A131243 -id:A131243 - cp's OEIS frontend

A131244 Row sums of triangle A131243.

1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, 3396, 7453, 16346, 35861, 78659, 172549, 378487, 830234, 1821136, 3994730, 8762543, 19220904, 42161568, 92482585, 202863051, 444985664, 976088107, 2141075804, 4696507779

Offset: 0

Gary W. Adamson, Jun 22 2007

A131246 is a companion sequence.

			a(4) = 30 = sum of row 4 terms of A131243: (8 + 7 + 10 + 4 + 1).

Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-1).

Cf. A131243, A131245, A131246.

A065941 := proc(n,k) binomial(n-floor((k+1)/2),floor(k/2)) ; end proc:
A131243 := proc(n,k) local a,j ; a := 0 ; for j from k to n do a := a+ A065941(n,j)*A065941(j,k) ; end do: a ; end proc:
A131244 := proc(n) add(A131243(n,k),k=0..n) ; end proc:
seq(A131244(n),n=0..50) ; # R. J. Mathar, Jan 29 2011

Vec((1+x-x^3-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Jun 12 2015

G.f. ( 1+x-x^3-2*x^2 ) / ( 1-2*x-2*x^2+3*x^3+x^4 ). - R. J. Mathar, Jan 29 2011

A131246 Row sums of triangle A131245.

Original entry on oeis.org

1, 3, 6, 13, 27, 57, 119, 250, 523, 1097, 2297, 4815, 10086, 21137, 44283, 92793, 194419, 407378, 853559, 1788481, 3747361, 7851867, 16451910, 34471669, 72228171, 151339401, 317100335, 664418698, 1392152131

Offset: 0

Gary W. Adamson, Jun 22 2007

nonn

A131244 is a companion sequence.

			a(3) = 13 = sum of row 3 terms of triangle A131245: (5 + 5 + 2 + 1)

Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1)

Cf. A131243, A131244, A131245, A065941, A046854.

A046854 := proc(n,k) binomial(floor((n+k)/2),k) ; end proc:
A131245 := proc(n,k) local a,j ; a := 0 ; for j from k to n do a := a+ A046854(n,j)*A046854(j,k) ;  end do: a ; end proc:
A131246 := proc(n) add(A131245(n,k),k=0..n) ; end proc:
seq(A131246(n),n=0..50) ; # R. J. Mathar, Jan 29 2011

CoefficientList[Series[-(1+x)(x^2-x-1)/(1-x-3x^2+x^3+x^4),{x,0,30}],x] (* or *) LinearRecurrence[{1,3,-1,-1},{1,3,6,13},30] (* Harvey P. Dale, Sep 07 2013 *)

G.f. -(1+x)*(x^2-x-1)/ ( 1-x-3*x^2+x^3+x^4 ). - R. J. Mathar, Jan 29 2011

a(0)=1, a(1)=3, a(2)=6, a(3)=13, a(n)=a(n-1)+3*a(n-2)-a(n-3)-a(n-4). - Harvey P. Dale, Sep 07 2013

A131245 A046854^2 as an infinite lower triangular matrix.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 5, 2, 1, 8, 9, 7, 2, 1, 13, 19, 13, 9, 2, 1, 21, 33, 34, 17, 11, 2, 1, 34, 65, 61, 53, 21, 13, 2, 1, 55, 111, 141, 97, 76, 25, 15, 2, 1, 89, 210, 248, 257, 141, 103, 29, 17, 2, 1, 144, 355, 534, 461, 421, 193, 134, 33, 19, 2, 1

Offset: 0

Gary W. Adamson, Jun 22 2007

Left border = Fibonacci numbers.
Row sums = A131246.
A131243 is the square of the reflection triangle to A046854: A065941.
Row sums of A131243 = (1, 3, 6, 14, 30, 67, 146, 322, 705, 1549, ...).

			First few rows of the triangle:
   1;
   2,  1;
   3,  2,  1;
   5,  5,  2,  1;
   8,  9,  7,  2,  1;
  13, 19, 13,  9,  2,  1;
  21, 33, 34, 17, 11,  2,  1;
  ...

Cf. A131243, A131244, A131246, A046854, A065941.

T(n, k) = binomial((n+k)\2, k);
row(n) = my(m=matrix(n+1, n+1, i, j, T(i-1,j-1))); vector(n+1, i, (m^2)[n+1,i]);
lista(nn) = for (n=0, nn, my(v=row(n)); for (i=1, #v, print1(v[i], ", "));); \\ Michel Marcus, Feb 28 2022

More terms from Michel Marcus, Feb 28 2022

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A131244 Row sums of triangle A131243.

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A131246 Row sums of triangle A131245.

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A131245 A046854^2 as an infinite lower triangular matrix.

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