A033537 -id:A033537 - cp's OEIS frontend

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

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

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

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

A139579 a(n) = 2n^2 + 15n.

Original entry on oeis.org

0, 17, 38, 63, 92, 125, 162, 203, 248, 297, 350, 407, 468, 533, 602, 675, 752, 833, 918, 1007, 1100, 1197, 1298, 1403, 1512, 1625, 1742, 1863, 1988, 2117, 2250, 2387, 2528, 2673, 2822, 2975, 3132, 3293, 3458, 3627, 3800, 3977, 4158, 4343, 4532, 4725, 4922, 5123, 5328, 5537

Offset: 0

Omar E. Pol, May 19 2008

Stefano Spezia, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139578, A139580, A139581.

LinearRecurrence[{3,-3,1},{0,17,38},50] (* Stefano Spezia, Oct 21 2023 *)

a(n)=2*n^2+15*n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 4*n + 13; a(0) = 0. - Vincenzo Librandi, Nov 24 2010
From Stefano Spezia, Oct 21 2023: (Start)
O.g.f.: x*(17 - 13*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(17 + 2*x). (End)
From Amiram Eldar, Nov 10 2023: (Start)
Sum_{n>=1} 1/a(n) = 182144/675675 - 2*log(2)/15.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/15 - Pi/30 + 67952/675675. (End)

A139576 a(n) = n(2n + 9).

Original entry on oeis.org

0, 11, 26, 45, 68, 95, 126, 161, 200, 243, 290, 341, 396, 455, 518, 585, 656, 731, 810, 893, 980, 1071, 1166, 1265, 1368, 1475, 1586, 1701, 1820, 1943, 2070, 2201, 2336, 2475, 2618, 2765, 2916, 3071, 3230, 3393, 3560, 3731, 3906

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139577, A139578, A139579, A139580, A139581.

Cf. A277979.

s=0;lst={s};Do[s+=n++ +11;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[Sum[(2*i + n - 1), {i, 4, n}], {n, 3, 45}] (* Zerinvary Lajos, Jul 11 2009 *)
Table[n(2n+9),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,11,26},50] (* Harvey P. Dale, Dec 18 2018 *)

a(n)=n*(2*n+9) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 9*n.
a(n) = a(n-1) + 4*n + 7 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(11 - 7*x)/(1-x)^3.
E.g.f.: exp(x)*x*(11 + 2*x).
a(n) = A277979(n)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139577 a(n) = n(2n + 11).

Original entry on oeis.org

0, 13, 30, 51, 76, 105, 138, 175, 216, 261, 310, 363, 420, 481, 546, 615, 688, 765, 846, 931, 1020, 1113, 1210, 1311, 1416, 1525, 1638, 1755, 1876, 2001, 2130, 2263, 2400, 2541, 2686, 2835, 2988, 3145, 3306, 3471, 3640, 3813, 3990

Offset: 0

Omar E. Pol, May 19 2008

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014105, A014106, A033537, A130861, A139576, A139578, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +13;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+11),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,30},50] (* Harvey P. Dale, Mar 17 2019 *)

a(n)=n*(2*n+11) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*n^2 + 11*n.
a(n) = a(n-1) + 4*n + 9 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(13 - 9*x)/(1-x)^3.
E.g.f.: exp(x)*x*(13 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139578 a(n) = n(2n + 13).

Original entry on oeis.org

0, 15, 34, 57, 84, 115, 150, 189, 232, 279, 330, 385, 444, 507, 574, 645, 720, 799, 882, 969, 1060, 1155, 1254, 1357, 1464, 1575, 1690, 1809, 1932, 2059, 2190, 2325, 2464, 2607, 2754, 2905, 3060, 3219, 3382, 3549, 3720, 3895, 4074, 4257, 4444, 4635, 4830, 5029

Offset: 0

Omar E. Pol, May 19 2008

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

Cf. A014105, A014106, A033537, A130861, A139576, A139577, A139579, A139580, A139581.

s=0;lst={s};Do[s+=n++ +15;AppendTo[lst, s], {n, 0, 7!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 19 2008 *)
Table[n(2n+13),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,15,34},50] (* Harvey P. Dale, Nov 22 2014 *)

a(n)=n*(2*n+13) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*n^2 + 13*n.
a(n) = a(n-1) + 4*n + 11 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: x*(15 - 11*x)/(1 - x)^3.
E.g.f.: exp(x)*x*(15 + 2*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Showing 1-10 of 24 results. Next

cp's OEIS Frontend

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

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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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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.

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A139579 a(n) = 2*n^2 + 15*n.

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A139576 a(n) = n*(2*n + 9).

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A139577 a(n) = n*(2*n + 11).

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

A139579 a(n) = 2n^2 + 15n.

A139576 a(n) = n(2n + 9).

A139577 a(n) = n(2n + 11).

A139578 a(n) = n(2n + 13).