A254875 a(n) = floor((10*n^3 + 57*n^2 + 102*n + 72) / 72).

1, 3, 8, 16, 28, 45, 68, 97, 134, 179, 233, 297, 372, 458, 557, 669, 795, 936, 1093, 1266, 1457, 1666, 1894, 2142, 2411, 2701, 3014, 3350, 3710, 4095, 4506, 4943, 5408, 5901, 6423, 6975, 7558, 8172, 8819, 9499, 10213, 10962, 11747, 12568, 13427, 14324, 15260

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..2500

Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Formula

G.f.: (1 + x + 2*x^2 + x^3) / ((1 - x)^2 * (1 - x^2) * (1 - x^3)).

a(n) - 2*a(n+1) + 2*a(n+3) - a(n+4) = -1 if n == 0 (mod 3) else -2 for all n in Z.

a(n) = -A254874(-4-n) for all n in Z.

Example

G.f. = 1 + 3*x + 8*x^2 + 16*x^3 + 28*x^4 + 45*x^5 + 68*x^6 + 97*x^7 + ...

Mathematica

a[ n_] := Quotient[ 10 n^3 + 57 n^2 + 102 n + 72, 72];
Table[Floor[(10n^3+57n^2+102n+72)/72], {n, 0, 60}] (* or *) LinearRecurrence[ {2, 0, -1, -1, 0, 2, -1}, {1, 3, 8, 16, 28, 45, 68}, 60] (* Harvey P. Dale, Jan 07 2017 *)

Prog

(PARI) {a(n) = (10*n^3 + 57*n^2 + 102*n + 72) \ 72};
(PARI) {a(n) = polcoeff( (-1)^(n<0) * (if( n<0, n = -4 - n; x, x^2) + 1 + x + x^2 + x^3) / ((1 - x)^2 * (1 - x^2) * (1 - x^ 3)) + x * O(x^n), n)};
(Magma) [Floor((10*n^3 +57*n^2 +102*n +72)/72): n in [0..30]]; // G. C. Greubel, Aug 03 2018

Crossrefs

Cf. A254874.

Sequence in context: A122796 A104249 A225253 * A025202 A131941 A009858

Adjacent sequences: A254872 A254873 A254874 * A254876 A254877 A254878

Keyword

nonn,easy

Author

Michael Somos, Feb 09 2015

Status

approved