A167499 -id:A167499 - cp's OEIS frontend

A001477 The nonnegative integers.

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, 74, 75, 76, 77

Offset: 0

N. J. A. Sloane

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

			Triangular view:
   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

Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

N. J. A. Sloane, Table of n, a(n) for n = 0..500000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
David Corneth, Counting to 13999 visualized | showing changes per digit, YouTube video, 2019.
Hans Havermann, Table giving n and American English name for n, for 0 <= n <= 100999, without spaces or hyphens
Hans Havermann, American English number names to one million, without spaces or hyphens
The IMO Compendium, Problem 6, 19th IMO 1977.
Tanya Khovanova, Recursive Sequences
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 12.
Eric Weisstein's World of Mathematics, Natural Number
Eric Weisstein's World of Mathematics, Nonnegative Integer
Index entries for "core" sequences
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).
Index to sequences related to Olympiads.

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
Cf. A000290.
Cf. A061579 (transposed matrix / reversed triangle).

a001477 = id
a001477_list = [0..]  -- Reinhard Zumkeller, May 07 2012

print([n for n in 0:280]) # Paul Muljadi, Apr 15 2024

Magma
```
[ n : n in [0..100]];
```
Maple
```
[ seq(n,n=0..100) ];
```

Table[n, {n, 0, 100}] (* Stefan Steinerberger, Apr 08 2006 *)
LinearRecurrence[{2, -1}, {0, 1}, 77] (* Robert G. Wilson v, May 23 2013 *)
CoefficientList[ Series[x/(x - 1)^2, {x, 0, 76}], x] (* Robert G. Wilson v, May 23 2013 *)
Range[0,100] (* Harvey P. Dale, Dec 29 2024 *)

PARI
```
A001477(n)=n /* first term is a(0) */
```

def a(n): return n
print([a(n) for n in range(78)]) # Michael S. Branicky, Nov 13 2022

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

A279967 Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.

Original entry on oeis.org

1, 1, 2, 2, 2, 7, 2, 9, 10, 15, 2, 10, 1, 13, 17, 8, 0, 13, 1, 14, 9, 8, 0, 13, 3, 30, 13, 10, 2, 16, 1, 23, 5, 7, 14, 15, 2, 8, 28, 32, 2, 23, 2, 9, 49, 12, 0, 48, 2, 11, 1, 20, 3, 18, 13, 28, 0, 4, 1, 56, 5, 8, 16, 35, 46, 4, 2, 6, 2, 10

Offset: 1

Alec Jones, Dec 24 2016

From Hartmut F. W. Hoft, Jan 23 2017: (Start)
Shown by induction and direct (modular) computations for
column 1: Every number is even, except for the first two 1's; in addition to row 3, value 2 occurs in rows 4*k and 4*k+1, and every value in rows 4*k+2 and 4*k+3 is divisible by 4, for all k>=1.
column 2: The first four entries, 2, 2, 9 and 10, contain the only odd number; no nonzero entry in row k>3 has 9 as a factor, and value 0 occurs in rows 4*k+1 and 4*k+2, for all k>=1.
Conjecture:
a({1, 6, 8, 9, 10, 15, 26, 45, 48, 84, 96, 112, 115, 252, 336, 343}) =
  {1, 7, 9,10, 15, 17, 30, 49, 48,104,117, 115, 122, 257, 343, 395} are the only numbers in the sequence with the property a(n) >= n (verified through n=500500, i.e., the triangle with 1000 antidiagonals).
This conjecture together with Bouniakowsky's conjecture that certain quadratic integer polynomials generate infinitely many primes (e.g. see A002496 for n^2+1 and A188382 for 2*n^2+n+1) implies that in every column in the triangle infinitely many prime sequence indices occur and therefore infinitely many 0's whenever the column contains no 1's. The proof is based on the fact that for a large enough prime sequence index p in whose prior column no 1 occurs then a(p)=0; therefore infinitely many 0's occur in that column. Obviously, once value 1 occurs in a column no 0 value can occur in a subsequent row.
Conjecture:
Every row in the triangle contains exactly two 1's.
(End)

			After 6 terms, the array looks like:
.
1   2   7
1   2
2
We have a(6) = 7 because a(1) = 1, a(3) = 2, a(4) = 2, and a(5) = 2 divide 6; 1 + 2 + 2 + 2 = 7.
From _Hartmut F. W. Hoft_, Jan 23 2017: (Start)
1   2   7  15  17   9  10  15  49  13   4  31  22
1   2  10  13  14  13  14   9  18  46  12  66
2   9   1   1  30   7   2   3  35  12   3
2  10  13   3   5  23  20  16  14  17
2   0  13  23   2   1   8  11   2
8   0   1  32  11   5   3   6
8  16  28   2  56  42   8
2   8  48   1   2 104
2   0   4  10   1
12   0   2  10
28   6   2
2  42
2
.
Expanded the triangle to the first 13 antidiagonals of the array, i.e. a(1) ... a(91), to show the start of the 2- and 0-value patterns in columns 1 and 2. The first 0 beyond column 2 is a(677) in row 27, column 11 of the triangle.
A188382(n)=2*n^2+n+1 for n>=0 are the alternate sequence indices for column 1 starting in row 1, 2*n^2+n+2 for n>=1 are the alternate sequence indices for column 2 starting in row 2, and 2*n^2+n+11 for n>=5 are the alternate sequence indices for column 11 starting in row 1.
The sequence indices in the triangle for row positions k>=1 in columns 1,..., 5 are given in sequences A000124(k), A152948(k+3), A152950(k+3), A145018(k+4) and A167499(k+4).
(End)

Peter Kagey, Table of n, a(n) for n = 1..5000

Cf. A279966 for the related sequence which counts prior terms.
Cf. A269347 for a one-dimensional version of this sequence.
Cf. also A279211, A279212.
Cf. A000124, A002496, A145018, A152948, A152950, A167499, A188382.

(*  printing of the triangle is commented out of function a279967[]  *)
pCol[{i_, j_}] := Map[{#, j}&, Range[1, i-1]]
pDiag[{i_, j_}] := If[j>=i, Map[{#, j-i+#}&, Range[1, i-1]], Map[{i-j+#, #}&, Range[1, j-1]]]
pRow[{i_, j_}] := Map[{i, #}&, Range[1, j-1]]
pAdiag[{i_, j_}] := Map[{i+j-#, #}&, Range[1, j-1]]
priorPos[{i_, j_}] := Join[pCol[{i, j}], pDiag[{i, j}], pRow[{i, j}], pAdiag[{i, j}]]
seqPos[{i_, j_}] := (i+j-2)(i+j-1)/2+j
antiDiag[k_] := Map[{k+1-#, #}&, Range[1, k]]
upperTriangle[k_] := Flatten[Map[antiDiag, Range[1, k]], 1]
a279967[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={1}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Apply[Plus, Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; AppendTo[seqL, val]; ms[[Apply[Sequence, pos]]]=val]; (* Print[TableForm[ms]]; *) seqL]
a279967[13] (* values in first 13 antidiagonals *)
(* Hartmut F. W. Hoft, Jan 23 2017 *)

A167487 a(n) = n*(n + 3)/2 + 8.

Original entry on oeis.org

8, 10, 13, 17, 22, 28, 35, 43, 52, 62, 73, 85, 98, 112, 127, 143, 160, 178, 197, 217, 238, 260, 283, 307, 332, 358, 385, 413, 442, 472, 503, 535, 568, 602, 637, 673, 710, 748, 787, 827, 868, 910, 953, 997, 1042, 1088, 1135, 1183, 1232, 1282, 1333, 1385, 1438, 1492

Offset: 0

Vincenzo Librandi, Nov 07 2009

2*a(i) + 3 is prime for i = 0..14. - Vincenzo Librandi, Jun 01 2014

Numbers m >= 8 such that 8*m - 55 is a square. - Bruce J. Nicholson, Jul 26 2017

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

Cf. A167499.

[n*(n+3)/2+8: n in [0..60]]; // Vincenzo Librandi, Sep 16 2013

Table[n (n + 3)/2 + 8, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
CoefficientList[Series[(8 - 14 x + 7 x^2) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Sep 16 2013 *)
LinearRecurrence[{3,-3,1},{8,10,13},60] (* Harvey P. Dale, Jul 05 2020 *)

a(n)=n*(n+3)/2+8 \\ Charles R Greathouse IV, Jun 16 2017

a(n) = n + a(n-1) + 1 with n > 1, a(1)=10.
G.f.: (8 - 14*x + 7*x^2)/(1 - x)^3. - Vincenzo Librandi, Sep 16 2013
a(n) = Sum_{i=n-5..n+7} i*(i+1)/26. - Bruno Berselli, Oct 20 2016
Sum_{n>=0} 1/a(n) = -1/7 + 2*Pi*tanh(sqrt(55)*Pi/2)/sqrt(55). - Amiram Eldar, Dec 13 2022
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(8 + 2*x + x^2/2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

Original entry on oeis.org

0, 1, 4, 3, 7, 12, 6, 11, 17, 24, 10, 16, 23, 31, 40, 15, 22, 30, 39, 49, 60, 21, 29, 38, 48, 59, 71, 84, 28, 37, 47, 58, 70, 83, 97, 112, 36, 46, 57, 69, 82, 96, 111, 127, 144, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 55, 67, 80, 94, 109, 125, 142, 160, 179, 199, 220

Offset: 0

Reinhard Zumkeller, Jul 17 2014

			First rows and their row sums (A245301):
   0                                                                  0;
   1,  4                                                              5;
   3,  7,  12                                                        22;
   6, 11,  17,  24                                                   58;
  10, 16,  23,  31,  40                                             120;
  15, 22,  30,  39,  49,  60                                        215;
  21, 29,  38,  48,  59,  71,  84                                   350;
  28, 37,  47,  58,  70,  83,  97, 112                              532;
  36, 46,  57,  69,  82,  96, 111, 127, 144                         768;
  45, 56,  68,  81,  95, 110, 126, 143, 161, 180                   1065;
  55, 67,  80,  94, 109, 125, 142, 160, 179, 199, 220              1430;
  66, 79,  93, 108, 124, 141, 159, 178, 198, 219, 241, 264         1870;
  78, 92, 107, 123, 140, 158, 177, 197, 218, 240, 263, 287, 312    2392.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A000217, A046092, A062725, A245301.

a245300 n k = (n + k) * (n + k + 1) `div` 2 + k
a245300_row n = map (a245300 n) [0..n]
a245300_tabl = map a245300_row [0..]
a245300_list = concat a245300_tabl

[k + Binomial(n+k+1,2): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 01 2021

Table[k + Binomial[n+k+1,2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 01 2021 *)

flatten([[k + binomial(n+k+1,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 01 2021

T(n, 0) = A000217(n).
T(n, n) = A046092(n).
T(2*n, n) = A062725(n) (central terms).
Sum_{k=0..n} T(n, k) = A245301(n).
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000124(n+1) = A134869(n+1), n >= 1.
T(n, 2) = A152948(n+4), n >= 2.
T(n, 3) = A152950(n+4), n >= 3.
T(n, 4) = A145018(n+5), n >= 4.
T(n, 5) = A167499(n+4), n >= 5.
T(n, 6) = A166136(n+5), n >= 6.
T(n, 7) = A167487(n+6), n >= 7.
T(n, n-1) = A056220(n), n >= 1.
T(n, n-2) = A142463(n-1), n >= 2.
T(n, n-3) = A054000(n-1), n >= 3.
T(n, n-4) = A090288(n-3), n >= 4.
T(n, n-5) = A268581(n-4), n >= 5.
T(n, n-6) = A059993(n-4), n >= 6.
T(n, n-7) = (-1)*A147973(n), n >= 7.
T(n, n-8) = A139570(n-5), n >= 8.
T(n, n-9) = A271625(n-5), n >= 9.
T(n, n-10) = A222182(n-4), n >= 10.
T(2*n, n-1) = A081270(n-1), n >= 1.
T(2*n, n+1) = A117625(n+1), n >= 1. (End)

A187710 a(n) = n^2 + n + 10.

Original entry on oeis.org

10, 12, 16, 22, 30, 40, 52, 66, 82, 100, 120, 142, 166, 192, 220, 250, 282, 316, 352, 390, 430, 472, 516, 562, 610, 660, 712, 766, 822, 880, 940, 1002, 1066, 1132, 1200, 1270, 1342, 1416, 1492, 1570, 1650, 1732, 1816, 1902, 1990, 2080, 2172, 2266, 2362, 2460

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 12 2011

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002061, A002378, A167499.

[n^2 + n + 10: n in [0..50]]; // G. C. Greubel, Nov 06 2018

f[n_] := n^2 + n + 10; f[Range[0, 100]]
LinearRecurrence[{3, -3, 1}, {10, 12, 16}, 50] (* Harvey P. Dale, Jan 18 2014 *)

a(n)=n^2+n+10 \\ Charles R Greathouse IV, Jun 17 2017

a(0)=10, a(1)=12, a(2)=16; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2014
From Bruno Berselli, Oct 20 2016: (Start)
G.f.: 2*(5 - 9*x + 5*x^2)/(1 - x)^3.
a(n) = 2*A167499(n-1) for n>0.
a(n) = Sum_{i=n-5..n+5} i*(i+1)/11. (End)
E.g.f.: (x^2 + 2*x + 10)*exp(x). - G. C. Greubel, Nov 06 2018
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*sqrt(39)/2)/sqrt(39). - Amiram Eldar, Jan 17 2021

Offset changed to 0 from Bruno Berselli, Oct 20 2016

A278436 Numbers such that A279966(n) = 0.

Original entry on oeis.org

17, 23, 47, 57, 93, 107, 173, 233, 353, 437, 467, 563, 677, 743, 817, 829, 851, 863, 955, 1037, 1187, 1213, 1277, 1387, 1433, 1487, 1549, 2089, 2147, 2213, 2287, 2293, 2417, 2473, 2689, 2777, 2911, 3083, 3323, 3391, 6691, 9337, 22969, 38557, 47347, 75391, 104999, 130927, 146719

Offset: 1

Alec Jones, Dec 24 2016

nonn

Not all numbers in this list are prime; nonprime elements include 57, 93, 437, 817, 851, 955, 1037, 1387, 2147.

This sequence is neither a subset nor a superset of sequence A281533 since 155 and 817 are the first numbers in one, but not the other, respectively. - Hartmut F. W. Hoft, Jan 23 2017

			Number 817 = 19*43, equivalent to array position (4, 37), is in the sequence since none of the numbers in the prior column, diagonal, row and antidiagonal contain the counts of 1, 19, 43 and 817. - _Hartmut F. W. Hoft_, Jan 23 2017

Cf. A279966, A000124, A145018, A152948, A152950, A167499, A279967.

(* support functions are in A279967 *)
a278436[k_] := Module[{ut=upperTriangle[k], ms=Table[" ", {i, 1, k}, {j, 1, k}], h, pos, val, seqL={}}, ms[[1, 1]]=1; For[h=2, h<=Length[ut], h++, pos=ut[[h]]; val=Length[Select[Map[ms[[Apply[Sequence, #]]]&, priorPos[pos]], #!=0 && Mod[seqPos[pos], #]==0&]]; If[val==0, AppendTo[seqL, h]]; ms[[Apply[Sequence, pos]]]=val]; seqL]
a278436[100] (* data through 3391. - Hartmut F. W. Hoft, Jan 23 2017 *)

A288937 Positions of 0 in A288936; complement of A288938.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 15, 20, 26, 33, 41, 50, 60, 71, 83, 96, 110, 125, 141, 158, 176, 195, 215, 236, 258, 281, 305, 330, 356, 383, 411, 440, 470, 501, 533, 566, 600, 635, 671, 708, 746, 785, 825, 866, 908, 951, 995, 1040, 1086, 1133, 1181, 1230, 1280, 1331

Offset: 1

Clark Kimberling, Jun 28 2017

The sequence (a(n)/n) diverges.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A167499, A288936, A288938.

z = 11; (* number of iterates *)
s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "011", "10" -> "011"}]
TableForm[Table[w[n], {n, 0, 10}]]
st = ToCharacterCode[w[z]] - 48   (* A288936 *)
Flatten[Position[st, 0]]  (* A288937 *)
Flatten[Position[st, 1]]  (* A288938 *)

A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 11, 13, 15, 17, 16, 18, 20, 22, 24, 22, 24, 26, 28, 30, 32, 29, 31, 33, 35, 37, 39, 41, 37, 39, 41, 43, 45, 47, 49, 51, 46, 48, 50, 52, 54, 56, 58, 60, 62, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87

Offset: 1

Torlach Rush, Aug 25 2022

The first column of the triangle is the Lazy Caterer's sequence A000124.
Each subsequent column starts with A000124(n) + (2 * (n-1)).
The first downward diagonal is A046691(n).
Columns and downward diagonals of the triangle identify many sequences (possibly shifted) in the database. Examples can be found in crossrefs below.
The sum of the n-th upward diagonal of the triangle is A356288(n).

			Triangle T(n,k) begins:
  n\k   1   2   3   4   5   6   7   8   9  10  11  ...
   1:   2
   2:   4   6
   3:   7   9  11
   4:  11  13  15  17
   5:  16  18  20  22  24
   6:  22  24  26  28  30  32
   7:  29  31  33  35  37  39  41
   8:  37  39  41  43  45  47  49  51
   9:  46  48  50  52  54  56  58  60  62
  10:  56  58  60  62  64  66  68  70  72  74
  11:  67  69  71  73  75  77  79  81  83  85  87
  ...

Cf. A000124, A004120, A046691, A051938, A055999, A056000, A155212, A167487, A167499, A167614, A246172, A334563, A356288.

Table[((n-1)(n+2))/2+2k,{n,20},{k,n}]//Flatten (* Harvey P. Dale, May 26 2023 *)

def T(n, k): return ((n-1) * (n+2))//2 + 2*k
for n in range(1, 12):
    for k in range(1,(n+1)): print(T(n,k), end = ', ')

# Indexed as a linear sequence.
def a000124(n): return n*(n+1)//2 + 1
def a(n):
    l = m = 0
    for k in range(1,n):
        lc = a000124(k - 1)
        if n >= lc:
            l = lc
            m = k
        else: break
    return n + m + (n - l)

T(n,k) = ((n-1) * (n+2))/2 + 2*k.

T(n,k+1) = T(n,k) + 2, k < n.

cp's OEIS Frontend

A001477 The nonnegative integers.

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A279967 Square array read by antidiagonals upwards in which each term is the sum of prior elements in the same row, column, diagonal, or antidiagonal that divide n; the array is seeded with an initial value a(1)=1.

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A167487 a(n) = n*(n + 3)/2 + 8.

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A245300 Triangle T(n,k) = (n+k)*(n+k+1)/2 + k, 0 <= k <= n, read by rows.

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A187710 a(n) = n^2 + n + 10.

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A278436 Numbers such that A279966(n) = 0.

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A288937 Positions of 0 in A288936; complement of A288938.

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A356754 Triangle read by rows: T(n,k) = ((n-1)(n+2))/2 + 2k.