A100041 -id:A100041 - cp's OEIS frontend

A014105 Second hexagonal numbers: a(n) = n(2n + 1).

0, 3, 10, 21, 36, 55, 78, 105, 136, 171, 210, 253, 300, 351, 406, 465, 528, 595, 666, 741, 820, 903, 990, 1081, 1176, 1275, 1378, 1485, 1596, 1711, 1830, 1953, 2080, 2211, 2346, 2485, 2628, 2775, 2926, 3081, 3240, 3403, 3570, 3741, 3916, 4095, 4278

Offset: 0

N. J. A. Sloane, Jun 14 1998

Note that when starting from a(n)^2, equality holds between series of first n+1 and next n consecutive squares: a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2*n)^2; e.g., 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Henry Bottomley, Jan 22 2001; with typos fixed by Zak Seidov, Sep 10 2015
a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002
Partial sums of odd numbers 3 mod 4, that is, 3, 3+7, 3+7+11, ... See A001107. - Jon Perry, Dec 18 2004
If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007
More generally (see the first comment), for n > 0, let b(n,k) = a(n) + k*(4*n + 1). Then b(n,k)^2 + (b(n,k) + 1)^2 + ... + (b(n,k) + n)^2 = (b(n,k) + n + 1 + 2*k)^2 + ... + (b(n,k) + 2*n + 2*k)^2 + k^2; e.g., if n = 3 and k = 2, then b(n,k) = 47 and 47^2 + ... + 50^2 = 55^2 + ... + 57^2 + 2^2. - Charlie Marion, Jan 01 2011
Sequence found by reading the line from 0, in the direction 0, 10, ..., and the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Nov 09 2011
a(n) is the number of positions of a domino in a pyramidal board with base 2n+1. - César Eliud Lozada, Sep 26 2012
Differences of row sums of two consecutive rows of triangle A120070, i.e., first differences of A016061. - J. M. Bergot, Jun 14 2013 [In other words, the partial sums of this sequence give A016061. - Leo Tavares, Nov 23 2021]
a(n)*Pi is the total length of half circle spiral after n rotations. See illustration in links. - Kival Ngaokrajang, Nov 05 2013
For corresponding sums in first comment by Henry Bottomley, see A059255. - Zak Seidov, Sep 10 2015
a(n) also gives the dimension of the simple Lie algebras B_n (n >= 2) and C_n (n >= 3). - Wolfdieter Lang, Oct 21 2015
With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for unsigned A130757, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 13 2015
Partial sums of squares with alternating signs, ending in an even term: a(n) = 0^2 - 1^2 +- ... + (2*n)^2, cf. Example & Formula from Berselli, 2013. - M. F. Hasler, Jul 03 2018
Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley, n > 0. (Cf. A237593.) - Omar E. Pol, Aug 28 2018
a(n) is the area of a triangle with vertices at (0,0), (2*n+1, 2*n), and ((2*n+1)^2, 4*n^2). - Art Baker, Dec 12 2018
This sequence is the largest subsequence of A000217 such that gcd(a(n), 2*n) = a(n) mod (2*n) = n, n > 0 up to a given value of n. It is the interleave of A033585 (a(n) is even) and A033567 (a(n) is odd). - Torlach Rush, Sep 09 2019
A generalization of Hasler's Comment (Jul 03 2018) follows. Let P(k,n) be the n-th k-gonal number. Then for k > 1, partial sums of {P(k,n)} with alternating signs, ending in an even term, = n*((k-2)*n + 1). - Charlie Marion, Mar 02 2021
Let U_n(H) = {A in M_n(H): A*A^H = I_n} be the group of n X n unitary matrices over the quaternions (A^H is the conjugate transpose of A. Note that over the quaternions we still have A*A^H = I_n <=> A^H*A = I_n by mapping A and A^H to (2n) X (2n) complex matrices), then a(n) is the dimension of its Lie algebra u_n(H) = {A in M_n(H): A + A^H = 0} as a real vector space. A basis is given by {(E_{st}-E_{ts}), i*(E_{st}+E_{ts}), j*(E_{st}+E_{ts}), k*(E_{st}+E_{ts}): 1 <= s < t <= n} U {i*E_{tt}, j*E_{tt}, k*E_{tt}: t = 1..n}, where E_{st} is the matrix with all entries zero except that its (st)-entry is 1. - Jianing Song, Apr 05 2021

			For n=6, a(6) = 0^2 - 1^2 + 2^2 - 3^2 + 4^2 - 5^2 + 6^2 - 7^2 + 8^2 - 9^2 + 10^2 - 11^2 + 12^2 = 78. - _Bruno Berselli_, Aug 29 2013

Louis Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Matthew Cho, Anton Dochtermann, Ryota Inagaki, Suho Oh, Dylan Snustad, and Bailee Zacovic, Chip-firing and critical groups of signed graphs, arXiv:2306.09315 [math.CO], 2023. See p. 22.
Robert FERREOL, Illustration: triangular numbers of even order
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Milan Janjic, Two Enumerative Functions, University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Kival Ngaokrajang, Illustration of half circle spiral.
Markus Scheuer, show that; strange sum yields triangular numbers, Mathematics StackExchange.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019).
Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).
Leo Tavares, Illustration: Squared Hexagons.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A000384, A002378, A002943, A100040, A100041, A081266, A144312.
Cf. A033567, A033585, A059255, A130757, A132440, A027907.
Second column of array A094416.
Equals A033586(n) divided by 4.
See Comments of A132124.
Second n-gonal numbers: A005449, A147875, A045944, A179986, A033954, A062728, A135705.
Row sums in triangle A253580.
Cf. A016061, A003215, A000290, A188859.

List([0..50],n->n*(2*n+1)); # Muniru A Asiru, Oct 31 2018

a014105 n = n * (2 * n + 1)
a014105_list = scanl (+) 0 a004767_list  -- Reinhard Zumkeller, Oct 03 2012

[ n*(2*n+1) : n in [0..50] ]; // Wesley Ivan Hurt, Jun 14 2014

seq(binomial(2*n+1,2), n=0..46); # Zerinvary Lajos, Jan 21 2007

Table[n*(2*n+1), {n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3,-3,1},{0,3,10},50] (* Harvey P. Dale, Feb 10 2015 *)
CoefficientList[Series[x*(3 + x)/(1 - x)^3,{x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

PARI
```
a(n)=n*(2*n+1)
```

[n*(2*n+1) for n in range(50)] # G. C. Greubel, Dec 16 2018

a(n) = 3*Sum_{k=1..n} tan^2(k*Pi/(2*(n + 1))). - Ignacio Larrosa Cañestro, Apr 17 2001
a(n)^2 = n*(a(n) + 1 + a(n) + 2 + ... + a(n) + 2*n); e.g., 10^2 = 2*(11 + 12 + 13 + 14). - Charlie Marion, Jun 15 2003
From N. J. A. Sloane, Sep 13 2003: (Start)
G.f.: x*(3 + x)/(1 - x)^3.
E.g.f.: exp(x)*(3*x + 2*x^2).
a(n) = A000217(2*n) = A000384(-n). (End)
a(n) = A084849(n) - 1; A100035(a(n) + 1) = 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = A126890(n, k) + A126890(n, n-k), 0 <= k <= n. - Reinhard Zumkeller, Dec 30 2006
a(2*n) = A033585(n); a(3*n) = A144314(n). - Reinhard Zumkeller, Sep 17 2008
a(n) = a(n-1) + 4*n - 1 (with a(0) = 0). - Vincenzo Librandi, Dec 24 2010
a(n) = Sum_{k=0.2*n} (-1)^k*k^2. - Bruno Berselli, Aug 29 2013
a(n) = A242342(2*n + 1). - Reinhard Zumkeller, May 11 2014
a(n) = Sum_{k=0..2} C(n-2+k, n-2) * C(n+2-k, n), for n > 1. - J. M. Bergot, Jun 14 2014
a(n) = floor(Sum_{j=(n^2 + 1)..((n+1)^2 - 1)} sqrt(j)). Fractional portion of each sum converges to 1/6 as n -> infinity. See A247112 for a similar summation sequence on j^(3/2) and references to other such sequences. - Richard R. Forberg, Dec 02 2014
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, with a(0) = 0, a(1) = 3, and a(2) = 10. - Harvey P. Dale, Feb 10 2015
Sum_{n >= 1} 1/a(n) = 2*(1 - log(2)) = 0.61370563888010938... (A188859). - Vaclav Kotesovec, Apr 27 2016
From Wolfdieter Lang, Apr 27 2018: (Start)
a(n) = trinomial(2*n, 2) = trinomial(2*n, 2*(2*n-1)), for n >= 1, with the trinomial irregular triangle A027907; i.e., trinomial(n,k) = A027907(n,k).
a(n) = (1/Pi) * Integral_{x=0..2} (1/sqrt(4 - x^2)) * (x^2 - 1)^(2*n) * R(4*(n-1), x), for n >= 0, with the R polynomial coefficients given in A127672, and R(-m, x) = R(m, x). [See Comtet, p. 77, the integral formula for q = 3, n -> 2*n, k = 2, rewritten with x = 2*cos(phi).] (End)
a(n) = A002943(n)/2. - Ralf Steiner, Jul 23 2019
a(n) = A000290(n) + A002378(n). - Torlach Rush, Nov 02 2020
a(n) = A003215(n) - A000290(n+1). See Squared Hexagons illustration. Leo Tavares, Nov 23 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/2 + log(2) - 2. - Amiram Eldar, Nov 28 2021

Link added and minor errors corrected by Johannes W. Meijer, Feb 04 2010

A084849 a(n) = 1 + n + 2*n^2.

Original entry on oeis.org

1, 4, 11, 22, 37, 56, 79, 106, 137, 172, 211, 254, 301, 352, 407, 466, 529, 596, 667, 742, 821, 904, 991, 1082, 1177, 1276, 1379, 1486, 1597, 1712, 1831, 1954, 2081, 2212, 2347, 2486, 2629, 2776, 2927, 3082, 3241, 3404, 3571, 3742, 3917, 4096, 4279, 4466

Offset: 0

Paul Barry, Jun 09 2003

Equals (1, 2, 3, ...) convolved with (1, 2, 4, 4, 4, ...). a(3) = 22 = (1, 2, 3, 4) dot (4, 4, 2, 1) = (4 + 8 + 6 + 4). - Gary W. Adamson, May 01 2009
a(n) is also the number of ways to place 2 nonattacking bishops on a 2 X (n+1) board. - Vaclav Kotesovec, Jan 29 2010
Partial sums are A174723. - Wesley Ivan Hurt, Apr 16 2016
Also the number of irredundant sets in the n-cocktail party graph. - Eric W. Weisstein, Aug 09 2017

G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Burrows and C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv:1502.06664 [math.CO], 2015.
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph.D. Thesis, Waterford Institute of Technology, 2011.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Irredundant Set.
Wikipedia, Alexander polynomial and Seifert surface. [See _Peter Bala_'s comment.]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000027, A000217, A001844, A004767 (first differences), A014105, A058331, A060884, A100036, A100037, A100038, A100039, A100040, A100041, A131901, A134082, A174723, A177342.

[1+n+2*n^2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016

A084849:=n->1+n+2*n^2: seq(A084849(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 200, 4}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
f[n_]:=(n*(2*n+1)+1);Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Table[1 + n + 2 n^2, {n, 0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{3, -3, 1}, {4, 11, 22}, {0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
CoefficientList[Series[(-1 - x - 2 x^2)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Aug 09 2017 *)

a(n)=1+n+2*n^2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = A058331(n) + A000027(n).
G.f.: (1 + x + 2*x^2)/(1 - x)^3.
a(n) = A014105(n) + 1; A100035(a(n)) = 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = ceiling((2*n + 1)^2/2) - n = A001844(n) - n. - Paul Barry, Jul 16 2006
From Gary W. Adamson, Oct 07 2007: (Start)
Row sums of triangle A131901.
(a(n): n >= 0) is the binomial transform of (1, 3, 4, 0, 0, 0, ...). (End)
Equals A134082 * [1,2,3,...]. -
a(n) = (1 + A000217(2*n-1) + A000217(2*n+1))/2. - Enrique Pérez Herrero, Apr 02 2010
a(n) = (A177342(n+1) - A177342(n))/2, with n > 0. - Bruno Berselli, May 19 2010
a(n) - 3*a(n-1) + 3*a(n-2) - a(n-3) = 0, with n > 2. - Bruno Berselli, May 24 2010
a(n) = 4*n + a(n-1) - 1 (with a(0) = 1). - Vincenzo Librandi, Aug 08 2010
With an offset of 1, the polynomial a(t-1) = 2*t^2 - 3*t + 2 is the Alexander polynomial (with negative powers cleared) of the 3-twist knot. The associated Seifert matrix S is [[-1,-1], [0,-2]]. a(n-1) = det(transpose(S) - n*S). Cf. A060884. - Peter Bala, Mar 14 2012
E.g.f.: (1 + 3*x + 2*x^2)*exp(x). - Ilya Gutkovskiy, Apr 16 2016

A100037 Positions of occurrences of the natural numbers as a second subsequence in A100035.

Original entry on oeis.org

4, 9, 18, 31, 48, 69, 94, 123, 156, 193, 234, 279, 328, 381, 438, 499, 564, 633, 706, 783, 864, 949, 1038, 1131, 1228, 1329, 1434, 1543, 1656, 1773, 1894, 2019, 2148, 2281, 2418, 2559, 2704, 2853, 3006, 3163, 3324, 3489, 3658, 3831, 4008, 4189, 4374, 4563

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

For n > 1, A100035(a(n)) = n and A100035(m) != n for a(n-1) <= m < a(n);

A100036(n) < a(n) < A100038(n) < A100039(n).

			First terms (10 = A, 11 = B, 12 = C) of A100035(a(n)):
...1....2........3............4................5......
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B;
a(1) = A084849(2) = 4, A100035(4) = 1;
a(2) = A014107(2) = 9, A100035(9) = 2;
a(3) = A033537(3) = 18, A100035(18) = 3;
a(4) = A100040(4) = 31, A100035(31) = 4;
a(5) = A100041(5) = 48, A100035(48) = 5.

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Cf. A014107, A033537, A084849, A100035, A100036, A100038, A100039, A100040, A100041.

a(n) = 2*n^2 - n + 3 (conjectured). - Ralf Stephan, May 15 2007

A100038 Positions of occurrences of the natural numbers as third subsequence in A100035.

Original entry on oeis.org

11, 20, 33, 50, 71, 96, 125, 158, 195, 236, 281, 330, 383, 440, 501, 566, 635, 708, 785, 866, 951, 1040, 1133, 1230, 1331, 1436, 1545, 1658, 1775, 1896, 2021, 2150, 2283, 2420, 2561, 2706, 2855, 3008, 3165, 3326, 3491, 3660, 3833, 4010, 4191, 4376, 4565

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

n>1: A100035(a(n))=n and A100035(m)<>n for a(n-1)<=m

A100036(n) < A100037(n) < a(n) < A100039(n).

			First terms (10=A,11=B,12=C) of A100035(a(n)):
..........1........2............3................4...
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1;
a(1) = A084849(3) = 11, A100035(11) = 1;
a(2) = A014107(3) = 20, A100035(20) = 2;
a(3) = A033537(4) = 33, A100035(33) = 3;
a(4) = A100040(5) = 50, A100035(50) = 4;
a(5) = A100041(6) = 71, A100035(71) = 5.

Cf. A100037.

a(n) = 2*n^2 + 3*n + 6 (conjectured). - Ralf Stephan, May 15 2007

A100039 Positions of occurrences of the natural numbers as fourth subsequence in A100035.

Original entry on oeis.org

22, 35, 52, 73, 98, 127, 160, 197, 238, 283, 332, 385, 442, 503, 568, 637, 710, 787, 868, 953, 1042, 1135, 1232, 1333, 1438, 1547, 1660, 1777, 1898, 2023, 2152, 2285, 2422, 2563, 2708, 2857, 3010, 3167, 3328, 3493, 3662, 3835, 4012, 4193, 4378, 4567, 4760

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

n>1: A100035(a(n))=n and A100035(m)<>n for a(n-1)<=m

A100036(n) < A100037(n) < A100038(n) < a(n).

			First terms (10=A,11=B,12=C) of A100035(a(n)):
.....................1............2................3....
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1;
a(1) = A084849(4) = 22, A100035(22) = 1;
a(2) = A014107(4) = 35, A100035(35) = 2;
a(3) = A033537(5) = 52, A100035(52) = 3;
a(4) = A100040(6) = 73, A100035(73) = 4;
a(5) = A100041(7) = 98, A100035(98) = 5.

2n^2 + 7n + 13 (conjectured). - Ralf Stephan, May 15 2007

A100040 a(n) = 2*n^2 + n - 5.

Original entry on oeis.org

-5, -2, 5, 16, 31, 50, 73, 100, 131, 166, 205, 248, 295, 346, 401, 460, 523, 590, 661, 736, 815, 898, 985, 1076, 1171, 1270, 1373, 1480, 1591, 1706, 1825, 1948, 2075, 2206, 2341, 2480, 2623, 2770, 2921, 3076, 3235, 3398, 3565, 3736, 3911, 4090, 4273

Offset: 0

Reinhard Zumkeller, Oct 31 2004

a(n) is the result of taking five consecutive numbers starting at n-2, then adding the products of the first and the last and of the second with the fourth and finally adding the middle term. That is, a(n) = (n^2-4) + (n^2-1) + n. - J. M. Bergot, Mar 06 2018

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. A100036, A100037, A100038, A100039.

List([0..50],n->2*n^2+n-5); # Muniru A Asiru, Mar 20 2018

Magma
```
[ 2*n^2+n-5: n in [0..50] ];
```

[seq(2*n^2+n-5,n=0..50)]; # Muniru A Asiru, Mar 20 2018

Table[2*n^2 + n - 5, {n, 0, 50}] (* G. C. Greubel, Jul 15 2017 *)
LinearRecurrence[{3,-3,1},{-5,-2,5},50] (* Harvey P. Dale, Sep 21 2017 *)

a(n)=2*n^2+n-5 \\ Charles R Greathouse IV, Sep 24 2015

A100035(a(n)) = 4 for n>3;
a(n) = A014105(n) - 5 = A084849(n) - 6 = A100041(n) + 2.
a(n) = 2*a(n-1)-a(n-2)+4; a(0)=-5, a(1)=-2. - Vincenzo Librandi, Dec 26 2010
G.f.: (-5 + 13*x - 4*x^2)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 25 2011
E.g.f.: (2*x^2 + 3*x - 5)*exp(x). - G. C. Greubel, Jul 15 2017

A100035 a(n+1) occurs not earlier as a neighbor of terms = a(n): either it is the greatest number < a(n) or, if no such number exists, the smallest number > a(n); a(1) = 1.

Original entry on oeis.org

1, 2, 3, 1, 4, 3, 5, 4, 2, 5, 1, 6, 5, 7, 6, 4, 7, 3, 6, 2, 7, 1, 8, 7, 9, 8, 6, 9, 5, 8, 4, 9, 3, 8, 2, 9, 1, 10, 9, 11, 10, 8, 11, 7, 10, 6, 11, 5, 10, 4, 11, 3, 10, 2, 11, 1, 12, 11, 13, 12, 10, 13, 9, 12, 8, 13, 7, 12, 6, 13, 5, 12, 4, 13, 3, 12, 2, 13, 1, 14, 13, 15, 14, 12, 15, 11, 14, 10

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

The natural numbers (A000027) occur infinitely many times as disjoint subsequences, see the example below and A100036, A100037, A100038 and A100039: exactly one k exists for all x < y such that a(k) = x and (a(k-1) = y or a(k+1) = y).
a(2*k^2 + k + 1) = a(A084849(k)) = 1 for k >= 0;
a(2*k^2 - 3*k) = a(A014107(k)) = 2 for k > 1;
a(2*k^2 + 5*k) = a(A033537(k)) = 3 for k > 1;
a(2*k^2 + k - 5) = a(A100040(k)) = 4 for k > 2;
a(2*k^2 + k - 7) = a(A100041(k)) = 5 for k > 3.

			First terms (10 = A, 11 = B, 12 = C) and some subsequences = A000027:
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1CBD
123.4.5....6.7........8.9............A.B................C.D.
...1....2........3............4................5..........
..........1........2............3................4......
.....................1............2................3....

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000027, A014107, A033537, A100036, A100037, A100038, A100039, A100040, A100041.

A100036 a(n) = smallest m such that A100035(m) = n.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 14, 23, 25, 38, 40, 57, 59, 80, 82, 107, 109, 138, 140, 173, 175, 212, 214, 255, 257, 302, 304, 353, 355, 408, 410, 467, 469, 530, 532, 597, 599, 668, 670, 743, 745, 822, 824, 905, 907, 992, 994, 1083, 1085, 1178, 1180, 1277, 1279, 1380

Offset: 1

Reinhard Zumkeller, Oct 31 2004

nonn

Smallest positions of occurrences of the natural numbers as subsequence in A100035;
A100035(a(n)) = n and A100035(m) <> n for m < a(n);
a(n) < A100037(n) < A100038(n) < A100039(n).

			First terms (10=A,11=B,12=C) of A100035(a(n)):
123.4.5....6.7........8.9............A.B................C.
1231435425165764736271879869584938291A9BA8B7A6B5A4B3A2B1CBD;
a(1) = A084849(1) = 1, A100035(1) = 1;
a(2) = A014107(1) = 2, A100035(2) = 2;
a(3) = A033537(1) = 3, A100035(3) = 3;
a(4) = A100040(1) = 5, A100035(5) = 4;
a(5) = A100041(1) = 7, A100035(7) = 5.

Conjecture: a(n) = partial sums of sequence [1,1,1,2,2,5,2,9,2,13,2,17,2,21,2,25,2,29,2,33,...2,n/2-7,2,...]. In other words, a(n) consists of the numbers 1,2,3 and the sequences A096376 and A096376+2 interspersed. - Ralf Stephan, May 15 2007

A185869 (Odd,even)-polka dot array in the natural number array A000027; read by antidiagonals.

Original entry on oeis.org

2, 7, 9, 16, 18, 20, 29, 31, 33, 35, 46, 48, 50, 52, 54, 67, 69, 71, 73, 75, 77, 92, 94, 96, 98, 100, 102, 104, 121, 123, 125, 127, 129, 131, 133, 135, 154, 156, 158, 160, 162, 164, 166, 168, 170, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 277, 279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 326, 328, 330, 332, 334, 336, 338, 340, 342, 344, 346, 348, 350, 379, 381, 383, 385, 387, 389, 391, 393, 395, 397, 399, 401, 403, 405

Offset: 1

Clark Kimberling, Feb 05 2011

This is the second of four polka dot arrays; see A185868.
row 1: A130883;
row 2: A100037;
row 3: A100038;
row 4: A100039;
col 1: A014107;
col 2: A033537;
col 3: A100040;
col 4: A100041;
diag (2,18,...): A077591;
diag (7,31,...): A157914;
diag (16,48,...): A035008;
diag (29,69,...): A108928;
antidiagonal sums: A033431;
antidiagonal sums: 2*(1^3, 2^3, 3^3, 4^3,...) = 2*A000578.
A060432(n) + n is odd if and only if n is in this sequence. - Peter Kagey, Feb 03 2016

			Northwest corner:
  2....7....16...29...46
  9....18...31...48...69
  20...33...50...71...96
  35...52...73...98...127

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000027 (as an array), A060432, A185868, A185870, A185871.

a185869 n = a185869_list !! (n - 1)
a185869_list = scanl (+) 2 $ a' 1
  where  a' n = 2 * n + 3 : replicate n 2 ++ a' (n + 1)
-- Peter Kagey, Sep 02 2016

f[n_,k_]:=2n-1+(2n+2k-3)(n+k-1);
TableForm[Table[f[n,k],{n,1,10},{k,1,15}]]
Table[f[n-k+1,k],{n,14},{k,n,1,-1}]//Flatten

from math import isqrt, comb
def A185869(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    x = n-comb(a,2)
    y = a-x+1
    return y*((y+(c:=x<<1)<<1)-5)+x*(c-3)+2 # Chai Wah Wu, Jun 18 2025

T(n,k) = 2n-1+(n+k-1)*(2n+2k-3), k>=1, n>=1.

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.

Original entry on oeis.org

1, 5, 6, 2, 3, 4, 12, 13, 14, 15, 7, 8, 9, 10, 11, 23, 24, 25, 26, 27, 28, 16, 17, 18, 19, 20, 21, 22, 38, 39, 40, 41, 42, 43, 44, 45, 29, 30, 31, 32, 33, 34, 35, 36, 37, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 80

Offset: 1

Boris Putievskiy, Feb 08 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.
Enumeration table T(n,k). The order of the list:
T(1,1)=1;
T(1,3), T(2,2), T(3,1);
T(1,2), T(2,1);
. . .
T(1,n), T(2,n-1), T(3,n-2), ... T(n,1);
T(1,n-1), T(2,n-3), T(3,n-4),...T(n-1,1);
. . .
First row  matches with the elements antidiagonal {T(1,n), ... T(n,1)},
second row matches with the elements antidiagonal {T(1,n-1), ... T(n-1,1)}.
Table contains:
row  1 is alternation of elements A130883 and A096376,
row  2  accommodates elements A033816 in even places,
row  3  accommodates elements A100037 in odd  places,
row  5  accommodates elements A100038 in odd  places;
column 1 is alternation of elements A084849 and  A000384,
column 2 is alternation of elements A014106 and  A014105,
column 3 is alternation of elements A014107 and  A091823,
column 4 is alternation of elements A071355 and |A168244|,
column 5 accommodates elements A033537 in even places,
column 7 is alternation of elements A100040 and  A130861,
column 9 accommodates elements A100041 in even places;
the main diagonal is A058331,
diagonal  1, located above the main diagonal is  A001844,
diagonal  2, located above the main diagonal is  A001105,
diagonal  3, located above the main diagonal is  A046092,
diagonal  4, located above the main diagonal is  A056220,
diagonal  5, located above the main diagonal is  A142463,
diagonal  6, located above the main diagonal is  A054000,
diagonal  7, located above the main diagonal is  A090288,
diagonal  9, located above the main diagonal is  A059993,
diagonal 10, located above the main diagonal is |A147973|,
diagonal 11, located above the main diagonal is  A139570;
diagonal  1, located under the main diagonal is  A051890,
diagonal  2, located under the main diagonal is  A005893,
diagonal  3, located under the main diagonal is  A097080,
diagonal  4, located under the main diagonal is  A093328,
diagonal  5, located under the main diagonal is  A137882.

			The start of the sequence as table:
  1....5...2..12...7..23..16...
  6....3..13...8..24..17..39...
  4...14...9..25..18..40..31...
  15..10..26..19..41..32..60...
  11..27..20..42..33..61..50...
  28..21..43..34..62..51..85...
  22..44..35..63..52..86..73...
  . . .
The start of the sequence as triangle array read by rows:
  1;
  5,6;
  2,3,4;
  12,13,14,15;
  7,8,9,10,11;
  23,24,25,26,27,28;
  16,17,18,19,20,21,22;
  . . .
Row number r matches with r numbers segment {(r+1)*r/2-r*(-1)^(r+1)-r+2,... (r+1)*r/2-r*(-1)^(r+1)+1}.

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 Weisstein's World of Mathematics, Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A130883, A096376, A033816, A100037, A100038, A084849, A000384, A014106, A014105, A014107, A091823, A071355, A168244, A033537, A100040, A130861, A100041, A058331, A001844, A001105, A046092, A056220, A142463, A054000, A090288, A059993, A147973, A139570, A051890, A005893, A097080, A093328, A137882.

T[n_, k_] := (n+k)(n+k-1)/2 - (-1)^(n+k)(n+k-1) - k + 2;
Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

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

T(n,k) = (k+n)*(k+n-1)/2-(k+n-1)*(-1)^(k+n)-k+2.
As linear sequence
a(n) = A003057(n)*A002024(n)/2- A002024(n)*(-1)^A003056(n)-A004736(n)+2.
a(n) = (t+2)*(t+1)/2 - (t+1)*(-1)^t-j+2, where j=(t*t+3*t+4)/2-n and t=int((math.sqrt(8*n-7) - 1)/ 2).

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A014105 Second hexagonal numbers: a(n) = n*(2*n + 1).

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A084849 a(n) = 1 + n + 2*n^2.

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A100037 Positions of occurrences of the natural numbers as a second subsequence in A100035.

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A100038 Positions of occurrences of the natural numbers as third subsequence in A100035.

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A100039 Positions of occurrences of the natural numbers as fourth subsequence in A100035.

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A100040 a(n) = 2*n^2 + n - 5.

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A100035 a(n+1) occurs not earlier as a neighbor of terms = a(n): either it is the greatest number < a(n) or, if no such number exists, the smallest number > a(n); a(1) = 1.

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A100036 a(n) = smallest m such that A100035(m) = n.

A014105 Second hexagonal numbers: a(n) = n(2n + 1).

A211394 T(n,k) = (k+n)(k+n-1)/2-(k+n-1)(-1)^(k+n)-k+2; n , k > 0, read by antidiagonals.