A256239 -id:A256239 - cp's OEIS frontend

A256235 Sum of all the parts in the partitions of 5n into 5 parts.

0, 5, 70, 450, 1680, 4800, 11310, 23590, 44600, 78615, 130550, 207075, 315600, 465790, 667940, 935250, 1281520, 1723970, 2280330, 2972455, 3822500, 4857510, 6104560, 7596325, 9365400, 11450750, 13890760, 16731225, 20017060, 23801315, 28135800, 33081495

Offset: 0

Colin Barker, Mar 20 2015

			For n=2 there are 7 partitions of 5*2 = 10, so a(2) = 7*10 = 70.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,-4,-1,-1,4,4,-1,-1,-4,0,1,1,1,-1).

Cf. A235988, A238328, A256225, A256239.

Plus @@ Total /@ IntegerPartitions[5 #, {5}] & /@ Range[0, 31] (* Michael De Vlieger, Mar 20 2015 *)
CoefficientList[Series[5 x (2 x^14 + 19 x^13 + 97 x^12 + 277 x^11 + 591 x^10 + 955 x^9 + 1267 x^8 + 1355 x^7 + 1217 x^6 + 880 x^5 + 520 x^4 + 231 x^3 + 75 x^2 + 13 x + 1) / ((x - 1)^6 (x + 1)^3 (x^2 + 1)^2 (x^2 + x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{1,1,1,0,-4,-1,-1,4,4,-1,-1,-4,0,1,1,1,-1},{0,5,70,450,1680,4800,11310,23590,44600,78615,130550,207075,315600,465790,667940,935250,1281520},40] (* Harvey P. Dale, Jun 14 2016 *)

concat(0, Vec(5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2) + O(x^100)))

a(n) = 5*n*A256225(n).

G.f.: 5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2).

A256251 First differences of A256250.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 36, 44, 52, 60, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100

Offset: 0

Omar E. Pol, Mar 20 2015

Number of cells turned ON at n-th stage in the structure of A256250.

Apart from the initial 1, four times A006257 (Josephus problem).

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,36,44,52,60;
4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124;
4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148,156,164,172,180,188,196,204,212,220,228,236,244,252;
...
Row sums give A000302.
Right border gives A173033.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata
Index entries for sequences related to the Josephus Problem

Cf. A000302, A006257, A011782, A028399, A139251, A147582, A162793, A169708, A256239.

a(n) = if(n, 8*(n - 2^logint(n,2)) + 4, 1)

[1] + [8*(n - 2^floor(log(n,base=2))) + 4 for n in range(1,77)] # Danny Rorabaugh, Apr 20 2015

a(0) = 1. For n >= 1; a(n) = 4*A006257(n).

For n>0, a(n) = 8*(n - 2^floor(log_2(n))) + 4 (by the formula of Gregory Pat Scandalis in A006257). - Danny Rorabaugh, Apr 20 2015

cp's OEIS Frontend

A256235 Sum of all the parts in the partitions of 5n into 5 parts.

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