A022856 -id:A022856 - cp's OEIS frontend

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 3, 2, 3, 3, 1, 0, 3, 4, 5, 4, 1, 4, 3, 5, 7, 8, 5, 1, 0, 4, 6, 9, 12, 12, 6, 1, 5, 4, 7, 11, 16, 20, 17, 7, 1, 0, 5, 8, 13, 20, 28, 32, 23, 8, 1, 6, 5, 9, 15, 24, 36, 48, 49, 30, 9, 1, 0, 6, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 7, 6, 11, 19, 32, 52, 80, 112, 129

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n,0)=T(n,2) by T(2n,0)=T(n,m) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058393, A058395, A057884 (and effectively A007318).

			Rows are (1,0,2,0,3,0,4,...), (1,1,2,2,3,3,...), (1,2,3,4,5,6,...), (1,3,5,7,9,11,...), etc.

Rows are A027656 (A000027 with zeros), A008619, A000027, A005408, A008574 etc. Columns are A000012, A001477, A022856 etc. Diagonals include A034007, A045891, A045623, A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 etc. The triangle A055249 also appears in half of the array.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(2n, 0)=T(n, 2) and T(2n+1, 0)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^2.

A067337 Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 9, 1, 4, 8, 14, 27, 1, 5, 12, 22, 41, 81, 1, 6, 17, 34, 63, 122, 243, 1, 7, 23, 51, 97, 185, 365, 729, 1, 8, 30, 74, 148, 282, 550, 1094, 2187, 1, 9, 38, 104, 222, 430, 832, 1644, 3281, 6561, 1, 10, 47, 142, 326, 652, 1262, 2476, 4925, 9842

Offset: 0

Henry Bottomley, Jan 15 2002

			Rows start 1; 1,1; 1,2,3; 1,3,5,9; 1,4,8,14,27; etc. T(4,0)=2*0+1-0=1; T(4,1)=2*1+3-1=4; T(4,2)=2*4+3-3=8; T(4,3)=2*8+1-3=14; T(4,4)=2*14+0-1=27.

Row sums are A025192. Columns include A000012, A000027 and A022856 (essentially). Right hand columns include A000244 (essentially), A007051 and A047926. Central diagonal is A067336.

T(n, k)=2*T(n, k-1)+A037012(n, k). T(n, k)=T(n-1, k-1)+T(n-1, k) if k0.

A354902 a(n) = 2n^2 - 6n + 11 for n > 1 with a(0)=1 and a(1)=3.

Original entry on oeis.org

1, 3, 7, 11, 19, 31, 47, 67, 91, 119, 151, 187, 227, 271, 319, 371, 427, 487, 551, 619, 691, 767, 847, 931, 1019, 1111, 1207, 1307, 1411, 1519, 1631, 1747, 1867, 1991, 2119, 2251, 2387, 2527, 2671, 2819, 2971, 3127, 3287, 3451, 3619, 3791, 3967, 4147, 4331, 4519, 4711, 4907

Offset: 0

Sumukh Patel, Jun 11 2022

a(n) is the minimum number of nodes required for a full binary tree where each node in all longest paths from the root node down to any leaf node is height-balanced and the root node has a height balance factor of 0.

Full binary tree: A binary tree is called a full binary tree if each node has exactly two children or no children.

			The diagrams below illustrate the terms a(3)=11 and a(4)=19.
           R                         R
          / \                       / \
         /   \                     /   \
        /     \                   /     \
       o       o                 /       \
      / \     / \               /         \
     o   N   N   o             /           \
    / \         / \           /             \
   N   N       N   N         o               o
                            / \             / \
                           /   \           /   \
                          /     \         /     \
                         o       o       o       o
                        / \     / \     / \     / \
                       o   N   N   N   N   N   o   N
                      / \                     / \
                     N   N                   N   N

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Lecture Notes for Computer Science 2530, Height-balanced trees
NIST, Root node
NIST, Leaf node
Wikipedia, Full binary tree
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A022856, A027688.

int a(int n){ return n>1 ? 2*(n*n) - 6*n + 11 : 2*n + 1; }

CoefficientList[Series[(1 + x^2 - 2 x^3 + 4 x^4)/(1 - x)^3, {x, 0, 51}], x] (* Michael De Vlieger, Jun 19 2022 *)

a(n) = 2*A027688(n-2) + 1, for n >= 2.
a(n) = 4*A022856(n+2) - 1, for n >= 1.
a(n) = a(n-1) + 4*(n-2) for n >= 3.
G.f.: (1 + x^2 - 2*x^3 + 4*x^4)/(1 - x)^3. - Stefano Spezia, Jun 12 2022
Sum_{n>=2} 1/a(n) = Pi*tanh(sqrt(13)*Pi/2)/(2*sqrt(13)). - Amiram Eldar, Jul 10 2022

A026041 a(n) = d(n)/2, where d = A026040.

Original entry on oeis.org

12, 20, 32, 49, 72, 102, 140, 187, 244, 312, 392, 485, 592, 714, 852, 1007, 1180, 1372, 1584, 1817, 2072, 2350, 2652, 2979, 3332, 3712, 4120, 4557, 5024, 5522, 6052, 6615, 7212, 7844, 8512, 9217, 9960, 10742, 11564, 12427, 13332, 14280, 15272, 16309, 17392, 18522, 19700, 20927, 22204, 23532, 24912, 26345

Offset: 4

Clark Kimberling

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

[n*(n^2-3*n+14)/6: n in [4..60]]; // Vincenzo Librandi, Oct 17 2013

CoefficientList[Series[-(7 x^3 - 24 x^2 + 28 x - 12)/(x - 1)^4, {x, 0, 60}], x] (* Vincenzo Librandi, Oct 17 2013 *)

a(n)=n*(n^2-3*n+14)/6 \\ Charles R Greathouse IV, Oct 18 2022

For n>4, a(n) = a(n-1)+A022856(n+2). - Levi R. Self (levi.r.self(AT)gmail.com), Aug 04 2007
G.f.: -x^4*(7*x^3-24*x^2+28*x-12)/(x-1)^4. - Colin Barker, Oct 07 2012
a(n) = n*(n^2-3*n+14)/6. - Vincenzo Librandi, Oct 17 2013

A318054 a(n) = n(n + 1)(n^2 + n + 22)/24.

Original entry on oeis.org

0, 2, 7, 17, 35, 65, 112, 182, 282, 420, 605, 847, 1157, 1547, 2030, 2620, 3332, 4182, 5187, 6365, 7735, 9317, 11132, 13202, 15550, 18200, 21177, 24507, 28217, 32335, 36890, 41912, 47432, 53482, 60095, 67305, 75147, 83657, 92872, 102830, 113570, 125132, 137557

Offset: 0

Luce ETIENNE, Aug 14 2018

			a(1) = 2; a(2)= 5+2 = 7; a(3) = 10+5+2 = 17; a(4) = 18+10+5+2 = 35; a(5) = 30+18+10+5+2 = 65; a(6) = 47+30+18+10+5+2 = 112.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Tenner, Bridget Eileen Reduced word manipulation: patterns and enumeration, J. Algebr. Comb. 46, No. 1, 189-217 (2017), w in S_n(231): l(w)=4.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A177787.

Cf. A089071, A022856, A152948.

List([0..30],n->n*(n+1)*(n^2+n+22)/24); # Muniru A Asiru, Aug 15 2018

seq(coeff(series(x*(2*x^2-3*x+2)/(1-x)^5, x,n+1),x,n),n=0..30); # Muniru A Asiru, Aug 15 2018

a(n) = n*(n+1)*(n^2+n+22)/24; \\ Michel Marcus, Aug 17 2018

G.f.: x*(2*x^2-3*x+2)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = (1/6)*Sum_{i=1..n} (n-i)*((n-i)^2+11), for n >= 1.

A062507 Table by antidiagonals related to partial sums and differences of Pell numbers (A000129).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 10, 7, 5, 3, 1, 0, 24, 17, 12, 8, 4, 1, 0, 58, 41, 29, 20, 12, 5, 1, 0, 140, 99, 70, 49, 32, 17, 6, 1, 0, 338, 239, 169, 119, 81, 49, 23, 7, 1, 0, 816, 577, 408, 288, 200, 130, 72, 30, 8, 1, 0, 1970, 1393, 985, 696, 488, 330, 202, 102

Offset: 0

Henry Bottomley, Jul 09 2001

			Rows start (0,1,0,2,4,10,...), (0,1,1,3,7,17,...), (0,1,2,5,12,29,...) etc.

Rows are effectively A052542, A001333, A000129, A048739, A048776. Columns are effectively A000004, A000012, A001477, A022856.

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

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 4, 3, 5, 3, 1, 0, 7, 8, 8, 4, 1, 8, 7, 15, 16, 12, 5, 1, 0, 15, 22, 31, 28, 17, 6, 1, 16, 15, 37, 53, 59, 45, 23, 7, 1, 0, 31, 52, 90, 112, 104, 68, 30, 8, 1, 32, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 0, 63, 114, 225

Offset: 0

Philippe Deléham, Jan 11 2014

Row sums are A007179(n+1).

			Triangle begins (0<=k<=n):
1
0, 1
2, 1, 1
0, 3, 2, 1
4, 3, 5, 3, 1
0, 7, 8, 8, 4, 1
8, 7, 15, 16, 12, 5, 1
0, 15, 22, 31, 28, 17, 6, 1

Cf. Columns: A077957, A052551, A077866.
Diagonals: A000012, A001477, A022856.
Cf. Similar sequences: A059260, A191582.

T(n,n)=1, T(2n,0)=2^n, T(2n+1,0)=0, T(n,k)=T(n-1,k-1)+T(n-1,k) for 0

T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-3,k)-2*T(n-3,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

T(n,n)=1, T(n+1,n)=n, T(n+2,n)=n*(n+1)/2 + 2.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3*x + 2*x^2/2! + x^3/3!) = 3*x + 8*x^2/2! + 16*x^3/3! + 28*x^4/4! + 45*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

Previous Showing 11-17 of 17 results.

cp's OEIS Frontend

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A067337 Triangle where T(n,k)=2*T(n,k-1)+C(n-1,k)-C(n-1,k-1) and n>=k>=0.

Views

Author

Keywords

Examples

Crossrefs

Formula

A354902 a(n) = 2*n^2 - 6*n + 11 for n > 1 with a(0)=1 and a(1)=3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Mathematica

Formula

A026041 a(n) = d(n)/2, where d = A026040.

Views

Author

Keywords

Links

Programs

Magma

Mathematica

PARI

Formula

A318054 a(n) = n*(n + 1)*(n^2 + n + 22)/24.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Maple

PARI

Formula

A062507 Table by antidiagonals related to partial sums and differences of Pell numbers (A000129).

Views

Author

Keywords

Examples

Crossrefs

Formula

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A354902 a(n) = 2n^2 - 6n + 11 for n > 1 with a(0)=1 and a(1)=3.

A318054 a(n) = n(n + 1)(n^2 + n + 22)/24.