A317300 Sequence obtained by taking the general formula for generalized k-gonal numbers: m*((k - 2)*m - k + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and k >= 5. Here k = 0.

0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35, -24, -48, -35, -63, -48, -80, -63, -99, -80, -120, -99, -143, -120, -168, -143, -195, -168, -224, -195, -255, -224, -288, -255, -323, -288, -360, -323, -399, -360, -440, -399, -483, -440, -528, -483, -575, -528, -624, -575, -675, -624, -728, -675, -783

Offset

0,3

Comments

Taking the same formula with k = 1 we have A317301.

Taking the same formula with k = 2 we have A001057 (canonical enumeration of integers).

Taking the same formula with k = 3 we have 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).

Taking the same formula with k = 4 we have A008794 (squares repeated) except the initial zero.

Taking the same formula with k >= 5 we have the generalized k-gonal numbers (see Crossrefs section).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = -A174474(n+1).

From Colin Barker, Aug 01 2018: (Start)

G.f.: x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

a(n) = -n*(n + 4) / 4 for n even.

a(n) = -(n - 3)*(n + 1) / 4 for n odd.

(End)

Prog

(PARI) concat(0, Vec(x*(1 - 4*x + x^2) / ((1 - x)^3*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 01 2018

Crossrefs

Row 0 of A303301.

Cf. A001057, A008795, A008794, A174474, A317301.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).

Sequence in context: A180661 A363053 A174474 * A007391 A247668 A359258

Adjacent sequences: A317297 A317298 A317299 * A317301 A317302 A317303

Keyword

sign,easy

Author

Omar E. Pol, Jul 29 2018

Status

approved