A026549 -id:A026549 - cp's OEIS frontend

A171142 Triangle T(n,k) of the coefficients [x^k] of the polynomial p_n(x), where p_n(x)=(1+x)*p_{n-1}(x) if n even, p_n(x) = (x^2+4x+1)^((n-1)/2) if n odd.

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 8, 18, 8, 1, 1, 9, 26, 26, 9, 1, 1, 12, 51, 88, 51, 12, 1, 1, 13, 63, 139, 139, 63, 13, 1, 1, 16, 100, 304, 454, 304, 100, 16, 1, 1, 17, 116, 404, 758, 758, 404, 116, 17, 1, 1, 20, 165, 720, 1770, 2424, 1770, 720, 165, 20, 1, 1, 21, 185, 885

Offset: 1

Roger L. Bagula and Gary W. Adamson, Dec 04 2009

Row sums are apparently in A026549.

			The triangle starts in row n=1 with column 0<=k<n as:
1;
1, 1;
1, 4, 1;
1, 5, 5, 1;
1, 8, 18, 8, 1;
1, 9, 26, 26, 9, 1;
1, 12, 51, 88, 51, 12, 1;
1, 13, 63, 139, 139, 63, 13, 1;
1, 16, 100, 304, 454, 304, 100, 16, 1;
1, 17, 116, 404, 758, 758, 404, 116, 17, 1;
1, 20, 165, 720, 1770, 2424, 1770, 720, 165, 20, 1;
1, 21, 185, 885, 2490, 4194, 4194, 2490, 885, 185, 21, 1;

Cf. A051159, A169623, A007318

A171142P := proc(n) option remember; if type(n,'even') then (x+1)*procname(n-1) ; else (x^2+4*x+1)^((n-1)/2) ; end if; expand(%) ;end proc:
A171142 := proc(n,k) coeff(A171142P(n,x),x,k) ; end proc:

Clear[p, n, x, a]
w = 4;
p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]];
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}];
Flatten[a]

A208131 Partial products of A052901.

Original entry on oeis.org

1, 3, 6, 12, 36, 72, 144, 432, 864, 1728, 5184, 10368, 20736, 62208, 124416, 248832, 746496, 1492992, 2985984, 8957952, 17915904, 35831808, 107495424, 214990848, 429981696, 1289945088, 2579890176, 5159780352, 15479341056, 30958682112, 61917364224, 185752092672

Offset: 0

Reinhard Zumkeller, Apr 04 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,12).

Cf. A001222, A052901, A026549, A026532.

a208131 n = a208131_list !! n
a208131_list = scanl (*) 1 $ a052901_list
-- Reinhard Zumkeller, Mar 29 2012

FoldList[Times,1,PadRight[{},30,{3,2,2}]] (* Harvey P. Dale, Mar 19 2013 *)

a(n+1) = a(n) * A052901(n).
A001222(a(n)) = n.
a(n) = 12^floor(n/3)*(r+1)*(r+2)/2 with r = n mod 3. G.f.: -(6*x^2+3*x+1) / (12*x^3-1). - Alois P. Heinz, Apr 05 2012
Sum_{n>=0} 1/a(n) = 18/11. - Amiram Eldar, Feb 13 2023

A164560 Partial sums of A164532.

Original entry on oeis.org

1, 5, 11, 35, 71, 215, 431, 1295, 2591, 7775, 15551, 46655, 93311, 279935, 559871, 1679615, 3359231, 10077695, 20155391, 60466175, 120932351, 362797055, 725594111, 2176782335, 4353564671, 13060694015, 26121388031, 78364164095

Offset: 1

Klaus Brockhaus, Aug 16 2009

Interleaving of A164559 and A024062 without initial term 0.

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (1,6,-6).

Cf. A164532, A164123 (partial sums of A162436), A164559 (6^n/3-1), A024062 (6^n-1), A026549.

T:=[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..28] ]; [ n eq 1 select T[1] else Self(n-1)+T[n]: n in [1..#T]];

a(n) = 6*a(n-2)+5 for n > 2; a(1) = 1, a(2) = 5.
a(n) = (3-(-1)^n)*6^(1/4*(2*n-1+(-1)^n))/2-1.
G.f.: x*(1+4*x)/((1-x)*(1-6*x^2)).
a(n) = A026549(n) - 1.

A329114 a(n) = floor(A026532(n)/5).

Original entry on oeis.org

0, 0, 1, 3, 7, 21, 43, 129, 259, 777, 1555, 4665, 9331, 27993, 55987, 167961, 335923, 1007769, 2015539, 6046617, 12093235, 36279705, 72559411, 217678233, 435356467, 1306069401, 2612138803, 7836416409, 15672832819, 47018498457, 94036996915, 282110990745

Offset: 1

Clark Kimberling, Nov 10 2019

Index entries for linear recurrences with constant coefficients, signature (0,7,0,-6).

Cf. A026532, A329115.

s[1] = 1; s[n_] := If[IntegerQ[n/2], 3*s[n - 1], 2*s[n - 1]]
Table[s[n], {n, 1, 20}] (* A026549 *)
Table[Floor[s[n]/5], {n, 1, 50}] (* A329114 *)

a(n+1) = 3*a(n) if n is odd, a(n+1) = 2*a(n)+1 if n is even.
a(n) = f(3^f(n/2) * 2^f((n-1)/2) / 5), where f = floor.
G.f.: (x^2 (1 + 3 x))/((-1 + x) (1 + x) (-1 + 6 x^2)).
a(n) = 7*a(n-2) - 6*a(n-4).

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 5, 0, 6, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 7, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 9, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

Contains no ones.

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The prime factors of 96 are {2,2,2,2,2,3}, with sum 13, and we have permutations such as (2,2,2,2,3,2), with run-compression (2,3,2), with sum 7, so a(96) = 13 - 7 = 6.

Positions of first appearances are A280286.
For least instead of greatest sum of run-compression we have A280292.
Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
For prime indices instead of factors we have A374248.
This is an opposite version of A374250, for prime indices A373956.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 is run-compression of prime indices, row-sums A066328.
A373949 counts compositions by sum of run-compression, opposite A373951.
A373957 gives greatest number of runs in a permutation of prime factors.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
A374252 counts permutations of prime factors by number of runs.
Cf. A000040, A026549, A037201, A046660, A124767, A246655, A333755, A373954, A374246, A374247.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Total[prifacs[n]]-Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]

a(n) = A001414(n) - A374250(n).

cp's OEIS Frontend

A171142 Triangle T(n,k) of the coefficients [x^k] of the polynomial p_n(x), where p_n(x)=(1+x)*p_{n-1}(x) if n even, p_n(x) = (x^2+4x+1)^((n-1)/2) if n odd.

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A208131 Partial products of A052901.

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A164560 Partial sums of A164532.

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Magma

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A329114 a(n) = floor(A026532(n)/5).

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A374255 Sum of prime factors of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime factors of n.

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Mathematica

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