A025192 -id:A025192 - cp's OEIS frontend

A153311 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)(p(x, n - 1) + 3^(n - 2)(x + x^Floor[n/2] + x^(n - 2))); Row sums are 2*3^n.

2, 3, 3, 2, 14, 2, 2, 25, 25, 2, 2, 36, 77, 45, 2, 2, 65, 167, 176, 74, 2, 2, 148, 313, 424, 412, 157, 2, 2, 393, 704, 980, 1079, 812, 402, 2, 2, 1124, 1826, 1684, 2788, 2620, 1943, 1133, 2, 2, 3313, 5137, 3510, 6659, 7595, 4563, 5263, 3322, 2, 2, 9876, 15011, 8647

Offset: 0

Roger L. Bagula, Dec 23 2008

Row sums:

{2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098,...}.

			{2},
{3, 3},
{2, 14, 2},
{2, 25, 25, 2},
{2, 36, 77, 45, 2},
{2, 65, 167, 176, 74, 2},
{2, 148, 313, 424, 412, 157, 2},
{2, 393, 704, 980, 1079, 812, 402, 2},
{2, 1124, 1826, 1684, 2788, 2620, 1943, 1133, 2},
{2, 3313, 5137, 3510, 6659, 7595, 4563, 5263, 3322, 2},
{2, 9876, 15011, 8647, 10169, 20815, 18719, 9826, 15146, 9885, 2}

A025192

Clear[p, n, m, x];
p[x, 0] = 2; p[x, 1] = 3*x + 3; p[x, 2] = 2*x^2 + 14*x + 2;
p[x_, n_] := p[x, n] = (x + 1)*(p[x, n - 1] + 3^(n - 2)*(x + x^Floor[n/2] + x^(n - 2)));
Table[ExpandAll[p[x, n]], {n, 0, 10}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

p(x,n)=(x + 1)*(p(x, n - 1) + 3^(n - 2)*(x + x^Floor[n/2] + x^(n - 2))).

A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 6, 1, 0, 18, 5, 0, 0, 54, 21, 1, 0, 0, 162, 81, 8, 0, 0, 0, 486, 297, 45, 1, 0, 0, 0, 1458, 1053, 216, 11, 0, 0, 0, 0, 4374, 3645, 945, 78, 1, 0, 0, 0, 0, 13122, 12393, 3888, 450, 14, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 14 2011

Riordan array ((1-x)/(1-3x), x^2/(1-3x)).
A skewed version of triangular array in A193723.
A202209*A007318 as infinite lower triangular matrices.

			Triangle begins:
  1
  2, 0
  6, 1, 0
  18, 5, 0, 0
  54, 21, 1, 0, 0
  162, 81, 8, 0, 0, 0
  486, 297, 45, 1, 0, 0, 0

Cf. A000244, A025192, A081038, A183188 (antidiagonal sums).

G.f.: (1-x)/(1-3*x-y*x^2).
T(n,k) = Sum_{j, j>=0} T(n-2-j,k-1)*3^j.
T(n,k) = 3*T(n-1,k) + T(n-2,k-1).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A057682(n+1), A000079(n), A122367(n), A025192(n), A052924(n), A104934(n), A202206(n), A122117(n), A197189(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5 respectively.

A290582 Numbers m > 2 such that every divisor > 2 is the sum of two or more consecutive divisors.

Original entry on oeis.org

6, 18, 54, 66, 162, 486, 726, 1458, 4374, 7986, 13122, 39366, 87846, 118098, 354294, 530226, 966306, 1062882, 3188646, 9565938, 10629366, 28697814, 43035786, 86093442, 116923026, 258280326, 578476566, 774840978, 1286153286, 2324522934, 6973568802, 14147686146

Offset: 1

Michel Lagneau, Aug 07 2017

nonn

a(n) is even because the divisors are {d(1), d(2), d(3), ...} with d(1) = 1, and if d(2) is odd then d(3) = d(2) + 1 is even, a contradiction. Therefore d(2) = 2 and d(3) = 3.
a(n) == 0 (mod 6).
The sequence is infinite because the numbers of the form 2*3^i (A025192) and the numbers of the form 6*11^i for i >= 1 are in the sequence.
The divisors of 2*3^i are {d(1), d(2), d(3), ...} = {1, 2, 3, 6, 9, 18, 27, 54, ...} where d(1 + 2i) = 3^i for i >= 0 and d(2i) = 2*3^(i-1) for i >= 1.
The divisors of the form 6*11^k are {d(1), d(2), d(3), ...} = {1, 2, 3, 6, 11, 22, 33, 66, 121, ...} where d(i + 4j) = i*11^j for i >= 1 and j >= 0 and where d(4j) = 6*11^(j-1) for j >= 1.
No term can be divisible by 4, 5, 7, 9, 13, 17, or 19. Up to 5*10^11, the only terms which are divisible by a prime > 11 are 530226 = 2*3^5*1091, 43035786 = 2*3^7*9839, 578476566 = 2*3^5*1091^2, and 2*3^7*9839^2. Larger such terms are 2*3*11^12*6904542428779 and 2*3^29*308836698141971. - Giovanni Resta, Aug 07 2017

			66 is in the sequence because the divisors are {1, 2, 3, 6, 11, 22, 33, 66} and:
3 = 2 + 1;
6 = 3 + 2 + 1;
11 = 6 + 3 + 2;
22 = 11 + 6 + 3 + 2;
33 = 22 + 11;
66 = 33 + 22 + 11.

Cf. A008776, A025192.

with(numtheory):nn:=10^5:
for n from 4 to nn do:
  d:=divisors(n):n1:=nops(d):it:=0:
   for k from 3 to n1 do:
     for j from 1 to k-1 do:
       s:=sum('d[k-i]', 'i'=1..j):
        if s=d[k]
         then
         it:=it+1:
         else
        fi:
     od:
    od:
     if n1>2 and it = n1-2
      then
      printf(`%d, `,n):
      else
     fi:
    od:

Select[Range[3, 10^5], Function[d, Function[t, AllTrue[ TakeWhile[ Reverse@ d, # > 2 &], MemberQ[t, #] &]]@ Union@ Flatten@ Array[Total /@ Partition[d, #, 1] &, Length@ d - 1, 2]]@ Divisors@ # &] (* Michael De Vlieger, Aug 07 2017 *)

isokds(k, v) = {vsmall = select(x->(x < k), v); for (i=1, #vsmall, s = v[i]; if (s > k, break); for (j=i+1, #vsmall, s += vsmall[j]; if (s > k, break, if (k == s, return(1))););); return (0);}
isok(n) = {if (n>2, my(d = divisors(n)); for (i=1, #d, if (d[i] > 2, if (! isokds(d[i], d), return (0)); ); ); return(1);)} \\ Michel Marcus, Aug 07 2017

Name corrected by Jon E. Schoenfield, Sep 11 2017

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 88, 192, 420, 922, 2024, 4440, 9736, 21352, 46832, 102720, 225298, 494144, 1083804, 2377112, 5213736, 11435312, 25081112, 55010496, 120654744, 264632554, 580419672, 1273036832, 2792156864, 6124049048, 13431901808, 29460245120, 64615275940

Offset: 0

Ilya Gutkovskiy, Nov 26 2019

nonn

Cf. A000122, A004402, A006456, A025192, A032803, A240944, A279225, A279226, A280542, A320654.

nmax = 32; CoefficientList[Series[1/(1 - 2 Sum[x^(k^2), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

G.f.: 1 / (2 - theta_3(x)), where theta_3() is the Jacobi theta function.

a(0) = 1; a(n) = Sum_{k=1..n} A000122(k) * a(n-k).

cp's OEIS Frontend

A153311 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)(p(x, n - 1) + 3^(n - 2)(x + x^Floor[n/2] + x^(n - 2))); Row sums are 2*3^n.

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A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A290582 Numbers m > 2 such that every divisor > 2 is the sum of two or more consecutive divisors.

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Maple

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Extensions

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

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cp's OEIS Frontend

A153311 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)*(p(x, n - 1) + 3^(n - 2)*(x + x^Floor[n/2] + x^(n - 2))); Row sums are 2*3^n.

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Mathematica

Formula

A183189 Triangle T(n,k), read by rows, given by (2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A290582 Numbers m > 2 such that every divisor > 2 is the sum of two or more consecutive divisors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A153311 Coefficient triangle sequence of a polynomial recursion: p(x,n)=(x + 1)(p(x, n - 1) + 3^(n - 2)(x + x^Floor[n/2] + x^(n - 2))); Row sums are 2*3^n.