A254874 -id:A254874 - cp's OEIS frontend

A254875 a(n) = floor((10n^3 + 57n^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

Michael Somos, Feb 09 2015

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

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).

Cf. A254874.

[Floor((10*n^3 +57*n^2 +102*n +72)/72): n in [0..30]]; // G. C. Greubel, Aug 03 2018

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 *)

{a(n) = (10*n^3 + 57*n^2 + 102*n + 72) \ 72};

{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)};

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.

A254877 Expansion of (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 1, 3, 4, 8, 9, 16, 18, 28, 31, 45, 49, 68, 73, 97, 104, 134, 142, 179, 189, 233, 245, 297, 311, 372, 388, 458, 477, 557, 578, 669, 693, 795, 822, 936, 966, 1093, 1126, 1266, 1303, 1457, 1497, 1666, 1710, 1894, 1942, 2142, 2194, 2411, 2467, 2701, 2762, 3014

Offset: 0

Michael Somos, Feb 09 2015

The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, and either u+v <= x+w or x+u+v+w is even.

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 9*x^5 + 16*x^6 + 18*x^7 + 28*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1).

Cf. A254874, A254875.

a[ n_] := Quotient[ 5 n^3 + If[ OddQ[n], 48 n^2 + 141 n + 162, 57 n^2 + 204 n + 288], 288];
a[ n_] := Module[{m = n}, SeriesCoefficient[ If[ n < 0, m = -7 - n; -1, 1] (1 - x^5)/((1 - x) (1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, ((u + v <= x + w && x + u + v + w == 2 k + 1) || x + u + v + w == 2 k)}, {x, u, v, w, k}, Integers, 10^9];

{a(n) = (5*n^3 + if( n%2, 48*n^2 + 141*n + 162, 57*n^2 + 204*n + 288 )) \ 288};

{a(n) = my(s=(-1)^(n<0)); if( n<0, n = -7-n); s * polcoeff( (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};

Euler transform of length 5 sequence [ 1, 2, 1, 1, -1].
a(n) = -a(-7-n) for all n in Z.
a(2*n) = A254875(n), a(2*n + 1) = A254874(n).

cp's OEIS Frontend

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

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A254877 Expansion of (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A254877 Expansion of (1 - x^5) / ((1 - x) * (1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

Views

Author

Keywords

Comments

Examples

Links

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Programs

Mathematica

PARI

PARI

Formula

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