A177747 Convolution of A008805 (triangular numbers repeated) with itself.

1, 2, 7, 12, 27, 42, 77, 112, 182, 252, 378, 504, 714, 924, 1254, 1584, 2079, 2574, 3289, 4004, 5005, 6006, 7371, 8736, 10556, 12376, 14756, 17136, 20196, 23256, 27132, 31008, 35853, 40698, 46683, 52668, 59983, 67298, 76153, 85008, 95634, 106260, 118910, 131560, 146510

Offset

0,2

Links

Bruno Berselli, Table of n, a(n) for n = 0..1000

Brian Hopkins and Aram Tangboonduangjit, Water Cells in Compositions of 1s and 2s, arXiv:2412.11528 [math.CO], 2024. See p. 3.

Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Formula

Square (1 + x + 3x^2 + 3x^3 + 6x^4 + 6x^5 + ...)

G.f.: 1/((x+1)^4*(x-1)^6). [Bruno Berselli, Mar 23 2012]

a(n) = (n+5)*(2*n*(n+10)*(n^2+10*n+35)+5*(2*n*(n+10)+39)*(-1)^n+573)/3840. [Bruno Berselli, Mar 23 2012]

Example

As a multiplication table array:
.
1, 1, 3, 3, 6,...
1, 1, 3, 3,......
3, 3, 9,.........
3, 3,............
6,...............
.
Then taking antidiagonal sums of terms, we obtain 1, (1 + 1) = 2, (3 + 1 + 3) = 7,  (3 + 3 + 3 + 3) = 12, (6, + 3 + 9 + 3 + 6) = 27, etc.

Mathematica

lst = CoefficientList[ Series[1/((1 - x) (1 - x^2)^2), {x, 0, 111}], x]; t[n_, k_] := lst[[n]] lst[[k]]; f[n_] := Sum[ t[n - m + 1, m], {m, n}]; Array[f, 45] (* Robert G. Wilson v, Dec 18 2010 *)
LinearRecurrence[{2, 3, -8, -2, 12, -2, -8, 3, 2, -1}, {1, 2, 7, 12, 27, 42, 77, 112, 182, 252}, 45] (* Bruno Berselli, Mar 23 2012 *)

Prog

(Magma) A008805:=func<i|(2*i^2+10*i+11+(2*i+5)*(-1)^i)/16>; [&+[A008805(i)*A008805(n-i): i in [0..n]]: n in [0..44]]; // Bruno Berselli, Mar 23 2012

Crossrefs

Cf. A008805.

Sequence in context: A059329 A242201 A350093 * A288888 A293621 A175879

Adjacent sequences: A177744 A177745 A177746 * A177748 A177749 A177750

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 17 2010

Extensions

More terms from Robert G. Wilson v, Dec 18 2010

Definition rewritten by Bruno Berselli, Mar 23 2012

Status

approved