A081489 -id:A081489 - cp's OEIS frontend

A121997 Count up to n, n times.

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

Offset: 1

Franklin T. Adams-Watters, Sep 11 2006

The n-th block consists of n subblocks, each of which counts from 1 to n.
This a fractal sequence: removing the first instance of each value leaves the original sequence.
The first comment implies that this gives the column index of the n-th element of a sequence whose terms are coefficients, read by rows, of a sequence of matrices of size 1 X 1, 2 X 2, 3 X 3, etc.; cf. example. The row index is given by A238013(n), and the size of the matrix by A074279(n). - M. F. Hasler, Feb 16 2014

			Sequence begins:
  1;
  1,2;
  1,2;
  1,2,3;
  1,2,3;
  1,2,3;
  ...
The blocks of n subblocks of n terms (n=1,2,3,...) can be cast into a square matrices of order n; then the terms are equal to the index of the column they fall into.

Cf. A081489 (locations of new values), A075349 (locations of 1's).
Cf. A000290 (row lengths), A002411 (row sums), A036740 (row products).
Cf. A002024 and references there, esp. in PROG section.
Cf. A238013.

A121997(N=9)=concat(vector(N,i,concat(vector(i,j,vector(i,k,k))))) \\ Note: this creates a vector; use A121997()[n] to get the n-th term. - M. F. Hasler, Feb 16 2014

from sympy import integer_nthroot
def A121997(n): return 1+(n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6-1)%k # Chai Wah Wu, Nov 04 2024

A145066 Partial sums of A002522, starting at n=1.

Original entry on oeis.org

2, 7, 17, 34, 60, 97, 147, 212, 294, 395, 517, 662, 832, 1029, 1255, 1512, 1802, 2127, 2489, 2890, 3332, 3817, 4347, 4924, 5550, 6227, 6957, 7742, 8584, 9485, 10447, 11472, 12562, 13719, 14945, 16242, 17612, 19057, 20579, 22180, 23862, 25627

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 30 2008

			a(2) = a(1) + 2^2 + 1 = 2 + 4 + 1 = 7; a(3) = a(2) + 3^2 + 1 = 7 + 9 + 1 = 17.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002522 (n^2 + 1), A005563 ((n+1)^2 - 1), A051925 (zero followed by partial sums of A005563), A081489 (partial sums of A002522 starting at n=0).

Accumulate[Range[50]^2+1] (* Harvey P. Dale, Dec 07 2016 *)

{a=0; for(n=1, 42, print1(a=a+n^2+1, ","))}

def A145066(n): return (n*(n*(2*n + 3) + 1))//6 + n # Chai Wah Wu, Oct 30 2024

a(1) = 2; a(n) = a(n-1) + n^2 + 1 for n > 1.
From Christoph Pacher (christoph.pacher(AT)ait.ac.at), Jul 23 2010: (Start)
a(n) = Sum_{k=1..n} (k^2 + 1).
a(n) = A000330(n) + n.
a(n) = n*(n+1)*(2*n+1)/6 + n. (End)
G.f.: x*(2-x+x^2)/(1-x)^4. - Colin Barker, Apr 04 2012
E.g.f.: (1/6)*x*(12 + 9*x + 2*x^2)*exp(x). - G. C. Greubel, Jul 22 2017
a(n) = A081489(n+1) - 1. - Jianing Song, Oct 10 2021

Edited by Klaus Brockhaus, Oct 17 2008

A081490 Leading term of n-th row of A081491.

Original entry on oeis.org

1, 2, 4, 9, 19, 36, 62, 99, 149, 214, 296, 397, 519, 664, 834, 1031, 1257, 1514, 1804, 2129, 2491, 2892, 3334, 3819, 4349, 4926, 5552, 6229, 6959, 7744, 8586, 9487, 10449, 11474, 12564, 13721, 14947, 16244, 17614, 19059, 20581, 22182, 23864, 25629

Offset: 1

Amarnath Murthy, Mar 25 2003

First differences are given by A002522 = n^2 + 1. Second differences are odd numbers given by A005408.

a(1)=1, a(2)=2, (a(n+1) -a(n)) - (a(n) -a(n-1)) = 2*(n-1)-1. - Ben Paul Thurston, Aug 22 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A005408, A081489, A081491, A081492.

List([1..50], n-> (2*n^3-9*n^2+19*n-6)/6); # G. C. Greubel, Aug 13 2019

[(2*n^3-9*n^2+19*n-6)/6: n in [1..50]]; // G. C. Greubel, Aug 13 2019

with (combinat):a:=n->sum(fibonacci(3,i), i=0..n):seq(a(n)+1, n=-1..42); # Zerinvary Lajos, Apr 25 2008

Rest[CoefficientList[Series[x (1-2x+2x^2+x^3)/(x-1)^4,{x,0,50}],x]] (* or *) LinearRecurrence[{4,-6,4,-1}, {1,2,4,9}, 50] (* Harvey P. Dale, Apr 30 2011 *)

vector(50, n, (2*n^3-9*n^2+19*n-6)/6) \\ G. C. Greubel, Aug 13 2019

[(2*n^3-9*n^2+19*n-6)/6 for n in (1..50)] # G. C. Greubel, Aug 13 2019

a(1) = 1, a(n) = A081489(n-1) + 1.
From R. J. Mathar, Feb 06 2010: (Start)
G..f: x*(1-2*x+2*x^2+x^3)/(x-1)^4.
a(n) = n*(2*n^2 -9*n +19)/6 -1. (End)
a(n) = (n-2)^2 + a(n-1)+1, n>1. - Gary Detlefs, Jun 29 2010
a(1)=1, a(2)=2, a(3)=4, a(4)=9, a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Harvey P. Dale, Apr 30 2011

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

A081491 Triangle read by rows in which the n-th row contains n terms of an arithmetic progression with a common difference of (n-1) and the first term of (n+1)-th row is 1 more than the last term of the n-th row.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 15, 18, 19, 23, 27, 31, 35, 36, 41, 46, 51, 56, 61, 62, 68, 74, 80, 86, 92, 98, 99, 106, 113, 120, 127, 134, 141, 148, 149, 157, 165, 173, 181, 189, 197, 205, 213, 214, 223, 232, 241, 250, 259, 268, 277, 286, 295, 296, 306, 316, 326, 336, 346

Offset: 1

Amarnath Murthy, Mar 25 2003

			1; 2,3; 4,6,8; 9,12,15,18; 19,23,27,31,35; 36,41,46,51,56,61; ...

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A002522, A005408, A081489, A081490, A081492.

Table[NestList[#+n-1&,(2n^3-9n^2+19n-6)/6,n-1],{n,11}]//Flatten (* Harvey P. Dale, Feb 12 2024 *)

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003

A081492 Sum of terms in n-th row of A081491.

Original entry on oeis.org

1, 5, 18, 54, 135, 291, 560, 988, 1629, 2545, 3806, 5490, 7683, 10479, 13980, 18296, 23545, 29853, 37354, 46190, 56511, 68475, 82248, 98004, 115925, 136201, 159030, 184618, 213179, 244935, 280116, 318960, 361713, 408629, 459970, 516006, 577015

Offset: 1

Amarnath Murthy, Mar 25 2003

For odd n a(n) is a multiple of n and a(n)/n is the middle term of the corresponding row.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002522, A005408, A081489, A081490, A081491.

List([1..40], n-> n*(2*(n-1)^3+7*n-1)/6); # G. C. Greubel, Aug 13 2019

[n*(2*(n-1)^3+7*n-1)/6: n in [1..40]]; // G. C. Greubel, Aug 13 2019

seq(n*(2*(n-1)^3+7*n-1)/6, n=1..40); # G. C. Greubel, Aug 13 2019

LinearRecurrence[{5,-10,10,-5,1},{1,5,18,54,135},40] (* Harvey P. Dale, Jul 01 2018 *)

vector(40, n, n*(2*(n-1)^3+7*n-1)/6) \\ G. C. Greubel, Aug 13 2019

[n*(2*(n-1)^3+7*n-1)/6 for n in (1..40)] # G. C. Greubel, Aug 13 2019

a(n) = n*(2*n^3 - 6*n^2 + 13*n - 3)/6.
G.f.: x*(1+x)*(1-x+4*x^2)/(1-x)^5. - Colin Barker, Jul 28 2012
E.g.f.: x*(6 +9*x +6*x^2 +2*x^3)/6. - G. C. Greubel, Aug 13 2019

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 29 2003.

Formula corrected by Colin Barker, Jul 28 2012

A116701 Number of permutations of length n that avoid the patterns 132, 4321.

Original entry on oeis.org

1, 1, 2, 5, 13, 31, 66, 127, 225, 373, 586, 881, 1277, 1795, 2458, 3291, 4321, 5577, 7090, 8893, 11021, 13511, 16402, 19735, 23553, 27901, 32826, 38377, 44605, 51563, 59306, 67891, 77377, 87825, 99298, 111861, 125581, 140527, 156770, 174383, 193441, 214021

Offset: 0

Lara Pudwell, Feb 26 2006

Also, number of permutations of length n which avoid the patterns 312, 1234, 4312; or avoid the patterns 132, 1324, 4321, etc.
a(n) is the number of Dyck n-paths (A000108) with <= 3 peaks. For example, a(4)=13 counts all 14 Dyck 4-paths except the "sawtooth" path UDUDUDUD which has 4 peaks. - David Callan, Oct 13 2012
Apparently the number of Dyck paths of semilength n+1 in which the sum of the first 3 ascents adds to n+1. (Nonexistent ascents count as zero length.) - David Scambler, Apr 22 2013

			a(4)=13 because we have 11 permutations of [4] that do not avoid the patterns 132 and 4321; namely: 1324, 1342, 1432, 4132, 1423, 3142, 2431, 2413, 2143, 1243 and 4321.

Christian Bean, Bjarki Gudmundsson and Henning Ulfarsson, Automatic discovery of structural rules of permutation classes, arXiv:1705.04109 [math.CO], 2017.
H. Cheballah, S. Giraudo and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
Lara Pudwell, Systematic Studies in Pattern Avoidance, 2005.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A000108, A001263.

G:=1+(x*(x^4-2*x^3+5*x^2-3*x+1))/(1-x)^5: Gser:=series(G,x=0,48): seq(coeff(Gser,x,n),n=0..45); # Emeric Deutsch, Jul 29 2006

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 2, 5, 13, 31}, 40] (* Jean-François Alcover, Jan 09 2019 *)

G.f.: -(3*x^4 - 5*x^3 + 7*x^2 - 4*x + 1)/(x-1)^5.
a(n) = (n^4 - 4n^3 + 11n^2 - 8n + 12)/12. - Franklin T. Adams-Watters, Sep 16 2006
Binomial transform of [1, 1, 2, 3, 2, 0, 0, 0, ...]. - Gary W. Adamson, Oct 23 2007
Equals A001263 * [1, 1, 1, 0, 0, 0, ...], where A001263 = the Narayana triangle. - Gary W. Adamson, Nov 19 2007
E.g.f.: (1/12)*exp(x)*(12 + 12*x + 12*x^2 + 6*x^3 + x^4). - Stefano Spezia, Jan 09 2019
a(n) = binomial(n+1,4) + binomial(n,4) + binomial(n,2) + 1. - Michael Coopman, Oct 24 2021
a(n)-a(n-1) = A081489(n-1). - R. J. Mathar, Mar 08 2025

Entry revised by N. J. A. Sloane, Jul 25 2006

a(0)=1 prepended by Alois P. Heinz, Nov 28 2021

A320388 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 14, 16, 19, 18, 21, 25, 23, 26, 31, 29, 33, 38, 36, 40, 46, 44, 49, 56, 53, 58, 66, 64, 70, 77, 76, 82, 92, 89, 96, 106, 104, 113, 123, 120, 130, 142, 141, 149, 162, 160, 172, 186, 184, 195, 211, 210, 223, 238

Offset: 0

Seiichi Manyama, Oct 12 2018

nonn

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.

			There are a(17) = 15 such partitions of 17:
  01: [17]
  02: [1, 16]
  03: [2, 15]
  04: [3, 14]
  05: [4, 13]
  06: [5, 12]
  07: [6, 11]
  08: [7, 10]
  09: [1, 6, 10]
  10: [8, 9]
  11: [1, 7, 9]
  12: [2, 6, 9]
  13: [2, 7, 8]
  14: [3, 6, 8]
  15: [4, 6, 7]
There are a(18) = 14 such partitions of 18:
  01: [18]
  02: [1, 17]
  03: [2, 16]
  04: [3, 15]
  05: [4, 14]
  06: [5, 13]
  07: [6, 12]
  08: [7, 11]
  09: [8, 10]
  10: [1, 7, 10]
  11: [1, 8, 9]
  12: [2, 7, 9]
  13: [3, 7, 8]
  14: [1, 4, 6, 7]

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 (terms 0..100 from Seiichi Manyama)

Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320387.

Cf. A081489.

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
end
def f(n)
  return 1 if n == 0
  cnt = 0
  partition(n, 1, n).each{|ary|
    ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
    cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
  }
  cnt
end
def A320388(n)
  (0..n).map{|i| f(i)}
end
p A320388(50)

A375041 Irregular triangular array T; row n shows the coefficients of the (n-1)-st polynomial in the obverse convolution s(x)**t(x), where s(x) = n^2 x and t(x) = x+1. See Comments.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 8, 17, 10, 1, 18, 97, 180, 100, 1, 35, 403, 1829, 3160, 1700, 1, 61, 1313, 12307, 50714, 83860, 44200, 1, 98, 3570, 60888, 506073, 1960278, 3147020, 1635400, 1, 148, 8470, 239388, 3550473, 27263928, 101160920, 158986400, 81770000, 1, 213

Offset: 1

Clark Kimberling, Sep 11 2024

See A374848 for the definition of obverse convolution and a guide to related sequences and arrays.

			First 3 polynomials in s(x)**t(x) are
  1 + x,
  1 + 3 x + 2 x^2,
  1 + 8 x + 17 x^2 + 10 x^3.
First 5 rows of array:
  1    1
  1    3     2
  1    8    17    10
  1   18    97   180   100
  1   35  4034  1829  3160  1700

Cf. A000290, A081489 (column 2), A101686 (T(n,n+1)), A374848, A375042, A375043.

s[n_] := n^2  x; t[n_] := 1 + x;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[Expand[u[n]], {n, 0, 10}]
Column[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]   (* array *)
Flatten[Table[CoefficientList[Expand[u[n]], x], {n, 0, 10}]]  (* sequence *)

A144337 Triangle read by rows, A000012 * (2*A144328 - A000012).

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 4, 3, 6, 5, 5, 4, 9, 10, 7, 6, 5, 12, 15, 14, 9, 7, 6, 15, 20, 21, 18, 11, 8, 7, 18, 25, 28, 27, 22, 13, 9, 8, 21, 30, 35, 36, 33, 26, 15, 10, 9, 24, 35, 42, 45, 44, 39, 30, 17, 11, 10, 27, 40, 49, 54, 55, 52, 45, 34, 19

Offset: 1

Gary W. Adamson and Roger L. Bagula, Sep 18 2008

When the first column is removed from this triangle (A144337), the result is triangle A101447. - Georg Fischer, Aug 10 2023

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  3;
   4,  3,  6,  5;
   5,  4,  9, 10,  7;
   6,  5, 12, 15, 14,  9;
   7,  6, 15, 20, 21, 18, 11;
   8,  7, 18, 25, 28, 27, 22, 13;
   9,  8, 21, 30, 35, 36, 33, 26, 15;
  10,  9, 24, 35, 42, 45, 44, 39, 30, 17;
  ...

Cf. A081489, A101447, A144328.

a(25) corrected by Georg Fischer, Aug 10 2023

A159938 The number of homogeneous trisubstituted linear alkanes.

Original entry on oeis.org

2, 6, 16, 36, 70, 122, 196, 296, 426, 590, 792, 1036, 1326, 1666, 2060, 2512, 3026, 3606, 4256, 4980, 5782, 6666, 7636, 8696, 9850, 11102, 12456, 13916, 15486, 17170, 18972, 20896, 22946, 25126, 27440, 29892, 32486

Offset: 2

Parthasarathy Nambi, Apr 26 2009

See the paper by Valentin Vankov Iliev for details.

This sequence is related to A152947 by a(n) = (n-1)*A152947(n) + sum( A152947(i), i=1..n-1 ). - Bruno Berselli, Dec 19 2013

			The number of homogeneous trisubstituted linear alkane with ten carbon atoms is 426.

Valentin Vankov Iliev, A mathematical characterization of the groups of substitution isomerism of the linear alkanes, J. Math. Chem. 47 (2010), 52-61.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002522, A033816, A081489, A152947, A159940, A159941.

a(n) = (1/3)*(2*n^3 - 9*n^2 + 19*n - 12), where n is the number of carbons.
a(n) = 2*A081489(n-1) = (n-1)*(2*n^2-7*n+12)/3. - R. J. Mathar, Apr 28 2009
G.f.: 2*x^2*(1-x+2*x^2)/(1-x)^4. - Colin Barker, Aug 06 2012

More terms from Colin Barker, Aug 06 2012

cp's OEIS Frontend

A121997 Count up to n, n times.

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A145066 Partial sums of A002522, starting at n=1.

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A081490 Leading term of n-th row of A081491.

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A081491 Triangle read by rows in which the n-th row contains n terms of an arithmetic progression with a common difference of (n-1) and the first term of (n+1)-th row is 1 more than the last term of the n-th row.

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A081492 Sum of terms in n-th row of A081491.

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A116701 Number of permutations of length n that avoid the patterns 132, 4321.

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A320388 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing.