A100036 -id:A100036 - cp's OEIS frontend

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

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

A014107 a(n) = n(2n-3).

Original entry on oeis.org

0, -1, 2, 9, 20, 35, 54, 77, 104, 135, 170, 209, 252, 299, 350, 405, 464, 527, 594, 665, 740, 819, 902, 989, 1080, 1175, 1274, 1377, 1484, 1595, 1710, 1829, 1952, 2079, 2210, 2345, 2484, 2627, 2774, 2925, 3080, 3239, 3402, 3569, 3740, 3915, 4094, 4277

Offset: 0

N. J. A. Sloane

Positive terms give a bisection of A000096. - Omar E. Pol, Dec 16 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
Emily Barnard and Nathan Reading, Coxeter-biCatalan combinatorics, arXiv:1605.03524 [math.CO], 2016. See p. 51.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000096, A000108, A014105, A033537, A097070, A100035, A100036, A100037, A100038, A100039, A100345, A002378, A000384.

A014107:=n->n*(2*n-3); seq(A014107(n), n=0..100); # Wesley Ivan Hurt, Nov 19 2013

Table[2n^2 - 3n, {n, 0, 49}] (* Alonso del Arte, Dec 15 2012 *)
LinearRecurrence[{3,-3,1},{0,-1,2},50] (* Harvey P. Dale, Sep 18 2018 *)

PARI
```
a(n)=n*(2*n-3)
```

a(n) = A100345(n, n - 3) for n > 2.
a(n) = A033537(n) - 8*n^2; A100035(a(n)) = 2 for n > 1. - Reinhard Zumkeller, Oct 31 2004
a(n) = A014106(-n) for all n in Z. - Michael Somos, Nov 06 2005
From Michael Somos, Nov 06 2005: (Start)
G.f.: x*(-1 + 5*x)/(1 - x)^3.
E.g.f: x*(-1 + 2*x)*exp(x). (End)
a(n) = A097070(n)/A000108(n - 2), n >= 2. - Philippe Deléham, Apr 12 2007
a(n) = 2*a(n-1) - a(n-2) + 4, n > 1; a(0) = 0, a(1) = -1, a(2) = 2. - Zerinvary Lajos, Feb 18 2008
a(n) = a(n-1) + 4*n - 5 with a(0) = 0. - Vincenzo Librandi, Nov 20 2010
a(n) = (2*n-1)*(n-1) - 1. Also, with an initial offset of -1, a(n) = (2*n-1)*(n+1) = 2*n^2 + n - 1. - Alonso del Arte, Dec 15 2012
(a(n) + 1)^2 + (a(n) + 2)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2n - 1)^2 starting with a(1) = -1. - Jeffreylee R. Snow, Sep 17 2013
a(n) = A014105(n-1) - 1 for all n in Z. - Michael Somos, Nov 23 2021
From Amiram Eldar, Feb 20 2022: (Start)
Sum_{n>=1} 1/a(n) = -2*(1 - log(2))/3.
Sum_{n>=1} (-1)^n/a(n) = Pi/6 + log(2)/3 + 2/3. (End)
For n > 0, A002378(a(n)) = A000384(n-1)*A000384(n). - Charlie Marion, May 21 2023

A033537 a(n) = n(2n+5).

Original entry on oeis.org

0, 7, 18, 33, 52, 75, 102, 133, 168, 207, 250, 297, 348, 403, 462, 525, 592, 663, 738, 817, 900, 987, 1078, 1173, 1272, 1375, 1482, 1593, 1708, 1827, 1950, 2077, 2208, 2343, 2482, 2625, 2772, 2923, 3078, 3237, 3400, 3567, 3738, 3913, 4092, 4275, 4462, 4653, 4848, 5047, 5250, 5457, 5668

Offset: 0

N. J. A. Sloane

Permutations avoiding 12-3 that contain the pattern 32-1 exactly once.
a(n) = A014107(n) + 8*n^2; A100035(a(n)) = 3 for n>1. - Reinhard Zumkeller, Oct 31 2004
If Y is a 3-subset of an (2n+1)-set X then, for n>=1, a(n-1) is the number of (2n-1)-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

G. C. Greubel, Table of n, a(n) for n = 0..5000
T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A100036, A100037, A100038, A100039.

List([0..60], n-> n*(2*n+5) ); # G. C. Greubel, Oct 14 2019

[n*(2*n+5): n in [0..60]]; // G. C. Greubel, Oct 14 2019

seq(n*(2*n+5), n=0..60); # G. C. Greubel, Oct 14 2019

Table[n*(2*n+5), {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
LinearRecurrence[{3,-3,1},{0,7,18},60] (* Harvey P. Dale, Nov 19 2021 *)

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

[n*(2*n+5) for n in (0..60)] # G. C. Greubel, Oct 14 2019

a(n) = a(n-1) + 4*n + 3 (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
From L. Edson Jeffery, Oct 14 2012: (Start)
G.f.: x*(7-3*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=7, a(2)=18. (End)
E.g.f.: x*(7 + 2*x)*exp(x). - G. C. Greubel, Jul 15 2017
From Amiram Eldar, Feb 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 46/75 - 2*log(2)/5.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/10 + log(2)/5 - 26/75. (End)

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

A100041 a(n) = 2*n^2 + n - 7.

Original entry on oeis.org

-7, -4, 3, 14, 29, 48, 71, 98, 129, 164, 203, 246, 293, 344, 399, 458, 521, 588, 659, 734, 813, 896, 983, 1074, 1169, 1268, 1371, 1478, 1589, 1704, 1823, 1946, 2073, 2204, 2339, 2478, 2621, 2768, 2919, 3074, 3233, 3396, 3563, 3734, 3909, 4088, 4271, 4458, 4649

Offset: 0

Reinhard Zumkeller, Oct 31 2004

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.

Table[2*n^2 + n - 7, {n, 0, 50}] (* G. C. Greubel, Jul 15 2017 *)
LinearRecurrence[{3,-3,1},{-7,-4,3},50] (* Harvey P. Dale, Mar 25 2021 *)

a(n)=2*n^2+n-7 \\ Charles R Greathouse IV, Jun 17 2017

A100035(a(n)) = 5 for n>3.
a(n) = A014105(n) - 7 = A084849(n) - 8 = A100040(n) - 2.
From G. C. Greubel, Jul 15 2017: (Start)
G.f.: (7 - 17 x + 6 x^2)/(-1 + x)^3.
E.g.f.: (2*x^2 + 3*x - 7)*exp(x). (End)

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.

Showing 1-9 of 9 results.

cp's OEIS Frontend

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

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A014107 a(n) = n*(2*n-3).

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A033537 a(n) = n*(2*n+5).

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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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A100041 a(n) = 2*n^2 + n - 7.

A014107 a(n) = n(2n-3).

A033537 a(n) = n(2n+5).