A158842 -id:A158842 - cp's OEIS frontend

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

For pairs instead of triples we have A002024.
Positions of first appearances are A050407(n+2) = A000292(n)+1.
The zero-based version is A056556.
The triples have sums A070770.
The second instead of first part is A194848.
The third instead of first part is A333516.
Concatenating all the triples gives A360240.
Cf. A000217, A003056, A056557, A056558, A069905, A158842, A331195.

nn=9;First/@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import comb
from sympy import integer_nthroot
def A360010(n): return (m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3)) # Chai Wah Wu, Nov 04 2024

a(n) = A056556(n) + 1 = A331195(3n) + 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Feb 18 2024
a(n) = m+1 if n>binomial(m+2,3) and a(n) = m otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 04 2024

A158841 Triangle read by rows, matrix product of A145677 * A004736.

Original entry on oeis.org

1, 3, 1, 7, 4, 2, 13, 9, 6, 3, 21, 16, 12, 8, 4, 31, 25, 20, 15, 10, 5, 43, 36, 30, 24, 18, 12, 6, 57, 49, 42, 35, 28, 21, 14, 7, 73, 64, 56, 48, 40, 32, 24, 16, 8, 91, 81, 72, 63, 54, 45, 36, 27, 18, 9

Offset: 1

Gary W. Adamson and Roger L. Bagula, Mar 28 2009

			First few rows of the triangle:
    1;
    3,   1;
    7,   4,   2;
   13,   9,   6,   3;
   21,  16,  12,   8,   4;
   31,  25,  20,  15,  10,  5;
   43,  36,  30,  24,  18, 12,  6;
   57,  49,  42,  35,  28, 21, 14,  7;
   73,  64,  56,  48,  40, 32, 24, 16,  8;
   91,  81,  72,  63,  54, 45, 36, 27, 18,  9;
  111, 100,  90,  80,  70, 60, 50, 40, 30, 20, 10;
  133, 121, 110,  99,  88, 77, 66, 55, 44, 33, 22, 11;
  157, 144, 132, 120, 108, 96, 84, 72, 60, 48, 36, 24, 12;
  ...

Cf. A145677, A002061 (column k=1), A158842 (row sums).

A145677 := proc(n,k)
        if n <0 or k < 0 or k > n then
                0;
        elif k = 0 then
                1;
        elif k = n then
                n ;
        else
                0 ;
        end if;
end proc:
A004736 := proc(n,k)
        if n <0 or k < 1 or k > n then
                0;
        else
                n-k+1 ;
        end if;
end proc:
A158841 := proc(n,k)
        add( A145677(n-1,j-1)*A004736(j,k),j=k..n) ;
end proc: # R. J. Mathar, Nov 05 2011

T(n,k) = Sum_{j=k..n} A145677(n-1,j-1)*A004736(j,k), assuming column enumeration k >= 1 in A004736. - R. J. Mathar, Nov 05 2011

A237622 Interpolation polynomial through n points (0,1), (1,1), ..., (n-2,1) and (n-1,n) evaluated at 2n, a(0)=1.

Original entry on oeis.org

1, 1, 5, 31, 169, 841, 3961, 18019, 80081, 350065, 1511641, 6466461, 27457585, 115892401, 486748081, 2035917451, 8485840801, 35263382881, 146157442201, 604404010981, 2494365759601, 10275832148401, 42264944401681, 173588164506901, 712027089322849

Offset: 0

Alois P. Heinz, Feb 10 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A002061 (evaluated at n), A158842 (at n+1), A237664 (n+1 points).

a:= proc(n) option remember; `if`(n<3, [1, 1, 5][n+1],
       (n*(15*n^3-44*n^2+43*n-18) *a(n-1)
        -2*(n-1)*(2*n-3)*(3*n^2-n+2) *a(n-2))/
        ((n-2)*(n+1)*(3*n^2-7*n+6)))
    end:
seq(a(n), n=0..30);

a[n_] := Module[{m}, If[n == 0, 1, InterpolatingPolynomial[Table[{k, If[k == n-1, n, 1]}, {k, 0, n-1}], m] /. m -> 2n]];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020 *)

E.g.f.: exp(x)+2*exp(2*x)*(BesselI(1,2*x)*(x-1)+x*BesselI(0,2*x)).

a(n) ~ sqrt(n)*4^n/sqrt(Pi). - Vaclav Kotesovec, Feb 14 2014

A250352 Number of length 3 arrays x(i), i=1..3 with x(i) in i..i+n and no value appearing more than 2 times.

Original entry on oeis.org

8, 26, 62, 122, 212, 338, 506, 722, 992, 1322, 1718, 2186, 2732, 3362, 4082, 4898, 5816, 6842, 7982, 9242, 10628, 12146, 13802, 15602, 17552, 19658, 21926, 24362, 26972, 29762, 32738, 35906, 39272, 42842, 46622, 50618, 54836, 59282, 63962, 68882, 74048

Offset: 1

R. H. Hardin, Nov 19 2014

a(n) = (n+1)^3 - (n-1), where (n+1)^3 is the number of ways of selecting a triple from n+1 numbers in these subintervals, and there are n-1 of these triples, (3,3,3) up to (n-2,n-2,n-2), where all values are the same, which are discarded. - R. J. Mathar, Oct 09 2020

			Some solutions for n=6:
  2  0  1  2  6  4  0  1  0  0  2  4  6  2  4  0
  4  4  7  7  2  4  2  3  1  6  1  2  3  6  5  5
  6  4  7  2  4  7  8  5  3  6  4  7  5  8  8  2

R. H. Hardin, Table of n, a(n) for n = 1..210
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Row 3 of A250351.

a(n) = n^3 + 3*n^2 + 2*n + 2 = 2*A158842(n+1).
From Colin Barker, Nov 12 2018: (Start)
G.f.: 2*x*(4 - 3*x + 3*x^2 - x^3) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)

A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 3, 2, 2, 3, 3, 1, 3, 3, 2, 3, 3, 3, 4, 1, 1, 4, 2, 1, 4, 2, 2, 4, 3, 1, 4, 3, 2, 4, 3, 3, 4, 4, 1, 4, 4, 2, 4, 4, 3, 4, 4, 4, 5, 1, 1, 5, 2, 1, 5, 2, 2, 5, 3, 1, 5, 3, 2, 5, 3, 3, 5, 4, 1, 5, 4, 2, 5, 4, 3

Offset: 1

Gus Wiseman, Feb 11 2023

nonn

			Triples begin: (1,1,1), (2,1,1), (2,2,1), (2,2,2), (3,1,1), (3,2,1), (3,2,2), (3,3,1), (3,3,2), (3,3,3), ...

The triples have sums A070770.
Positions of first appearances are A158842.
For pairs instead of triples we have A330709 + 1.
The zero-based version is A331195.
- The first part is A360010 = A056556 + 1.
- The second part is A194848 = A056557 + 1.
- The third part is A333516 = A056558 + 1.
Cf. A002024, A003056, A069905.

nn=9;Join@@Select[Tuples[Range[nn],3],GreaterEqual@@#&]

from math import isqrt, comb
from sympy import integer_nthroot
def A360240(n): return (m:=integer_nthroot((n-1<<1)+6,3)[0])+(n>3*comb(m+2,3)) if (a:=n%3)==1 else (k:=isqrt(r:=(b:=(n-1)//3)+1-comb((m:=integer_nthroot((n-1<<1)-1,3)[0])-(b(k<<2)*(k+1)+1) if a==2 else 1+(r:=(b:=(n-1)//3)-comb((m:=integer_nthroot((n-1<<1)-3,3)[0])+(b>=comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2) # Chai Wah Wu, Jun 07 2025

a(n) = A331195(n-1) + 1.

A072277 Smallest integer > 1 which is both n-gonal and centered n-gonal.

Original entry on oeis.org

10, 25, 51, 91, 148, 225, 325, 451, 606, 793, 1015, 1275, 1576, 1921, 2313, 2755, 3250, 3801, 4411, 5083, 5820, 6625, 7501, 8451, 9478, 10585, 11775, 13051, 14416, 15873, 17425, 19075, 20826, 22681, 24643, 26715, 28900, 31201, 33621, 36163

Offset: 3

David W. Wilson, Jul 09 2002

a(n) is the (n-1)-th centered n-gonal number. The n-th centered n-gonal number is A100119(n) and the (n+1)-th centered n-gonal number is A158842(n). - Mohammed Yaseen, Jun 06 2021

			a(4) = 25 is both square and centered square.

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

Cf. A100119, A158842, A162607.

LinearRecurrence[{4,-6,4,-1},{10,25,51,91},50] (* or *) Table[(n^3-n^2+ 2)/2,{n,3,50}] (* Harvey P. Dale, Aug 19 2011 *)

a(n) = (n^3 - n^2 + 2)/2.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), with a(3)=10, a(4)=25, a(5)=51, a(6)=91. - Harvey P. Dale, Aug 19 2011
G.f.: x^3*(-3*x^3 + 11*x^2 - 15*x + 10)/(x-1)^4. - Harvey P. Dale, Aug 19 2011

A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 6, 7, 7, 1, 5, 8, 10, 11, 11, 1, 6, 10, 13, 15, 16, 16, 1, 7, 12, 16, 19, 21, 22, 22, 1, 8, 14, 19, 23, 26, 28, 29, 29, 1, 9, 16, 22, 27, 31, 34, 36, 37, 37, 1, 10, 18, 25, 31, 36, 40, 43, 45, 46, 46, 1, 11, 20, 28, 35, 41

Offset: 0

Franck Maminirina Ramaharo, Apr 20 2018

Columns are linear recurrence sequences with signature (3,-3,1).
8*T(n,k) + A166147(k-1) are squares.
Columns k are binomial transforms of [1, k, 1, 0, 0, 0, ...].
Antidiagonals sums yield A116731.

			The array T(n,k) begins
1    1    1    1    1    1    1    1    1    1    1    1    1  ...  A000012
1    2    3    4    5    6    7    8    9   10   11   12   13  ...  A000027
2    4    6    8   10   12   14   16   18   20   22   24   26  ...  A005843
4    7   10   13   16   19   22   25   28   31   34   37   40  ...  A016777
7   11   15   19   23   27   31   35   39   43   47   51   55  ...  A004767
11  16   21   26   31   36   41   46   51   56   61   66   71  ...  A016861
16  22   28   34   40   46   52   58   64   70   76   82   88  ...  A016957
22  29   36   43   50   57   64   71   78   85   92   99  106  ...  A016993
29  37   45   53   61   69   77   85   93  101  109  117  125  ...  A004770
37  46   55   64   73   82   91  100  109  118  127  136  145  ...  A017173
46  56   66   76   86   96  106  116  126  136  146  156  166  ...  A017341
56  67   78   89  100  111  122  133  144  155  166  177  188  ...  A017401
67  79   91  103  115  127  139  151  163  175  187  199  211  ...  A017605
79  92  105  118  131  144  157  170  183  196  209  222  235  ...  A190991
...
The inverse binomial transforms of the columns are
1    1    1    1    1    1    1    1    1    1    1    1    1  ...
0    1    2    3    4    5    6    7    8    9   10   11   12  ...
1    1    1    1    1    1    1    1    1    1    1    1    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  ...
0    0    0    0    0    0    0    0    0    0    0    0    0  ...
...
T(k,n-k) = A087401(n,k) + 1 as triangle
1
1   1
1   2   2
1   3   4   4
1   4   6   7   7
1   5   8  10  11  11
1   6  10  13  15  16  16
1   7  12  16  19  21  22  22
1   8  14  19  23  26  28  29  29
1   9  16  22  27  31  34  36  37  37
1  10  18  25  31  36  40  43  45  46  46
...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

Cf. A085475, A086270, A086271, A086272, A086273, A130154, A159798, A162609, A162610, A300401.

T := (n, k) -> binomial(n, 2) + k*n + 1;
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

Table[With[{n = m - k}, Binomial[n, 2] + k n + 1], {m, 0, 11}, {k, m, 0, -1}] // Flatten (* Michael De Vlieger, Apr 21 2018 *)

T(n, k) := binomial(n, 2)+ k*n + 1$
for n:0 thru 20 do
    print(makelist(T(n, k), k, 0, 20));

T(n,k) = binomial(n, 2) + k*n + 1;
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 17 2018

G.f.: (3*x^2*y - 3*x*y + y - 2*x^2 + 2*x - 1)/((x - 1)^3*(y - 1)^2).
E.g.f.: (1/2)*(2*x*y + x^2 + 2)*exp(y + x).
T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k), with T(0,k) = 1, T(1,k) = k + 1 and T(2,k) = 2*k + 2.
T(n,k) = T(n-1,k) + n + k - 1.
T(n,k) = T(n,k-1) + n, with T(n,0) = 1.
T(n,0) = A152947(n+1).
T(n,1) = A000124(n).
T(n,2) = A000217(n).
T(n,3) = A034856(n+1).
T(n,4) = A052905(n).
T(n,5) = A051936(n+4).
T(n,6) = A246172(n+1).
T(n,7) = A302537(n).
T(n,8) = A056121(n+1) + 1.
T(n,9) = A056126(n+1) + 1.
T(n,10) = A051942(n+10) + 1, n > 0.
T(n,11) = A101859(n) + 1.
T(n,12) = A132754(n+1) + 1.
T(n,13) = A132755(n+1) + 1.
T(n,14) = A132756(n+1) + 1.
T(n,15) = A132757(n+1) + 1.
T(n,16) = A132758(n+1) + 1.
T(n,17) = A212427(n+1) + 1.
T(n,18) = A212428(n+1) + 1.
T(n,n) = A143689(n) = A300192(n,2).
T(n,n+1) = A104249(n).
T(n,n+2) = T(n+1,n) = A005448(n+1).
T(n,n+3) = A000326(n+1).
T(n,n+4) = A095794(n+1).
T(n,n+5) = A133694(n+1).
T(n+2,n) = A005449(n+1).
T(n+3,n) = A115067(n+2).
T(n+4,n) = A133694(n+2).
T(2*n,n) = A054556(n+1).
T(2*n,n+1) = A054567(n+1).
T(2*n,n+2) = A033951(n).
T(2*n,n+3) = A001107(n+1).
T(2*n,n+4) = A186353(4*n+1) (conjectured).
T(2*n,n+5) = A184103(8*n+1) (conjectured).
T(2*n,n+6) = A250657(n-1) = A250656(3,n-1), n > 1.
T(n,2*n) = A140066(n+1).
T(n+1,2*n) = A005891(n).
T(n+2,2*n) = A249013(5*n+4) (conjectured).
T(n+3,2*n) = A186384(5*n+3) = A186386(5*n+3) (conjectured).
T(2*n,2*n) = A143689(2*n).
T(2*n+1,2*n+1) = A143689(2*n+1) (= A030503(3*n+3) (conjectured)).
T(2*n,2*n+1) = A104249(2*n) = A093918(2*n+2) = A131355(4*n+1) (= A030503(3*n+5) (conjectured)).
T(2*n+1,2*n) = A085473(n).
a(n+1,5*n+1)=A051865(n+1) + 1.
a(n,2*n+1) = A116668(n).
a(2*n+1,n) = A054569(n+1).
T(3*n,n) = A025742(3*n-1), n > 1 (conjectured).
T(n,3*n) = A140063(n+1).
T(n+1,3*n) = A069099(n+1).
T(n,4*n) = A276819(n).
T(4*n,n) = A154106(n-1), n > 0.
T(2^n,2) = A028401(n+2).
T(1,n)*T(n,1) = A006000(n).
T(n*(n+1),n) = A211905(n+1), n > 0 (conjectured).
T(n*(n+1)+1,n) = A294259(n+1).
T(n,n^2+1) = A081423(n).
T(n,A000217(n)) = A158842(n), n > 0.
T(n,A152947(n+1)) = A060354(n+1).
floor(T(n,n/2)) = A267682(n) (conjectured).
floor(T(n,n/3)) = A025742(n-1), n > 0 (conjectured).
floor(T(n,n/4)) = A263807(n-1), n > 0 (conjectured).
ceiling(T(n,2^n)/n) = A134522(n), n > 0 (conjectured).
ceiling(T(n,n/2+n)/n) = A051755(n+1) (conjectured).
floor(T(n,n)/n) = A133223(n), n > 0 (conjectured).
ceiling(T(n,n)/n) = A007494(n), n > 0.
ceiling(T(n,n^2)/n) = A171769(n), n > 0.
ceiling(T(2*n,n^2)/n) = A046092(n), n > 0.
ceiling(T(2*n,2^n)/n) = A131520(n+2), n > 0.

cp's OEIS Frontend

A360010 First part of the n-th weakly decreasing triple of positive integers sorted lexicographically. Each n > 0 is repeated A000217(n) times.

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A158841 Triangle read by rows, matrix product of A145677 * A004736.

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A237622 Interpolation polynomial through n points (0,1), (1,1), ..., (n-2,1) and (n-1,n) evaluated at 2n, a(0)=1.

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A250352 Number of length 3 arrays x(i), i=1..3 with x(i) in i..i+n and no value appearing more than 2 times.

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A360240 Weakly decreasing triples of positive integers sorted lexicographically and concatenated.

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A072277 Smallest integer > 1 which is both n-gonal and centered n-gonal.

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A303273 Array T(n,k) = binomial(n, 2) + k*n + 1 read by antidiagonals.

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