A036561 -id:A036561 - cp's OEIS frontend

A349544 Smallest possible value of |Sum_{k=0..n} (+-) 2^k * 3^(n-k)|, where each (+-) can be either plus or minus sign, independently for each term in the sum.

1, 1, 1, 5, 1, 19, 7, 5, 65, 61, 73, 227, 257, 5, 439, 1253, 2425, 2035, 833, 2677, 10591, 6509, 32071, 41173, 77263, 114323, 18145, 129685, 321151, 15757, 645449, 113957, 50735, 477653, 24295, 5089013, 3743881, 4809115, 12209455, 8216179, 32894927, 80299843, 45673913

Offset: 0

Vladimir Reshetnikov, Nov 21 2021

All terms are positive odd integers.

			For n = 3, there are 2^3 = 8 possible choices of signs: 3^3 + 2*3^2 + 2^2*3 + 2^3 = 65, 3^3 + 2*3^2 + 2^2*3 - 2^3 = 49, 3^3 + 2*3^2 - 2^2*3 + 2^3 = 41, 3^3 + 2*3^2 - 2^2*3 - 2^3 = 25, 3^3 - 2*3^2 + 2^2*3 + 2^3 = 29, 3^3 - 2*3^2 + 2^2*3 - 2^3 = 13, 3^3 - 2*3^2 - 2^2*3 + 2^3 = 5, and 3^3 - 2*3^2 - 2^2*3 - 2^3 = -11. The smallest absolute value is 5, so a(3) = 5.

Math StackExchange, Minimizing a generalized partial sum of a geometric progression

Cf. A001047, A036561, A232111, A232112.

b:= proc(k, n) option remember; `if`(k<0, {0}, map(x->
     (t-> [x+t, abs(x-t)][])(2^(n-k)*3^k), b(k-1, n)))
    end:
a:= n-> min(b(n$2)):
seq(a(n), n=0..18);  # Alois P. Heinz, Nov 21 2021

Min@*Abs/@FoldList[Join[3 #1 + 2^#2, 3 #1 - 2^#2] &, {1}, Range[25]]

def f(k,n):
    if k == 0 and n == 0: return (x for x in (1,))
    if k < n: return (y*3 for y in f(k,n-1))
    return (abs(x+y) for x in f(k-1,n) for y in (2**n,-2**n))
def A349544(n): return min(f(n,n)) # Chai Wah Wu, Nov 24 2021

a(33)-a(35) from Chai Wah Wu, Nov 24 2021

a(36)-a(42) from Martin Ehrenstein, Nov 26 2021

A136099 Triangle read by rows: the number of ways to factor 52^(n-k)3^k, columns 0<=k<=n, rows n>=0.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 7, 11, 11, 7, 12, 21, 26, 21, 12, 19, 38, 52, 52, 38, 19, 30, 64, 98, 109, 98, 64, 30, 45, 105, 171, 212, 212, 171, 105, 45, 67, 165, 289, 382, 424, 382, 289, 165, 67, 97, 254, 467, 662, 783, 783, 662, 467, 254, 97, 139, 381, 737, 1097, 1386, 1481, 1386, 1097, 737, 381, 139

Offset: 0

Alford Arnold, Dec 15 2007

Second in the series of arrays beginning with A054225.

			5*A036561(2,1) = 5*6 = 30 and there are five ways to factor 30.
Triangle begins:
   1;
   2,  2;
   4,  5,  4;
   7, 11, 11,   7;
  12, 21, 26,  21, 12;
  19, 38, 52,  52, 38, 19;
  30, 64, 98, 109, 98, 64, 30;
  ...

Cf. A001055, A036561, A054225, A129306, A131419.

T(n,k) = A001055(5*A036561(n,k)).

A230435 Triangle by rows, A001047 convolved with A000079.

Original entry on oeis.org

1, 2, 5, 4, 10, 19, 8, 20, 38, 65, 16, 40, 76, 130, 211, 32, 80, 152, 260, 422, 665, 64, 160, 304, 520, 844, 1330, 2059, 128, 320, 608, 1040, 1688, 2660, 4118, 6305, 256, 640, 1216, 2080, 3376, 5320, 8236, 12610, 19171

Offset: 0

Christopher Tompkins, Nov 18 2013

Generated from Running Total of each row of A036561.
Left edge is A000079 (offset 0): (1, 2, 4, 8, 16, 32, 64, ...)
Right edge is A001047 (offset 1): (1, 5, 19, 65, 211, 665, ...)
Row sums are A066810 (offset 2): (1, 7, 33, 131, 473, 1611, ...)

			The start of the sequence as a triangle read by rows:
  1;
  2,  5;
  4,  10, 19;
  8,  20, 38,  65;
  16, 40, 76,  130, 211;
  32, 80, 152, 260, 422, 665;
  ...

Cf. A000079, A001047, A036561, A066810.

T[n_,k_]:=Sum[3^j*2^(n-j),{j,0,k}];Flatten[Table[T[n,k],{n,0,8},{k,0,n}]] (* Detlef Meya, Dec 20 2023 *)

T(n,k) = Sum_{j=0..k} 3^j*2^(n-j). - Detlef Meya, Dec 20 2023

T(n,k) = 2^n*(3*(3/2)^k-2). - Alois P. Heinz, Dec 20 2023

A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 8, 13, 20, 30, 21, 29, 42, 62, 92, 34, 55, 84, 126, 188, 280, 89, 123, 178, 262, 388, 576, 856, 144, 233, 356, 534, 796, 1184, 1760, 2616, 377, 521, 754, 1110, 1644, 2440, 3624, 5384, 8000, 610, 987, 1508, 2262, 3372, 5016, 7456, 11080, 16464, 24464

Offset: 1

Michel Lagneau, Jan 02 2019

Conjecture: Let F(i) be the i-th Fibonacci number. Each number of T(n, k), k = 1, 2, 3 is the difference between two Fibonacci numbers F(i) - F(j) for some i, j, where F(i) is the smallest Fibonacci number greater than T(n, k). The case T(n, 1) is trivial. Examples: 10 = 13 - 3, 29 = 34 - 5, 20 = 21 - 1, 42 = 55 - 13, 84 = 89 - 5, ...
We observe interesting properties:
T(n,1) = A117647(n) = 1, 2, 5, 8, 21, ... where n = 1, 2, ...
T(2n,2) = A033887(n) = 3, 13, 55, ... (Fibonacci(3n+1)), and T(2n+1,2) = A048876(n) = 7, 29, 123, ... (Generalized Pell equation with second term of 7) where n = 1, 2, ...
T(3n,3) = 10, 84, 754, 6388,... If n = 2m - 1, T(6m - 3, 3) = F(9m - 2) - F(9m - 5) and if n = 2m, T(6m, 3) =  F(9m + 2) - F(9m - 4).
T(3n+1,3) = 20, 178, 1508, 13530, ... If n = 2m - 1, T(6m - 2, 3) = F(9m - 1) - F(9m - 7) and if n = 2m, T(6m+1, 3) =  F(9m + 4) - F(9m + 1).
T(3n+2,3) = 42, 356, 3194, 27060, ... If n = 2m - 1, T(6m - 1, 3) = F(9m + 1) - F(9m - 2) and if n = 2m, T(6m + 2, 3) =  F(9m + 5) - F(9m - 1).
Other property:
T(2m, 1) + T(2m, 2) = T(2m +1, 1) with T(2m, 1)= F(3m), T(2m, 2) = F(3m + 1) and T(2m + 1, 1) = F(3m + 2).
T(2m + 1, 1) + T(2m + 1, 2) = F(3m + 4) - F(3m - 1).

			The start of the sequence as a triangular array T(n, k) read by rows:
   1;
   2,   3;
   5,   7,  10;
   8,  13,  20,   30;
  21,  29,  42,   62,   92;
  34,  55,  84,  126,  188,  280;
  ...

Cf. A000045, A002878, A033887, A035312, A036561, A117647.

with(combinat,fibonacci):
lst:={1}:lst2:=lst:
for n from 2 to 15 do :
lst1:={}:ii:=0:
  for j from 1 to 1000 while(ii=0) do:
     i:=fibonacci(j):
     if {i} intersect lst2 = {} and {i+lst[1]} intersect lst2 = {}
      then
      lst1:=lst1 union {i}:ii:=1:
      else
     fi:
   od:
    for k from 1 to n-1 do:
      lst1:=lst1 union {lst1[k]+lst[k]}:
    od:
    lst:=lst1:lst2:=lst2 union lst:
    print(lst1):
   od:

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171

Offset: 0

Paul Curtz, Jun 20 2025

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025

T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025

t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).

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A349544 Smallest possible value of |Sum_{k=0..n} (+-) 2^k * 3^(n-k)|, where each (+-) can be either plus or minus sign, independently for each term in the sum.

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A136099 Triangle read by rows: the number of ways to factor 5*2^(n-k)*3^k, columns 0<=k<=n, rows n>=0.

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A230435 Triangle by rows, A001047 convolved with A000079.

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A323013 Form of Zorach additive triangle T(n,k) (see A035312) where each number is sum of west and northwest numbers, with the additional condition that the first element T(n,1) is a Fibonacci number.

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Maple

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

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A136099 Triangle read by rows: the number of ways to factor 52^(n-k)3^k, columns 0<=k<=n, rows n>=0.