
Empirical: T(n,3) = (1/6)*n^3 + 2*n^2 + (35/6)*n.
Empirical: T(n,4) = (1/24)*n^4 + (11/12)*n^3 + (155/24)*n^2 + (163/12)*n - 6 for n > 1.
Empirical: T(n,5) = (1/120)*n^5 + (7/24)*n^4 + (89/24)*n^3 + (473/24)*n^2 + (1877/60)*n - 33 for n > 2.
Empirical: T(n,6) = (1/720)*n^6 + (17/240)*n^5 + (203/144)*n^4 + (647/48)*n^3 + (2659/45)*n^2 + (1379/20)*n - 143 for n > 3.
Empirical: T(n,7) = (1/5040)*n^7 + (1/72)*n^6 + (143/360)*n^5 + (53/9)*n^4 + (33667/720)*n^3 + (12679/72)*n^2 + (9439/70)*n - 572 for n > 4.
Empirical: T(n,8) = (1/40320)*n^8 + (23/10080)*n^7 + (17/192)*n^6 + (269/144)*n^5 + (43949/1920)*n^4 + (228401/1440)*n^3 + (1054411/2016)*n^2 + (9941/56)*n - 2210 for n > 5.

cp's OEIS Frontend

A224038 T(n,k)=Number of nXk 0..1 arrays with antidiagonals unimodal and rows and diagonals nondecreasing.

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A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A027379 Expansion of (1+x^2-x^3)/(1-x)^3.

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A209268 Inverse permutation A054582.

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A267633 Expansion of (1 - 4t)/(1 - x + t x^2): a Fibonacci-type sequence of polynomials.

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A101494 Triangle, read by rows, where T(n,k) = Sum_{j=0..n-k-1} C(j+k,j)*T(n-1,j+k) for n>k>=0 with T(n,n)=1.

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A101881 Write two numbers, skip one, write two, skip two, write two, skip three ... and so on.

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A262221 a(n) = 25n(n + 1)/2 + 1.