A130519 -id:A130519 - cp's OEIS frontend

A214734 Sum_{k=1..n} floor(kp/q), where (p,q) are either coprime positive integers or q=1 or p=1, np>=q, ordered by (n + p + q) ascending, then n ascending, then p ascending.

1, 2, 3, 3, 1, 6, 6, 1, 4, 9, 2, 12, 10, 5, 1, 4, 12, 1, 18, 4, 20, 15, 1, 2, 6, 15, 3, 8, 24, 2, 30, 6, 30, 21, 1, 7, 1, 3, 7, 18, 30, 1, 5, 14, 40, 3, 45, 9, 42, 28, 1, 3, 8, 1, 4, 21, 1, 3, 7, 14, 36, 50, 2, 8, 21, 60, 5, 63, 12, 56, 36

Offset: 1

Renzo Benedetti, Jul 27 2012

nonn

Since this is a sequence with 3 indexes (n,p,q), then the order proposed is an ordering by planes of 3D-discrete points (similar to a diagonal ordering of 2D-discrete points). It is not possible to order by rows, columns since n, p, q are boundless.

This sequence generalizes other sequences like A130518, A001840, A058937, A130519, A001972 and maybe others (most of those sequences are replica of each other up to an offset), by providing a closed formula (see formulas).

			a(n, 1, 3) = n*(n+1)/ 6 - floor(n/3) - Sum_{k=1..(n mod 3)} (k mod 3) = n*(n+1)/ 6 - floor(n/3) - (4 mod 3)/3 = A130518(n).
Example of the ordering (n,p,q): (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,1,3), (1,3,1), (2,1,2), (2,2,1), (3,1,1), (1,1,4), ...

a(n, p, q) = Sum_{k=1..n} floor(k*p/q) defines the sequence.

a(n, p, q) = n*(n+1)*p/q/2 - floor(n/q) * (q-1)/2 - Sum_{k=1...(n mod q)} (k*p mod q)/q (the remaining sum has at most q-1 terms, and can assume at most q values when n varies, i.e., that sum for n is equal to the sum for n+q, so the computation of a(n, p, q) requires adding at most (q+1) terms). [Renzo Benedetti, Jul 27 2012]

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

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

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A182568 a(n) = 2floor(n/4)(n - 2*(1 + floor(n/4))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 6, 8, 12, 16, 20, 24, 30, 36, 42, 48, 56, 64, 72, 80, 90, 100, 110, 120, 132, 144, 156, 168, 182, 196, 210, 224, 240, 256, 272, 288, 306, 324, 342, 360, 380, 400, 420, 440, 462, 484, 506, 528, 552, 576, 600, 624, 650, 676, 702, 728, 756, 784, 812, 840, 870, 900, 930, 960, 992, 1024, 1056, 1088, 1122, 1156, 1190, 1224, 1260, 1296, 1332, 1368

Offset: 0

N. J. A. Sloane, May 05 2012

Pak Tung Ho, The toroidal crossing number of K_{4,n}, Discrete Math. 309 (2009), no. 10, 3238--3248. MR2526742(2010i:05088).
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Toroidal Crossing Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A001972, A130519.

Table[2 Floor[n/4] (n - 2 (1 + Floor[n/4])), {n, 0, 20}] (* or *)
Table[(5 - (-1)^n + 2 (n - 4) n - 4 Cos[n Pi/2])/8, {n, 0, 20}] (* or *)
Table[(5 - (-1)^n - 2 (-I)^n - 2 I^n - 8 n + 2 n^2)/8, {n, 0, 20}] (* or *)
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 2}, 80] (* or *)
CoefficientList[Series[-2 x^5/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)

From R. J. Mathar, Jun 28 2012: (Start)
G.f. -2*x^5 / ( (x + 1)*(x^2 + 1)*(x - 1)^3 ).
a(n) = 2*A001972(n-5) = 2*A130519(n-1). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6). - Eric W. Weisstein, Sep 11 2018

A269445 a(n) = Sum_{k=0..n} floor(k/13).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175

Offset: 0

Ilya Gutkovskiy, Feb 27 2016

Partial sums of A090620.

More generally, the ordinary generating function for the Sum_{k=0..n} floor(k/m) is x^m/((1 - x^m)*(1 - x)^2).

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

Cf. A090620.

Cf. similar sequences of Sum_{k=0..n} floor(k/m): A002620 (m=2), A130518 (m=3), A130519 (m=4), A130520 (m=5), A174709 (m=6), A174738 (m=7), A118729 (m=8), A218470 (m=9), A131242 (m=10), A218530 (m=11), A221912 (m=12), this sequence (m=13).

Table[Sum[Floor[k/13], {k, 0, n}], {n, 0, 73}]
LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2}, 74]

G.f.: x^13/((1 - x^13)*(1 - x)^2).

a(n) = 2*a(n-1) - a(n-2) + a(n-13) - 2*a(n-14) + a(n-15).

cp's OEIS Frontend

A214734 Sum_{k=1..n} floor(kp/q), where (p,q) are either coprime positive integers or q=1 or p=1, np>=q, ordered by (n + p + q) ascending, then n ascending, then p ascending.

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A221912 Partial sums of floor(n/12).

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A182568 a(n) = 2floor(n/4)(n - 2*(1 + floor(n/4))).

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A269445 a(n) = Sum_{k=0..n} floor(k/13).

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cp's OEIS Frontend

A214734 Sum_{k=1..n} floor(k*p/q), where (p,q) are either coprime positive integers or q=1 or p=1, n*p>=q, ordered by (n + p + q) ascending, then n ascending, then p ascending.

Views

Author

Keywords

Comments

Examples

Formula

A221912 Partial sums of floor(n/12).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A182568 a(n) = 2*floor(n/4)*(n - 2*(1 + floor(n/4))).

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Author

Keywords

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Programs

Mathematica

Formula

A269445 a(n) = Sum_{k=0..n} floor(k/13).

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Author

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Mathematica

Formula

A214734 Sum_{k=1..n} floor(kp/q), where (p,q) are either coprime positive integers or q=1 or p=1, np>=q, ordered by (n + p + q) ascending, then n ascending, then p ascending.

A182568 a(n) = 2floor(n/4)(n - 2*(1 + floor(n/4))).