A004652 -id:A004652 - cp's OEIS frontend

A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

1, 5, 2, 11, 6, 3, 19, 12, 8, 4, 29, 20, 15, 10, 7, 41, 30, 24, 18, 14, 9, 55, 42, 35, 28, 23, 17, 13, 71, 56, 48, 40, 34, 27, 22, 16, 89, 72, 63, 54, 47, 39, 33, 26, 21, 109, 90, 80, 70, 62, 53, 46, 38, 32, 25, 131, 110, 99, 88, 79, 69, 61, 52, 45, 37, 31, 155, 132, 120, 108

Offset: 1

Clark Kimberling, Jul 24 2009

A permutation of the natural numbers.
Beginning at row 6, the columns obey a 3rd-order recurrence:
c(n)=c(n-1)+c(n-2)-c(n-3)+1.
Except for initial terms, the first seven rows are A028387, A002378, A005563, A028552, A008865, A014209, A028873, and the first column, A004652.

			Corner:
1....5...11...19
2....6...12...20
3....8...15...24
4...10...18...28
The double interspersion A161179 begins thus:
1....4....7...12...17...24
2....3....8...11...18...23
5....6...13...16...25...30
9...10...19...22...33...38
Expel the odd-numbered columns and then swap rows 1 and 2, leaving
3....11...23...39
4....12...24...40
6....16...30...48
10...22...38...58
Then replace each of those numbers by its rank when all the numbers are jointly ranked.

Clark Kimberling, Doubly interspersed sequences, double interspersions and fractal sequences, The Fibonacci Quarterly 48 (2010) 13-20.

Cf. A163253, A163254, A163255, A163256, A163258.

Let S(n,k) denote the k-th term in the n-th row. Four cases:
S(1,k)=k^2+k-1
S(2,k)=k^2+k
if n>1 is odd, then S(n,k)=k^2+(n-1)k+(n-1)(n-3)/4
if n>2 is even, then S(n,k)= k^2+(n-1)k+n(n-4)/4.

A175287 Partial sums of ceiling(n^2/4).

Original entry on oeis.org

0, 1, 2, 5, 9, 16, 25, 38, 54, 75, 100, 131, 167, 210, 259, 316, 380, 453, 534, 625, 725, 836, 957, 1090, 1234, 1391, 1560, 1743, 1939, 2150, 2375, 2616, 2872, 3145, 3434, 3741, 4065, 4408, 4769, 5150, 5550, 5971, 6412, 6875, 7359, 7866, 8395, 8948, 9524, 10125, 10750

Offset: 0

Mircea Merca, Dec 03 2010

a(n) is the number of 1243-avoiding odd Grassmannian permutations of size n+1. Avoiding any of the patterns 2134, 2341, or 4123, gives the same sequence. - Juan B. Gil, Mar 09 2023

Conjecture: a(n) is the number of perimeter-magic (hollow) triangles of order 3 with magic sum n+2. Order 3 means each of the 3 edges has 3 elements >=1; the triangle has 6 elements. The elements do not need to be distinct, and triangles obtained by rotations are counted only once. The triangle (read ccw) for magic sum 3 has elements 1 1 1 1 1 1. The 2 triangles with magic sum 4 are 1 1 2 1 1 2 and 1 2 1 2 1 2. - R. J. Mathar, Mar 08 2025

			a(4) = ceil(0/4)+ceil(1/4)+ceil(4/4)+ceil(9/4)+ceil(16/4) = 0+1+1+3+4=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Partial sums of A004652.

Cf. A361272.

[Floor((n+1)*(2*n^2+n+9)/24): n in [0..60]]; // Vincenzo Librandi, Jun 22 2011

a:= n-> round((2*n^(3)+3*n^(2)+10*n)/24): seq(a(n), n=0..20);

Table[Sum[Ceiling[i^2/4], {i, 0, n}], {n, 0, 49}] (* or *) Table[(2n(2n^2 + 3n + 10) -9(-1)^n + 9)/48, {n, 0, 49}] (* Alonso del Arte, Dec 03 2010 *)
CoefficientList[Series[(x^3 - x^2 + x)/(x^5 - 3 x^4 + 2 x^3 + 2 x^2 - 3 x + 1), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
Accumulate[Ceiling[Range[0,50]^2/4]] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,2,5,9},60] (* Harvey P. Dale, Nov 19 2014 *)

x='x+O('x^99); concat(0, Vec((x^3-x^2+x)/ (x^5-3*x^4+2*x^3+2*x^2-3*x+1))) \\ Altug Alkan, Apr 05 2016

a(n) = round((2*n+1)*(2*n^2+2*n+9)/48).
a(n) = floor((n+1)*(2*n^2+n+9)/24).
a(n) = ceiling((2*n^3+3*n^2+10*n)/24).
a(n) = round((2*n^3+3*n^2+10*n)/24).
a(n) = a(n-4)+n^2-3*n+5 , n>3.
G.f.: x*(1-x+x^2) / ( (1+x)*(x-1)^4 ).
a(n) = (2*n*(2*n^2+3*n+10)-9*(-1)^n+9)/48. - Bruno Berselli, Dec 03 2010
a(n)+a(n+1) = A004006(n+1). - R. J. Mathar, Mar 08 2025

A035107 First differences give (essentially) A028242.

Original entry on oeis.org

3, 9, 17, 29, 44, 64, 88, 118, 153, 195, 243, 299, 362, 434, 514, 604, 703, 813, 933, 1065, 1208, 1364, 1532, 1714, 1909, 2119, 2343, 2583, 2838, 3110, 3398, 3704, 4027, 4369, 4729, 5109, 5508, 5928, 6368, 6830, 7313, 7819, 8347, 8899, 9474

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A028242, A004652, A035104, A035106.

[(4*n^3+54*n^2+212*n+153-9*(-1)^n)/48: n in [0..50]]; // Vincenzo Librandi, Oct 21 2013

LinearRecurrence[{3,-2,-2,3,-1},{3,9,17,29,44},50] (* Harvey P. Dale, Oct 20 2013 *)
CoefficientList[Series[(2 x^3 - 4 x^2 + 3)/((x - 1)^4 (x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 21 2013 *)

a(n) = (4*n^3 +54*n^2 +212*n +153 -9*(-1)^n)/48.

G.f.: (2*x^3-4*x^2+3) / ((x-1)^4*(x+1)). - Colin Barker, Mar 04 2013

A275799 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and three squares have one of the colors.

Original entry on oeis.org

1, 22, 140, 578, 1785, 4612, 10416, 21340, 40425, 72010, 121836, 197582, 308945, 468328, 690880, 995352, 1404081, 1944030, 2646700, 3549370, 4694921, 6133292, 7921200, 10123828, 12814425, 16076242, 20001996, 24696070, 30273825, 36864080

Offset: 2

Wolfdieter Lang, Oct 03 2016

See the k=3 column of table A054772(n, k), with more explanations there.

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3,-8,14,0,-14,8,3,-4,1).

Cf. A054772, A000012 (k=0), A004652 (k=1), A212714 (k=2).

Vec(x^2*(1+18*x+55*x^2+92*x^3+55*x^4+18*x^5+x^6)/((1-x)^7*(1+x)^3) + O(x^40)) \\ Colin Barker, Oct 16 2016

a(n) = A054772(n, 3) = A054772(n, n^2-3), n >= 2.
From Colin Barker, Oct 09 2016: (Start)
G.f.: x^2*(1+18*x+55*x^2+92*x^3+55*x^4+18*x^5+x^6) / ((1-x)^7*(1+x)^3).
a(n) = (n^6-3*n^4+2*n^2)/24 for n even.
a(n) = (n^6-3*n^4+5*n^2-3)/24 for n odd. (End)
From Stefan Hollos, Oct 16 2016: (Start)
a(n) = C(n^2,3)/4 for n even,
a(n) = (C(n^2,3) + (n^2-1)/2)/4 for n odd. (End)

A280340 a(n) = a(n-1) + 10^n * a(n-2) with a(0) = 1 and a(1) = 1.

Original entry on oeis.org

1, 1, 101, 1101, 1011101, 111111101, 1011212111101, 1112122222111101, 101122323232322111101, 1112223344434333322111101, 1011224344546565545343322111101, 111223345667777878776655443322111101, 1011224455769911213121200887756443322111101

Offset: 0

Seiichi Manyama, Dec 31 2016

nonn

The Rogers-Ramanujan continued fraction is defined by R(q) = q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+ ... )))). The limit of a(n)/A015468(n+2) is 10^(-1/5) * R(10).

a(n) has A004652(n+1) digits. The last n digits are the same as the last n digits of a(n-1). - Robert Israel, Jan 12 2017

			1/1 = a(0)/A015468(2).
1/(1+10/1) = 1/11 = a(1)/A015468(3).
1/(1+10/(1+10^2/1)) = 101/111 = a(2)/A015468(4).
1/(1+10/(1+10^2/(1+10^3/1))) = 1101/11111 = a(3)/A015468(5).

Seiichi Manyama, Table of n, a(n) for n = 0..62
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A004652, A015468, A128915.

Cf. similar sequences with the recurrence a(n-1) + q^n * a(n-2) for n>1, a(0)=1 and a(1)=1: A280294 (q=2), A279543 (q=3), this sequence (q=10).

A[0]:= 1: A[1]:= 1:
for n from 2 to 20 do A[n]:= A[n-1]+10^n*A[n-2] od:
seq(A[i],i=0..20); # Robert Israel, Jan 12 2017

RecurrenceTable[{a[0]==a[1]==1,a[n]==a[n-1]+10^n a[n-2]},a,{n,15}] (* Harvey P. Dale, Jul 12 2020 *)

a(n) a(n-3) = 10 a(n-2) a(n-1) - 10 a(n-2)^2 + a(n-1) a(n-3). - Robert Israel, Jan 12 2017

A097066 Expansion of (1-2x+2x^2)/((1+x)*(1-x)^3).

Original entry on oeis.org

1, 0, 2, 2, 5, 6, 10, 12, 17, 20, 26, 30, 37, 42, 50, 56, 65, 72, 82, 90, 101, 110, 122, 132, 145, 156, 170, 182, 197, 210, 226, 240, 257, 272, 290, 306, 325, 342, 362, 380, 401, 420, 442, 462, 485, 506, 530, 552, 577, 600, 626, 650, 677, 702, 730, 756, 785, 812

Offset: 0

Paul Barry, Jul 22 2004

Partial sums of A097065. Pairwise sums are A000124, with extra leading 1.

Binomial transform is 1, 1, 3, 9, 26, ..., A072863 with extra leading 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A000124, A072863, A097065.

Cf. A004526, A004652.

List([0..70], n-> (2*n^2 +3 +5*(-1)^n)/8); # G. C. Greubel, Jun 30 2019

[(2*n^2 +3 +5*(-1)^n)/8: n in [0..70]]; // G. C. Greubel, Jun 30 2019

CoefficientList[Series[(1-2x+2x^2)/((1+x)(1-x)^3), {x, 0, 70}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {1, 0, 2, 2}, 70] (* Harvey P. Dale, Apr 08 2014 *)
Table[(2n^2 +3 +5(-1)^n)/8, {n,0,70}] (* Vincenzo Librandi, Apr 09 2014 *)

vector(70, n, n--; (2*n^2 +3 +5*(-1)^n)/8) \\ G. C. Greubel, Jun 30 2019

[(2*n^2 +3 +5*(-1)^n)/8 for n in (0..70)] # G. C. Greubel, Jun 30 2019

G.f.: (1-2*x+2*x^2)/((1-x^2)*(1-x)^2).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(n) = 5*(-1)^n/8 + (2*n^2+3)/8.
a(n) = A004652(n+1) - A004526(n+1) = ceiling(((n+1)/2)^2) - floor((n+1)/2). - Ridouane Oudra, Jun 22 2019
E.g.f.: ((4+x+x^2)*cosh(x) - (1-x-x^2)*sinh(x))/4. - G. C. Greubel, Jun 30 2019

A135840 A135839 * A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 2, 1, 1, 4, 3, 2, 2, 1, 1, 4, 3, 3, 2, 2, 1, 1, 5, 4, 3, 3, 2, 2, 1, 1, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1, 8, 7, 6, 6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 1

Gary W. Adamson, Dec 01 2007

nonn

Row sums = A004652 starting (1, 3, 4, 7, 9, 13, 16, 21, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  3, 2, 2, 1, 1;
  4, 3, 2, 2, 1, 1;
  4, 3, 3, 2, 2, 1, 1;
  5, 4, 3, 3, 2, 2, 1, 1;
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows

Cf. A135839, A004652, A135841.

T[1, 1] := 1; T[n_, 1] := Floor[(n + 2)/2]; T[n_, n_] := 1; T[n_, k_] := Floor[(n - k + 2)/2]; Table[T[n, k], {n, 1, 8}, {k, 1, n}]//Flatten (* G. C. Greubel, Dec 05 2016 *)

T(1, 1) = 1, T(n, 1) = floor((n + 2)/2), T(n, n) = 1, T(n, k) = floor((n - k + 2)/2). - G. C. Greubel, Dec 05 2016

A167753 Hankel transform of A167751.

Original entry on oeis.org

1, 1, 0, -1, -1, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1

Offset: 0

Paul Barry, Nov 10 2009

sign

The nonzero terms appear to be indexed by ceiling((n+1)^2/4) - 0^n (see A004652).

Cf. A167752.

A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

Original entry on oeis.org

0, 1, 1, 3, 4, 8, 10, 15, 19, 26, 31, 39, 46, 56, 64, 75, 85, 98, 109, 123, 136, 152, 166, 183, 199, 218

Offset: 3

Kival Ngaokrajang, May 25 2013

From the drawings as shown in links, it can be separated the distinct quadrilaterals into 3 types:
Q1: Quadrilaterals which have at least one side equal to n-gon sides length.
Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than n-gon sides length.
Q3: Quadrilaterals which have no parallel sides and all sides are longer than n-gon side length.
Q1(n) = A004652(n-3); Q2(n) = A001917(n-6), Q2(3) = 0, Q2(4) = 0; Q3(n) = A005232(n-10), Q3(3) = 0, Q3(4) = 0, Q3(5) = 0, Q3(6) = 0, Q3(7) = 0, Q3(8) = 0, Q3(9) = 0.
a(n) = Q1(n) + Q2(n). The total distinct quadrilaterals is Q1 + Q2 + Q3. Also the total distinct quadrilaterals = A005232(n-4), for n>=4. Also a(n) = A005232(n-4) - A005232(n-10), for n>=10.

			For a pentagon, there are 5 quadrilaterals which are the same size and shape. Therefore a(5) = 1.

Kival Ngaokrajang, The distinct quadrilaterals for n = 4..9
Kival Ngaokrajang, The distinct quadrilaterals for n = 10
Kival Ngaokrajang, Q1 & Q2 for n = 23
Kival Ngaokrajang, Q3 for n = 23

Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n-3) is number of distinct triangles in an n-gon.

Empirical g.f.: -x^4*(x^2-x+1)^2*(x^2+x+1) / ((x-1)^3*(x+1)*(x^2+1)). - Colin Barker, Oct 31 2013

A138585 The sequence is formed by concatenating subsequences S1, S2, ... each of finite length. S1 consists of the element 1. The n-th subsequence consist of numbers {(n/2)(n/2 - 1)+ 1, ..., (n/2)(n/2 + 1)} for n even, {((n-1)/2)^2, ..., (n-1)/2 * ((n-1)/2 + 2)} for n odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 4, 5, 6, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 9, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 17, 18, 19, 20, 16, 17, 18, 19, 20, 21, 22, 23, 24, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 31, 32, 33, 34, 35

Offset: 1

Ctibor O. Zizka, May 13 2008

A generalized Connell sequence.

Except for the first term the first element of each subsequence Sn (equivalently, each row of the triangle) gives A004652 (offset by 1), and the last element is A035106.

			S1: {1}
S2: {1,2}
S3: {1,2,3,}
S4: {3,4,5,6}
S5: {4,5,6,7,8}
S6: {7,8,9,10,11,12}, etc.
so concatenation of S1/S2/S3/S4/S5/S6/... gives:
1,1,2,1,2,3,3,4,5,6,4,5,6,7,8,7,8,9,10,11,12,...

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Douglas E. Iannucci and Donna Mills-Taylor. On Generalizing the Connell Sequence. Journal of Integer Sequences 2 (1999), Article 99.1.7.

Cf. A001614.

S := proc(n) local s: if(n=1)then s:=1: elif(n mod 2 = 0)then s:=(n/2)*(n/2 -1)+1: else s:=((n-1)/2)^2: fi: seq(k,k=s..s+n-1): end: seq(S(n),n=1..12); # Nathaniel Johnston, Oct 01 2011

Corrected and edited by D. S. McNeil, Dec 12 2010

cp's OEIS Frontend

A163257 An interspersion: the order array of the even-numbered columns (after swapping the first two rows) of the double interspersion at A161179.

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A175287 Partial sums of ceiling(n^2/4).

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A035107 First differences give (essentially) A028242.

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A275799 Number of inequivalent (modulo C_4 rotations) square n X n grids with squares coming in two colors and three squares have one of the colors.

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A280340 a(n) = a(n-1) + 10^n * a(n-2) with a(0) = 1 and a(1) = 1.

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A097066 Expansion of (1-2*x+2*x^2)/((1+x)*(1-x)^3).

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A135840 A135839 * A000012 as infinite lower triangular matrices.

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A097066 Expansion of (1-2x+2x^2)/((1+x)*(1-x)^3).

A138585 The sequence is formed by concatenating subsequences S1, S2, ... each of finite length. S1 consists of the element 1. The n-th subsequence consist of numbers {(n/2)(n/2 - 1)+ 1, ..., (n/2)(n/2 + 1)} for n even, {((n-1)/2)^2, ..., (n-1)/2 * ((n-1)/2 + 2)} for n odd.