A024462 -id:A024462 - cp's OEIS frontend

A038765 Next-to-last diagonal of A024462.

1, 2, 7, 24, 81, 270, 891, 2916, 9477, 30618, 98415, 314928, 1003833, 3188646, 10097379, 31886460, 100442349, 315675954, 990074583, 3099363912, 9685512225, 30218798142, 94143178827, 292889889684, 910050728661, 2824295364810

Offset: 0

N. J. A. Sloane, May 03 2000

If w is a binary string of length 2n-1 and v(w) is a vector of the Hamming weights of each substring of length n, then a(n) is the number of distinct v(w) for all possible w. - Orson R. L. Peters, Jun 01 2017

S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Maths.SE, Number of different counts of 1s in sliding windows.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A024462.

[1] cat [3^(n-2)*(n+5): n in [1..30]]; // Vincenzo Librandi, Oct 22 2013

Maple
```
seq(ceil(1/9*3^n*(5+n)),n=0..50);
```

CoefficientList[Series[(1 - 2 x)^2/(1 - 3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{6,-9},{1,2,7},30] (* Harvey P. Dale, Jul 04 2018 *)

G.f.: (1-2*x)^2/(1-3*x)^2. [Detlef Pauly (dettodet(AT)yahoo.de), Mar 03 2003]

a(n) = 6*a(n-1)-9*a(n-2) for n>2. a(n) = 3^(n-2)*(n+5) for n>0. [Colin Barker, Jun 25 2012]

More terms from James Sellers, May 03 2000

A038764 a(n) = (9n^2 + 3n + 2)/2.

Original entry on oeis.org

1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, 562, 667, 781, 904, 1036, 1177, 1327, 1486, 1654, 1831, 2017, 2212, 2416, 2629, 2851, 3082, 3322, 3571, 3829, 4096, 4372, 4657, 4951, 5254, 5566, 5887, 6217, 6556, 6904, 7261, 7627, 8002, 8386, 8779, 9181

Offset: 0

N. J. A. Sloane, May 03 2000

Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004

a(n) is also the least weight of self-conjugate partitions having n+1 different parts such that each part is congruent to 1 modulo 3. The first such self-conjugate partitions, corresponding to a(n) = 0, 1, 2, 3, are 1, 4+3, 7+4+4+4+3, 10+7+7+7+4+4+4+3. - Augustine O. Munagi, Dec 18 2008

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Reflection of A060544 in A081272.
Second column of A024462. Also = A064641(n+1, 2).
Shallow diagonal of triangular spiral in A051682.
Cf. A027468, A080855, A283394.
Partial sums of A122709.

LinearRecurrence[{3, -3, 1}, {1, 7, 22}, 50] (* Paolo Xausa, Jul 03 2025 *)

a(n)=n*(9*n+3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jan 22 2018

a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014

a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).
From Paul Barry, Mar 15 2003: (Start)
G.f.: (1 + 2*x)^2/(1 - x)^3.
Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. -  Colin Barker, Jan 22 2018
a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = hypergeometric([-n, -2], [1], 3). - Peter Luschny, Nov 19 2014
E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - Stefano Spezia, Mar 07 2023

More terms from James Sellers, May 03 2000

Entry revised by N. J. A. Sloane, Jan 23 2018

A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 7, 15, 9, 1, 10, 36, 54, 27, 1, 13, 66, 162, 189, 81, 1, 16, 105, 360, 675, 648, 243, 1, 19, 153, 675, 1755, 2673, 2187, 729, 1, 22, 210, 1134, 3780, 7938, 10206, 7290, 2187, 1, 25, 276, 1764, 7182, 19278, 34020, 37908, 24057, 6561, 1, 28, 351, 2592, 12474, 40824, 91854, 139968, 137781, 78732, 19683

Offset: 0

N. J. A. Sloane, May 03 2000

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005
Triangle read by rows, n-th row = X^(n-1) * [1, 1, 0, 0, 0, ...] where X = an infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (3,3,3,...) in the subdiagonal; given row 0 = 1. - Gary W. Adamson, Jul 19 2008
Fusion of polynomial sequences P and Q given by p(n,x)=(x+2)^n and q(n,x)=(2x+1)^n; see A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. - Clark Kimberling, Aug 04 2011

			Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1,  7,  15,   9;
  1, 10,  36,  54,   27;
  1, 13,  66, 162,  189,   81;
  1, 16, 105, 360,  675,  648,  243;
  1, 19, 153, 675, 1755, 2673, 2187, 729;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

Cf. A000007, A000012, A001296, A006138, A006234, A016777, A024462.
Cf. A051836, A062741, A080420, A080421, A080422, A080423, A081294.
Cf. A084938, A098399, A110523, A133494, A136158, A193722.

A038763:= func< n,k | n eq 0 select 1 else 3^(k-1)*(3*n-2*k)*Binomial(n,k)/n >;
[A038763(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 27 2023

A038763[n_,k_]:= If[n==0, 1, 3^(k-1)*(3*n-2*k)*Binomial[n,k]/n];
Table[A038763[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2023 *)

T(n,k) = if ((n<0) || (k<0), return(0)); if ((n==0) && (k==0), return(1)); if (n==1, if (k<=1, return(1))); T(n-1,k) + 3*T(n-1,k-1);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "))); \\ Michel Marcus, Jul 25 2023

def A038763(n,k): return 1 if (n==0) else 3^(k-1)*(3*n-2*k)*binomial(n,k)/n
flatten([[A038763(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 27 2023

T(n, 0)=1; T(1, 1)=1; T(n, k)=0 for k>n; T(n, k) = T(n-1, k-1)*3 + T(n-1, k) for n >= 2.
Sum_{k=0..n} T(n,k) = A081294(n). - Philippe Deléham, Sep 22 2006
T(n, k) = A136158(n, n-k). - Philippe Deléham, Dec 17 2007
G.f.: (1-2*x*y)/(1-(3*y+1)*x). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Dec 27 2023: (Start)
T(n, 0) = A000012(n).
T(n, 1) = A016777(n-1).
T(n, 2) = A062741(n-1).
T(n, 3) = 9*A002411(n-2).
T(n, 4) = 27*A001296(n-3).
T(n, 5) = 81*A051836(n-4).
T(n, n) = A133494(n).
T(n, n-1) = A006234(n+2).
T(n, n-2) = A080420(n-2).
T(n, n-3) = A080421(n-3).
T(n, n-4) = A080422(n-4).
T(n, n-5) = A080423(n-5).
T(2*n, n) = 4*A098399(n-1) + (2/3)*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A006138(n-1) + (2/3)*[n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A110523(n-1) + (4/3)*[n=0]. (End)

More terms from Michel Marcus, Jul 25 2023

A081892 Second binomial transform of C(n+2,2).

Original entry on oeis.org

1, 5, 22, 90, 351, 1323, 4860, 17496, 61965, 216513, 747954, 2558790, 8680203, 29229255, 97785144, 325241892, 1076168025, 3544180029, 11622614670, 37967207922, 123587135991, 400980206115, 1297083797172, 4184141281200

Offset: 0

Paul Barry, Mar 30 2003

Binomial transform of A049611(n+1).
2nd binomial transform of C(n+2,2), A000217.
3rd binomial transform of (1,2,1,0,0,0,.....)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27).

Cf. A081893.

A right-edge column of triangle A024462.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x)^2/(1-3*x)^3)); // G. C. Greubel, Oct 18 2018

LinearRecurrence[{9, -27, 27}, {1, 5, 22}, 50] (* G. C. Greubel, Oct 18 2018 *)

x='x+O('x^30); Vec((1-2*x)^2/(1-3*x)^3) \\ G. C. Greubel, Oct 18 2018

a(n) = 3^(n - 2)*(n + 2)*(n + 9)/2 = 3^n*(n^2 + 11*n + 18)/18.
G.f.: (1 - 2*x)^2/(1 - 3*x)^3.
E.g.f.: (2 + 4*x + x^2)*exp(3*x)/2. - G. C. Greubel, Oct 18 2018

A323943 Trapezoidal matrix T(n,k) (n>=1, 1<=k<=n+2) read by rows, arising in enumeration of unbranched k-4-catafusenes.

Original entry on oeis.org

1, 2, 1, 3, 7, 5, 1, 9, 24, 22, 8, 1, 27, 81, 90, 46, 11, 1, 81, 270, 351, 228, 79, 14, 1, 243, 891, 1323, 1035, 465, 121, 17, 1, 729, 2916, 4860, 4428, 2430, 828, 172, 20, 1, 2187, 9477, 17496, 18144, 11718, 4914, 1344, 232, 23, 1, 6561, 30618, 61965, 71928, 53298, 26460, 8946, 2040, 301, 26, 1, 19683

Offset: 1

N. J. A. Sloane, Feb 09 2019

Rows sums are powers of 4.

			Matrix begins:
1, 2, 1,
3, 7, 5, 1,
9, 24, 22, 8, 1,
27, 81, 90, 46, 11, 1,
81, 270, 351, 228, 79, 14, 1,
...

S. J. Cyvin, B. N. Cyvin and J. Brunvoll, Isomer enumeration of some polygonal systems representing polycyclic conjugated hydrocarbons, Journal of molecular structure 376.1-3 (1996): 495-505. See Section 7.3.

Cf. A024462 (reversed rows).

Cf. A000244 (column 1), A038765 (column 2), A081892 (column 3)

A323943 := proc(n,k)
    option remember;
    if n = 1 then
        if k>=1 and k<=3 then
            op(k,[1,2,1]) ;
        else
            0;
        end if;
    else
        3*procname(n-1,k)+procname(n-1,k-1) ;
    end if;
end proc:
seq(seq(A323943(n,k),k=1..n+2),n=1..12) ; # R. J. Mathar, May 08 2019

T[n_, k_] := T[n, k] = If[n == 1, If[k >= 1 && k <= 3, {1, 2, 1}[[k]], 0], 3*T[n - 1, k] + T[n - 1, k - 1]];
Table[Table[T[n, k], {k, 1, n + 2}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Nov 08 2023, after R. J. Mathar *)

T(1,1)=T(1,3)=1, T(1,2)=2; thereafter T(n+1,k) = 3*T(n,k)+T(n,k-1).

cp's OEIS Frontend

A038765 Next-to-last diagonal of A024462.

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A038764 a(n) = (9*n^2 + 3*n + 2)/2.

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A038763 Triangular matrix arising in enumeration of catafusenes, read by rows.

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Magma

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A081892 Second binomial transform of C(n+2,2).

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Magma

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A323943 Trapezoidal matrix T(n,k) (n>=1, 1<=k<=n+2) read by rows, arising in enumeration of unbranched k-4-catafusenes.

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Comments

Examples

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Programs

Maple

Mathematica

Formula

A038764 a(n) = (9n^2 + 3n + 2)/2.