A057089 -id:A057089 - cp's OEIS frontend

A180167 a(0) = 1, a(1) = 7; a(n)= 6a(n-1) + 6a(n-2) for n>1.

1, 7, 48, 330, 2268, 15588, 107136, 736344, 5060880, 34783344, 239065344, 1643092128, 11292944832, 77616221760, 533454999552, 3666427327872, 25199293964544, 173194327754496, 1190361730314240, 8181336348412416, 56230188472359936, 386469148924634112

Offset: 0

Gary W. Adamson, Aug 14 2010

			a(4) = 2268 = 6*a(3) + 6*a(2) = 6*330 + 6*48.
Using the INVERT transform operation, a(3) = 330 = (205, 35, 6, 1) dot
(1, 1, 7, 48) = (205 + 35 + 42 + 48), where (1, 6, 35, 205, 1200, ...) = A180033.
G.f. = 1 + 7*x + 48*x^2 + 330*x^3 + 2268*x^4 + 15588*x^5 + 107136*x^6 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 14.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (6,6).

Cf. A057089, A180033.

CoefficientList[Series[(1 + x)/(1 - 6 x - 6 x^2), {x, 0, 21}], x] (* Michael De Vlieger, Dec 16 2021 *)

Vec((1 + x)/(1 - 6*x - 6*x^2) + O(x^50)) \\ Colin Barker, May 13 2016

G.f.: (1 + x)/(1 - 6*x - 6*x^2); = INVERT transform of A180033
a(n) = ((3-sqrt(15))^n*(-4+sqrt(15))+(3+sqrt(15))^n*(4+sqrt(15)))/(2*sqrt(15)). - Alexander R. Povolotsky, Aug 22 2010, corrected by Colin Barker, May 13 2016
a(n) = A057089(n) + A057089(n-1). - R. J. Mathar, Apr 04 2012
E.g.f.: (4*sqrt(15)*sinh(sqrt(15)*x) + 15*cosh(sqrt(15)*x))*exp(3*x)/15. - Ilya Gutkovskiy, May 13 2016

A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, 0, -1, 1, 1, -1, -1, 1, -1, 3, -2, -1, 1, 0, -2, 5, -3, -1, 1, 1, -2, -2, 7, -4, -1, 1, -1, 5, -7, -1, 9, -5, -1, 1, 0, -3, 12, -15, 1, 11, -6, -1, 1, 1, -3, -3, 21, -26, 4, 13, -7, -1, 1, -1, 7, -15, 3, 31, -40, 8, 15, -8, -1, 1, 0, -4, 22, -42

Offset: 0

Philippe Deléham, Nov 12 2011

			Triangle begins :
1
-1, 1
0, -1, 1
1, -1, -1, 1
-1, 3, -2, -1, 1
0, -2, 5, -3, -1, 1
1, -2, -2, 7, -4, -1, 1
-1, 5, -7, -1, 9, -5, -1, 1

Cf. A026729, A063967, A129267, A176971 (diagonals sums).

T(n,k)=T(n-1,k-1)+T(n-2,k-1)-T(n-1,k)-T(n-2,k), T(0,0)=1.
G.f.: 1/(1-(y-1)*x-(y-1)*x^2).
Sum_{k, 0<=k<=n}T(n,k)*x^k = A000748(n), A108520(n), A049347(n), A000007(n), A000045(n+1), A002605(n+1), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = -2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 5, 1, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 3, 11, 3, 0, 0, 0, 0, 0, 0, 1, 13, 13, 1, 0, 0, 0, 0, 0, 0, 0, 9, 26, 9, 0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 32, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 26, 63, 26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 80, 80, 14, 0, 0, 0, 0, 0

Offset: 1

Gerald McGarvey, Jan 15 2007

It appears that the main diagonal (1,1,2,5,11,...) is A051286 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes) and that the n-th antidiagonal sum = A094686(n-1) (a Fibonacci convolution). The n-th row sum = A002605(n).

			Array starts as:
1 0 0 0  0  0  0 ...
0 1 1 0  0  0  0 ...
0 1 2 2  1  0  0 ...
0 0 2 5  5  3  1   0 ...
0 0 1 5 11 13  9   4   1   0...
0 0 0 3 13 26 32  26  14   5   1  0 ...
0 0 0 1  9 32 63  80  71  45  20  6  1 0 ...
0 0 0 0  4 26 80 153 201 191 135 71 27 7 1 0 ...
...

Cf. A051286, A110320, A002605, A026729, A030528, A057089, A109466.

T[n_, k_] := Sum[Binomial[i, n-i] Binomial[i, k-i], {i, Floor[(n+1)/2], k}];
Table[T[n-k, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 12 2019 *)

A=matrix(22,22);A[1,1]=1;A[2,2]=1;A[2,1]=0;A[1,2]=0;A[3,2]=1;A[2,3]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))

A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
From Peter Bala, Nov 07 2017: (Start)
T(n,k) = Sum_{i = floor((n+1)/2)..k} binomial(i,n-i)* binomial(i,k-i).
Square array = A026729 * transpose(A026729), where A026729 is viewed as a lower unit triangular array. Omitting the first row and column of square array = A030528 * transpose(A030528).
O.g.f. 1/(1 - t*(1 + t)*x - t*(1 + t)*x^2) = 1 + (t + t^2)*x + (t + 2*t^2 + 2*t^3 + t^4)*x^2 + .... Cf. A109466 with o.g.f. 1/(1 - t*x - t*x^2).
The n-th row polynomial R(n,t) satisfies R(n,t) = R(n,-1 - t).
R(n,t) = (-1)^n*sqrt(-t*(1 + t))^n*U(n, 1/2*sqrt(-t*(1 + t))), where U(n,x) denotes the n-th Chebyshev polynomial of the second kind.
The sequence of row polynomials R(n,t) is a divisibility sequence of polynomials, that is, if m divides n then R(m,t) divides R(n,t) in the polynomial ring Z[t].
R(n,1) = A002605; R(n,2) = A057089. (End)

A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

Original entry on oeis.org

1, 6, 41, 281, 1931, 13271, 91211, 626891, 4308611, 29613011, 203529731, 1398856451, 9614317091, 66079041251, 454160150051, 3121435147811, 21453571787171, 147450041609891, 1013421680382371, 6965230331953571, 47871912074015651, 329022854435815331, 2261368599058985891

Offset: 0

Philippe Deléham, Mar 19 2024

x = A092294 = 3+sqrt(15) = 6.872983346...

			a(0) = 1;
a(1) = floor(x) = 6 where x = 3+sqrt(15);
a(2) = floor(6*x) = 41;
a(3) = floor(41*x) = 281.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,0,-6).

Cf. A057089, A092294, A370174.

NestList[Floor[(Sqrt[15]+3)*#] &, 1, 25] (* or *)
LinearRecurrence[{7, 0, -6}, {1, 6, 41}, 25] (* Paolo Xausa, Mar 31 2024 *)

a(n) = 7*a(n-1) - 6*a(n-3), a(0) = 1, a(1) = 6, a(2) = 41.
a(n) = 6*a(n-1) + 6*a(n-2) - 1.
a(n) = ((30-7*sqrt(15))*(3-sqrt(15))^n + (30+7*sqrt(15))*(3+sqrt(15))^n + 6)/66.
G.f.: (1-x-x^2)/(1-7*x+6*x^3).
a(n) = Sum_{k = 0..n} A370174(n,k)*5^k.
a(n) = (10*A057089(n) + 5*A057089(n-1) + 1)/11.

cp's OEIS Frontend

A180167 a(0) = 1, a(1) = 7; a(n)= 6*a(n-1) + 6*a(n-2) for n>1.

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A199324 Triangle T(n,k), read by rows, given by (-1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

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A370176 a(n) = floor(x*a(n-1)) for n > 0 where x = 3+sqrt(15), a(0) = 1.

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A180167 a(0) = 1, a(1) = 7; a(n)= 6a(n-1) + 6a(n-2) for n>1.