A062741 -id:A062741 - cp's OEIS frontend

A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

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

Offset: 0

Mark Dols, Sep 01 2009

Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

			As triangle:
    1;
    1,   -1;
    3,   -4,    1;
    9,  -15,    7,   -1;
   27,  -54,   36,  -10,    1;
   81, -189,  162,  -66,   13,   -1;
  243, -648,  675, -360,  105,  -16,    1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.
Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.
Cf. A133494, A136158, A164942.

A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
[A164948(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023

A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2023 *)

def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Sum_{k=0..n} T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
From Philippe Deléham, Oct 09 2011: (Start)
T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.
Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
G.f.: (1-2*x)/(1-3*x+x*y). - R. J. Mathar, Aug 12 2015
From G. C. Greubel, Dec 26 2023: (Start)
T(n, k) = (-1)^k * A136158(n, k).
T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
T(n, 0) = A133494(n).
T(n, 1) = -A006234(n+2), n >= 1.
T(n, 2) = A080420(n-2), n >= 2.
T(n, 3) = -A080421(n-3), n >= 3.
T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.
T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
T(n, n) = (-1)^n.
Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)

More terms from Philippe Deléham, Oct 09 2011

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A180568 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the grid P_3 x P_n (1<=k<=n), where P_j denotes the path graph on j nodes.

Original entry on oeis.org

2, 1, 7, 6, 2, 12, 14, 8, 2, 17, 22, 17, 8, 2, 22, 30, 26, 17, 8, 2, 27, 38, 35, 26, 17, 8, 2, 32, 46, 44, 35, 26, 17, 8, 2, 37, 54, 53, 44, 35, 26, 17, 8, 2, 42, 62, 62, 53, 44, 35, 26, 17, 8, 2, 47, 70, 71, 62, 53, 44, 35, 26, 17, 8, 2, 52, 78, 80, 71, 62, 53, 44, 35, 26, 17, 8, 2, 57

Offset: 1

Emeric Deutsch, Sep 28 2010

Row n contains n+1 entries.
Sum of entries in row n = (3/2)n(3n-1)=A062741(n).
The entries in row n are the coefficients of the Wiener polynomial of the grid P_3 x P_n.
Sum(k*T(n,k),k=1..n+1)=(1/2)n(n+3)(3n-1)=A180569(n) = the Wiener index of the grid P_3 x P_n.
The average of all distances in the grid P_3 x P_n is (n+3)/3.

			T(1,1)=2, T(1,2)=1 because in P_3 x P_1 = P_3 there are 2 pairs of nodes at distance 1 and one pair at distance 2.
Triangle starts:
2,1;
7,6,2;
12,14,8,2;
17,22,17,8,2;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Grid Graph.

Cf. A180569

p := proc (n) options operator, arrow: (t^(n+1)*(3+4*t+2*t^2)+(5*n-3)*t-(2*n+4)*t^2-(2*n+2)*t^3-n*t^4)/(1-t)^2 end proc: for n to 12 do f[n] := sort(expand(simplify(p(n)))) end do: for n to 12 do seq(coeff(f[n], t, k), k = 1 .. n+1) end do; # yields sequence in triangular form

The row generating polynomials p(n)=p(n,t) satisfy the recurrence relation p(n)=p(n-1)=2t+t^2+t(3+4t+2t^2)*sum(t^j,j=0..n-2) (these are the Wiener polynomials of the corresponding graphs).
The generating polynomial of row n is p(n; t)=[t^{n+1}*(3+4t+2t^2)+(5n-3)t-2(n+2)t^2-2(n+1)t^3-nt^4]/(1-t)^2.
G.f. = G(t,z)=Sum(T(n,k)*t^k*z^n, k>=1, n>=1) = tz(zt^2+2tz+t+3z+2)/[(1-tz)(1-z)^2].

A180575 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the prism graph C_n X P_3 with the edges of the outer cycle removed (called a web graph). Equivalently, the graph is obtained by attaching a pendant edge to each node of the outer cycle of the circular ladder (prism) C_n X P_2. C_n denotes the cycle graph on n nodes and P_n denotes the path graph on n nodes.

Original entry on oeis.org

12, 15, 9, 16, 24, 20, 6, 20, 35, 35, 15, 24, 42, 48, 30, 9, 28, 49, 63, 49, 21, 32, 56, 72, 64, 40, 12, 36, 63, 81, 81, 63, 27, 40, 70, 90, 90, 80, 50, 15, 44, 77, 99, 99, 99, 77, 33, 48, 84, 108, 108, 108, 96, 60, 18, 52, 91, 117, 117, 117, 117, 91, 39, 56, 98, 126, 126

Offset: 3

Emeric Deutsch, Sep 19 2010

The entries in row n are the coefficients of the Wiener polynomial of the graph.
Number of entries in row n is 2+floor(n/2).
The entries in row n are the coefficients of the Wiener polynomial of the corresponding graph.
Sum of entries in row n is 3n(3n-1)/2 = A062741.
Sum_{k>=1} k*T(n,k) = A180576(n) = the Wiener index of the corresponding graph.

			The triangle starts:
  12,15,9;
  16,24,20,6;
  20,35,35,15;
  24,42,48,30,9;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
Eric Weisstein's World of Mathematics, Web Graph

Cf. A062741, A180576.

P := proc (n) if `mod`(n, 2) = 1 then sort(expand(simplify(n*(4*t+3*t^2+2*t^3-2*t^((1/2)*n+1/2)-4*t^((1/2)*n+3/2)-3*t^((1/2)*n+5/2))/(1-t)))) else sort(expand(simplify((1/2)*n*(8*t+6*t^2+4*t^3-2*t^((1/2)*n)-6*t^((1/2)*n+1)-7*t^((1/2)*n+2)-3*t^((1/2)*n+3))/(1-t)))) end if end proc: for n from 3 to 14 do seq(coeff(P(n), t, j), j = 1 .. 2+floor((1/2)*n)) end do; # yields sequence in triangular form

The generating polynomial of row 2*n+1 (which is also the Wiener polynomial of the corresponding graph) is (2*n+1)*(4*t+3*t^2+2*t^3-2*t^(n+1)-4*t^(n+2)-3*t^(n+3))/(1-t).

The generating polynomial of row 2*n (which is also the Wiener polynomial of the corresponding graph) is n*(8*t+6*t^2+4*t^3-2*t^n-6*t^(n+1)-7*t^(n+2)-3*t^(n+3))/(1-t).

A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

Original entry on oeis.org

2, 1, 5, 6, 4, 8, 12, 12, 4, 11, 18, 21, 12, 4, 14, 24, 30, 21, 12, 4, 17, 30, 39, 30, 21, 12, 4, 20, 36, 48, 39, 30, 21, 12, 4, 23, 42, 57, 48, 39, 30, 21, 12, 4, 26, 48, 66, 57, 48, 39, 30, 21, 12, 4, 29, 54, 75, 66, 57, 48, 39, 30, 21, 12, 4, 32, 60, 84, 75, 66, 57, 48, 39, 30, 21, 12, 4

Offset: 1

Emeric Deutsch, Jun 25 2011

Row n contains n+1 entries.
Sum of entries in row n is 3*n*(3*n-1)/2=A062741(n).
Sum(k*T(n,k),k>=1)=A192025(n) (the Wiener indices).

			T(2,1)=5, T(2,2)=6, T(2,3)=4 because in the graph \/_\/ there are 5 pairs of nodes at distance 1, 6 pairs at distance 2, and 4 pairs at distance 3.
Triangle starts:
2,1;
5,6,4;
8,12,12,4;
11,18,21,12,4;

B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.

Cf. A062741, A192025

Q := proc (n) options operator, arrow: n*(t^2+2*t)+t*(1+2*t)^2*(sum((n-j)*t^(j-1), j = 1 .. n-1)) end proc: for n to 11 do P[n] := sort(expand(Q(n))) end do: for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n+1) end do; # yields sequence in triangular form

G.f.: G(t,z)=t*z*(2+t+z+2*t*z+3*t^2*z)/((1-t*z)*(1-z)^2).
G.f. of column 1: z*(2+z)/(1-z)^2.
G.f. of column 2: z*(1+4*z+z^2)/(1-z)^2.
G.f. of  column k>=3: z^(k-1)*(4+4*z+z^2)/(1-z)^2.
The generating polynomial of row n (i.e. the Wiener polynomial of the double-comb with 3n nodes) is n*(2*t +t^2)+t*(1+2*t)^2*(n*(1-t)-(1-t^n))/(1-t)^2 or, equivalently, n*(2*t+t^2)+t*(1+2*t)^2*Sum((n-j)*t^(j-1), j=1..n-1).

A253672 Another fractal t(h)ree.

Original entry on oeis.org

0, 1, 2, 0, 1, 3, 4, 5, 2, 0, 1, 3, 4, 6, 7, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 14, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 17, 14, 11, 8, 5, 2, 0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 20

Offset: 0

Eric Angelini and Reinhard Zumkeller, Jan 08 2015

A008585(n+1) = length of row n; A062741(n+1) = sum of row n;

the fractal nature is illustrated by the following manipulation: remove from all rows the first two terms and also the last one, after subtracting all terms by 3, the original t(h)ree will reappear.

			.  0:                                     | 0  1  2|
.  1:                                0  1 | 3  4  5|  2
.  2:                          0  1  3  4 | 6  7  8|  5  2
.  3:                    0  1  3  4  6  7 | 9 10 11|  8  5  2
.  4:              0  1  3  4  6  7  9 10 |12 13 14| 11  8  5  2
.  5:          0 1 3  4  6  7  9 10 12 13 |15 16 17| 14 11  8  5 2
.  6:      0 1 3 4 6  7  9 10 12 13 15 16 |18 19 20| 17 14 11  8 5 2
.  7:  0 1 3 4 6 7 9 10 12 13 15 16 18 19 |21 22 23| 20 17 14 11 8 5 2 .

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
Éric Angelini, More (and more) fractal trees - and erasures, SeqFan list, Jan 08 2015.

Cf. A008585, A062741.

a253672 n k = a253672_tabf !! n !! k
a253672_row n = a253672_tabf !! n
a253672_tabf = [0,1,2] : f [] [0,1,2] [] (iterate (map (+ 3)) [3..5]) where
   f as bs cs (uvws:uvwss) = (as' ++ uvws ++ cs') : f as' uvws cs' uvwss
     where as' = as ++ [u,v]; cs' = [w] ++ cs
           [u,v,w] = bs
a253672_list = concat a253672_tabf

T(n,0) = 0; T(n,1)=1; T(n,2*n-1) = 2; T(n+1,k+2) = T(n,k)+3, k = 0..3*n-1.

A236311 Riordan array ((1-x)/(1-3x+3x^2), x/(1-3x+3x^2)).

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 3, 15, 8, 1, 0, 33, 36, 11, 1, -9, 54, 117, 66, 14, 1, -27, 54, 297, 282, 105, 17, 1, -54, -27, 594, 945, 555, 153, 20, 1, -81, -297, 864, 2583, 2295, 963, 210, 23, 1, -81, -891, 513, 5778, 7803, 4725, 1533, 276, 26, 1, 0, -1863, -1944, 10098

Offset: 0

Philippe Deléham, Jan 21 2014

Row sums are 3^n = A000244(n).
Diagonals sums are 2^n = A000079(n).
T(n,n) = A000012(n).
T(n+1,n) = A016789(n).
T(n+2,n) = A062741(n+1).
T(n+3,n) = 3*A004188(n+1).
T(n,0) = A057682(n+1).

			Triangle begins :
1;
2, 1;
3, 5, 1;
3, 15, 8, 1;
0, 33, 36, 11, 1;
-9, 54, 117, 66, 14, 1;
-27, 54, 297, 282, 105, 17, 1;

Cf. A057083, A057682, A000012, A016789, A062741

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

cp's OEIS Frontend

A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

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A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

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A180568 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the grid P_3 x P_n (1<=k<=n), where P_j denotes the path graph on j nodes.

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A192024 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the double-comb graph \/\/\/...\/_\/ with 3n (n>=1) nodes. The entries in row n are the coefficients of the corresponding Wiener polynomial.

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A253672 Another fractal t(h)ree.

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A236311 Riordan array ((1-x)/(1-3*x+3*x^2), x/(1-3*x+3*x^2)).

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A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

A236311 Riordan array ((1-x)/(1-3x+3x^2), x/(1-3x+3x^2)).