A275874 -id:A275874 - cp's OEIS frontend

A137742 a(n) = (n-1)(n+4)(n+6)/6 for n > 1, a(1)=1.

1, 8, 21, 40, 66, 100, 143, 196, 260, 336, 425, 528, 646, 780, 931, 1100, 1288, 1496, 1725, 1976, 2250, 2548, 2871, 3220, 3596, 4000, 4433, 4896, 5390, 5916, 6475, 7068, 7696, 8360, 9061, 9800, 10578, 11396, 12255, 13156, 14100, 15088, 16121, 17200, 18326, 19500

Offset: 1

M. F. Hasler, Feb 10 2008

Also the number of different strings of length n+3 obtained from "123...n" by iteratively duplicating any substring (see A137743 for comments and examples). This is the principal (although not simplest) definition of this sequence and explains why a(1)=1 and not 0.

For n >= 3, sequence appears (not yet proved by induction) to give the number of multiplications between two nonzero matrix elements in calculating the product of two n X n Hessenberg matrices (square matrices which have 0's below the subdiagonal, other elements being in general nonzero). - John M. Coffey, Jun 21 2016

			a(5) = (5-1)*(5+4)*(5+6)/6 = 4*9*11/6 = 66. - _Michael B. Porter_, Jul 02 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for doubling substrings
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A137740-A137743, A135473, A137744-A137748.

See A275874 for another version.

[1] cat [(n^3+9*n^2+14*n-24)/6: n in [2..46]];  // Bruno Berselli, Aug 23 2011

Join[{1},Table[Binomial[n,3]-2*n,{n,6,60}]] (*or*) Join[{1},Table[(n-1)(n+4)(n+6)/6,{n,2,56}]] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2011 *)

A137742(n)=if(n<2,1,n=A135473(n+3,n);n[ #n]) /* function A135473 defined in A137743 */

A137742(n)=if(n<2,1,(n - 1)*(n + 4)*(n + 6)/6)

From Bruno Berselli, Aug 23 2011: (Start)
G.f.: x*(1+4*x-5*x^2+x^4)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(-n-7) = -A000297(n).  (End)
From Ilya Gutkovskiy, Jul 01 2016: (Start)
E.g.f.: 4 + x + (-24 + 24*x + 12*x^2 + x^3)*exp(x)/6.
Sum_{n>=1} 1/a(n) = 1542/1225. (End)
a(n) = binomial(n+4,3) - 2*(n+4) for n > 1. - Michael Chu, Dec 09 2021

A323221 a(n) = n(n + 5)(n + 7)/6 + 1.

Original entry on oeis.org

1, 9, 22, 41, 67, 101, 144, 197, 261, 337, 426, 529, 647, 781, 932, 1101, 1289, 1497, 1726, 1977, 2251, 2549, 2872, 3221, 3597, 4001, 4434, 4897, 5391, 5917, 6476, 7069, 7697, 8361, 9062, 9801, 10579, 11397, 12256, 13157, 14101, 15089, 16122, 17201, 18327, 19501

Offset: 0

Peter Luschny, Jan 25 2019

a(n) is related to the total angular defect of certain polytopes. See Hilton and Pedersen, Cor. 1; compare A275874.

			For n = 2 the sum formula gives:
I(2) = {{0,0}, {0,1}, {1,0}, {0,2}, {1,1}, {2,0}, {0,3}, {1,2}, {2,1}, {3,0}};
a(2) = 1 + 1 + 1 + 2 + 1 + 2 + 5 + 2 + 2 + 5 = 22.

P. Hilton and J. Pedersen, Descartes, Euler, Poincaré, Pólya and Polyhedra, L'Enseign. Math., 27 (1981), 327-343.

Çf. A323224 (column 4), A323233 (row 4), A034856 (first difference), A275874.

a := n -> n*(35 + 12*n + n^2)/6 + 1:
seq(a(n), n = 0..45);

a[n_] := n (35 + 12 n + n^2)/6 + 1;
Table[a[n], {n, 0, 45}]

Let I(n) denote the set of all tuples of length n with elements from {0, 1, 2, 3} with sum <= 3 and C(m) denote the m-th Catalan number. Then for n > 0
a(n) = Sum_{(j1,...,jn) in I(n)} C(j1)*C(j2)*...*C(jn).
a(n) = [x^n] (3*x^3 - 8*x^2 + 5*x + 1)/(x - 1)^4.
a(n) = n! [x^n] exp(x)*(x^3 + 15*x^2 + 48*x + 6)/6.
a(n) = a(n - 1)*(n*(n + 5)*(n + 7) + 6)/(n*(n + 2)*(n + 7) - 18) for n > 0.
a(n) = A323224(n, 4).
a(n) = A275874(n+4) + 1.

A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

Original entry on oeis.org

1, 1, 1, 1, 2, -1, 1, 3, -1, 1, 1, 4, 0, 0, -1, 1, 5, 2, -2, 1, 1, 1, 6, 5, -4, 3, -2, -1, 1, 7, 9, -5, 3, -3, 3, 1, 1, 8, 14, -4, 0, 0, 2, -4, -1, 1, 9, 20, 0, -6, 6, -4, 0, 5, 1, 1, 10, 27, 8, -14, 12, -10, 8, -3, -6, -1, 1, 11, 35, 21, -22, 14, -10, 10, -11, 7, 7, 1

Offset: 0

Kevin Ryde, Aug 21 2021

Doche and Mendès France form polynomials P_n(y) = Sum_{j=0..2^n-1} GRS(j) * y^wt(j) and here row n is the coefficients of P_n starting from the constant term, so P_n(y) = Sum_{k=0..n} T(n,k)*y^k. They conjecture that the number of real roots of P_n is A285869(n).

Row sum n is the sum of GRS terms from j = 0 to 2^n-1 inclusive, which Brillhart and Morton (Beispiel 6 page 129) show is A020986(2^n-1) = 2^ceiling(n/2) = A060546(n). The same follows by substituting y=1 in the P_n recurrence or the generating function.

			Triangle begins
        k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=0:   1
  n=1:   1,  1
  n=2:   1,  2, -1
  n=3:   1,  3, -1,  1
  n=4:   1,  4,  0,  0, -1
  n=5:   1,  5,  2, -2,  1,  1
  n=6:   1,  6,  5, -4,  3, -2, -1
  n=7:   1,  7,  9, -5,  3, -3,  3,  1
For T(5,3), those j in the range 0 <= j < 2^5 with wt(j) = 3 are
  j      =  7 11 13 14 19 21 22 25 26 28
  GRS(j) = +1 -1 -1 +1 -1 +1 -1 -1 -1 +1 total -2 = T(5,3)

Kevin Ryde, Table of n, a(n) for rows 0 to 100, flattened
John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, Illinois Journal of Mathematics, volume 22, issue 1, 1978, pages 126-148.
Christophe Doche and Michel Mendès France, An Exercise on the Average Number of Real Zeros of Random Real Polynomials, Finite and Infinite Combinatorics conference, Budapest, 2001, pages 1-14, see Rudin-Shapiro example page 9.

Cf. A020985 (GRS), A020986 (GRS partial sums), A000120 (binary weight), A285869.
Columns k=0..3: A000012, A001477, A000096, A275874.
Cf. A165326 (main diagonal), A248157 (second diagonal negated).
Cf. A060546 (row sums), A104969 (row sums squared terms).
Cf. A329301 (antidiagonal sums).
Cf. A104967 (rows reversed, up to signs).

my(M=Mod('x, 'x^2-(1-'y)*'x-2*'y)); row(n) = Vecrev(subst(lift(M^n),'x,'y+1));

T(n,k) = T(n-1,k) - T(n-1,k-1) + 2*T(n-2,k-1) for n>=2, and taking T(n,k)=0 if k<0 or k>n.
T(n,k) = (-1)^k * A104967(n,n-k).
Row polynomial P_n(y) = (1-y)*P_{n-1}(y) + 2*y*P_{n-2}(y) for n>=2. [Doche and Mendès France]
G.f.: (1 + 2*x*y)/(1 + x*(y-1) - 2*x^2*y).
Column g.f.: C_k(x) = 1/(1-x) for k=0 and C_k(x) = x^k * (2*x-1)^(k-1) / (1-x)^(k+1) for k>=1.

cp's OEIS Frontend

A137742 a(n) = (n-1)(n+4)(n+6)/6 for n > 1, a(1)=1.

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A323221 a(n) = n(n + 5)(n + 7)/6 + 1.

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A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A137742 a(n) = (n-1)*(n+4)*(n+6)/6 for n > 1, a(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A323221 a(n) = n*(n + 5)*(n + 7)/6 + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A137742 a(n) = (n-1)(n+4)(n+6)/6 for n > 1, a(1)=1.

A323221 a(n) = n(n + 5)(n + 7)/6 + 1.