A256832 -id:A256832 - cp's OEIS frontend

A270617 Primes p such that A256832(p) is divisible by p.

2, 5, 7, 13, 17, 23, 29, 31, 37, 41, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 103, 109, 113, 127, 137, 149, 151, 157, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 311, 313, 317, 337, 349, 353, 359, 367, 373, 379, 383, 389, 397

Offset: 1

Altug Alkan, Mar 20 2016

nonn

Sequence focuses on the prime numbers because of the complement of this sequence. Primes that are listed in this sequence cannot be generated by function which is related with A213891. See comment section of A213891.

			5 is a term because A256832(5) = 3480 is divisible by 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000129, A256832, A213891.

nn = 400; s = FoldList[Times, LinearRecurrence[{2, 1}, {1, 2}, nn]]; Select[Prime@ Range@ PrimePi@ nn, Divisible[s[[#]], #] &] (* Michael De Vlieger, Mar 27 2016, after Harvey P. Dale at A256832 *)

a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
t(n) = prod(k=1, n, Mod(a000129(k), n));
forprime(p=2, 1e3, if(lift(t(p)) == 0, print1(p, ", ")));

is(n)=my(a=Mod(1,n),b=Mod(2,n)); for(i=2,n, if(b==0, return(isprime(n))); [a,b]=[b,2*b+a]); 0 \\ Charles R Greathouse IV, Mar 31 2016

list(lim)=my(v=List([2]), G=factorback(primes([2,lim])), a=1, b=2, t=2, p=2); forprime(q=3,lim, for(n=p+1,q, [a,b]=[b,2*b+a]; t=gcd(t*b, G)); if(t%q==0, listput(v, q)); G/=q; p=q); Vec(v) \\ Charles R Greathouse IV, Mar 31 2016

A270834 Numbers n such that A256832(n)/A000129(n-1) is not divisible by n.

Original entry on oeis.org

3, 7, 9, 11, 19, 23, 31, 43, 47, 67, 71, 83, 107, 127, 131, 139, 151, 163, 167, 191, 211, 263, 271, 283, 307, 311, 331, 347, 359, 367, 383, 431, 439, 463, 467, 479, 491, 499, 503, 523, 547, 563, 571, 587, 619, 631, 647, 659, 691, 719, 727, 739, 743, 787, 811, 823, 839, 859, 863, 883, 887

Offset: 1

Altug Alkan, Mar 23 2016

nonn

The computation of integers n such that A256832(n) is not divisible by n, leads to A213891. This sequence contains A213891 as a subsequence.
It appears that 9 is the only composite number in this sequence.
No composites below 10^7. - Charles R Greathouse IV, Apr 20 2016
No composites below 2*10^7. - Charles R Greathouse IV, May 06 2016

			7 is a term because 1*2*5*12*29*169 = 588120 is not divisible by 7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000129, A213891, A256832, A270617.

With[{s = Sqrt@ 2}, Select[Range[2, 90], ! Divisible[Product[Expand[((1 + s)^k - (1 - s)^k)/2 s], {k, #}]/Simplify[((1 + s)^(# - 1) - (1 - s)^(# -
1))/(2 s)], #] &]] (* Michael De Vlieger, Mar 24 2016, after Vaclav Kotesovec at A256832 and Michael Somos at A000129 *)

a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
t(n) = Mod((prod(k=1, n, a000129(k)) / a000129(n-1)), n);
for(n=2, 1e3, if(lift(t(n)) != 0, print1(n, ", ")));

is(n)=my(a,b=Mod(1,n),t=b); for(k=2,n-2,[a,b]=[b,a+2*b]; t*=b; if(t==0, return(0))); t*(2*a+5*b) && n>2 \\ Charles R Greathouse IV, Mar 24 2016

A270474 Integers k such that A256832(k) is not divisible by k*(k+1)/2.

Original entry on oeis.org

2, 3, 10, 11, 18, 19, 42, 43, 66, 67, 82, 83, 106, 107, 130, 131, 138, 139, 162, 163, 210, 211, 282, 283, 306, 307, 330, 331, 346, 347, 466, 467, 490, 491, 498, 499, 522, 523, 546, 547, 562, 563, 570, 571, 586, 587, 618, 619, 658, 659, 690, 691, 738, 739, 786, 787, 810, 811, 858, 859

Offset: 1

Altug Alkan, Mar 17 2016

nonn

It appears that the odd numbers in the sequence are prime.

This holds at least up to a million. - Charles R Greathouse IV, Feb 24 2022

			3 is a term because (1*2*5) is not divisible by (1+2+3).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000129, A000217, A213891, A256832.

nn = 10^3; Function[k, Select[Range@ nn, ! Divisible[k[[#]], # (# + 1)/2] &]]@ FoldList[Times, LinearRecurrence[{2, 1}, {1, 2}, nn]] (* Michael De Vlieger, Mar 19 2016, after Harvey P. Dale at A256832 *)

a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
f(n) = prod(k=1, n, a000129(k)); \\ A256832
for(n=1, 1e3, if(f(n) % (n*(n+1)/2) != 0, print1(n, ", ")));

{g(n) = my(t, m=1);if( n<2, 0, while(1, t=contfracpnqn(concat([n,vector(m,i,2),n])); t=contfrac(n*t[1,1]/t[2, 1]); if(t[1]Bill McEachen, Feb 14 2022 (from A213891 code, faster)

is(n)=my(m=n^2+n,q=Mod([2, 1; 1, 0],m),Q=q,P=Mod(1,m)); for(k=2,n, P*=(Q*=q)[2,1]; if(P==0, return(0))); 1 \\ Charles R Greathouse IV, Feb 14 2022

A270491 a(n) = A256832(n) mod A003266(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 11138400, 2194264800, 970377408000, 194939999654400, 23386660116019200, 63018468582765696000, 81934202708323789824000, 118589068612624434080256000, 230237098382438262288036864000

Offset: 1

Altug Alkan, Mar 18 2016

nonn

			a(5) = 0 because (1*2*5*12*29) mod (1*1*2*3*5) = 0.

Cf. A000040, A000129, A003266, A256832.

Table[Mod[Product[Expand[((1 + Sqrt@ 2)^j - (1 - Sqrt@ 2)^j)/(2 Sqrt@ 2)], {j, n}], Product[Fibonacci@ k, {k, n}]], {n, 18}] (* Michael De Vlieger, Mar 18 2016, after Vaclav Kotesovec at A256832 *)

a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
a256832(n) = prod(k=1, n, a000129(k));
a003266(n) = prod(k=1, n, fibonacci(k));
for(n=1, 20, print1(a256832(n) % a003266(n), ", "));

A099927 Pellonomial triangle P(k,n) read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 30, 12, 1, 1, 29, 174, 174, 29, 1, 1, 70, 1015, 2436, 1015, 70, 1, 1, 169, 5915, 34307, 34307, 5915, 169, 1, 1, 408, 34476, 482664, 1166438, 482664, 34476, 408, 1, 1, 985, 200940, 6791772, 39618670, 39618670, 6791772, 200940, 985, 1

Offset: 0

Ralf Stephan, Nov 03 2004

Also (signed) coefficients of solutions to 0 = Sum[i=0..k+1, x(i)*Pell(m+i)^k ].

Sagan and Savage give two combinatorial interpretations for entry T(n,k) in terms of statistics on integer partitions fitting inside a k x (n-k) rectangle. They also relate the values T(n,k) to q-binomial coefficients evaluated at q = -(3 + 2*sqrt(2)). - Peter Bala, Mar 15 2013

			Triangle starts:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    5,     1;
  1,  12,   30,    12,     1;
  1,  29,  174,   174,    29,    1;
  1,  70, 1015,  2436,  1015,   70,   1;
  1, 169, 5915, 34307, 34307, 5915, 169, 1;
  ...

Alois P. Heinz, Rows n = 0..56, flattened
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, arXiv:0911.3159 [math.CO], 2009.
B. Sagan and C. Savage, Combinatorial Interpretations of Binomial Coefficient Analogues Related to Lucas Sequences, Integers 10 (2010), 697-703, A52.

Columns include A000129, A084158, A099930, A099931, A383719.
Row sums are in A099928. Central column is in A099929.
Cf. A010048, A256832, A383715.

p:= proc(n) p(n):= `if`(n<2, n, 2*p(n-1)+p(n-2)) end:
f:= proc(n) f(n):= `if`(n=0, 1, p(n)*f(n-1)) end:
T:= (n, k)-> f(n)/(f(k)*f(n-k)):
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 15 2013

p[n_] := p[n] = If[n<2, n, 2*p[n-1] + p[n-2]]; f[n_] := f[n] = If[n == 0, 1, p[n] * f[n-1]]; T[n_, k_] := f[n]/(f[k]*f[n-k]); Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

P(k, n) = Prod[i=k-n+1..k, Pell(i)] / Prod[i=1..n, Pell(i)], with Pell(n) = A000129(n).
From Peter Bala, Mar 15 2013: (Start)
In terms of the Pell numbers, Pell(n) = A000129(n), the triangle entry T(n,k) = [n]!/([k]!*[n-k]!), where [n]! := Pell(1)*...*Pell(n) for n >= 1, with the convention [0]! = 1.
Define E(x) = 1 + sum {n>=0} x^n/[n]!. Then a generating function for this triangle is E(z)*E(x*z) = 1 + (1 + x)*z + (1 + 2*x + x^2)*z^2/[2]! + (1 + 5*x + 5*x^2 + x^3)*z^3/[3]! + ... (End)
G.f. of column k: x^k * exp( Sum_{j>=1} Pell((k+1)*j)/Pell(j) * x^j/j ). - Seiichi Manyama, May 07 2025

A256831 Decimal expansion of Pell factorial constant.

Original entry on oeis.org

1, 1, 4, 1, 9, 8, 2, 5, 6, 9, 6, 6, 7, 7, 9, 1, 2, 0, 6, 0, 2, 8, 0, 4, 3, 3, 3, 8, 3, 6, 7, 8, 6, 0, 1, 5, 0, 8, 6, 4, 7, 3, 0, 4, 8, 2, 4, 0, 8, 5, 4, 0, 7, 9, 1, 5, 5, 6, 2, 5, 4, 3, 5, 2, 4, 4, 9, 8, 4, 3, 7, 8, 5, 4, 8, 0, 6, 2, 0, 8, 6, 0, 7, 8, 2, 5, 0, 6, 3, 7, 0, 6, 0, 9, 2, 5, 3, 3, 4, 7, 8, 1, 6, 3, 6

Offset: 1

Vaclav Kotesovec, Apr 10 2015

			1.141982569667791206028043338367860150864730482408540791556...

Cf. A000129, A099929, A256832.

Cf. A062073, A218490, A253924.

RealDigits[N[QPochhammer[2*Sqrt[2]-3], 105]][[1]]

Equals limit n->infinity A256832(n) / ((1+sqrt(2))^(n*(n+1)/2) / 2^(3*n/2)).

A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

Original entry on oeis.org

1, 1, 6, 203, 40222, 46410442, 312163223724, 12237378320283699, 2796071362211148193590, 3723566980632561787914135870, 28901575272390972687956930234335380, 1307480498356321410289575304307661963042110, 344746842780849469098742541704318199701366091840620

Offset: 0

Tom Edgar, Apr 10 2015

nonn

One definition of the Catalan numbers is binomial(2*n,n) / (n+1); the current sequence models this definition using the generalized binomial coefficients arising from Pell numbers (A000129).

			a(5) = Pell(10)..Pell(7)/Pell(5)..Pell(1) = (2378*985*408*169)/(29*12*5*2*1) = 46410442.
a(3) = A099927(6,3)/Pell(3) = 2436/12 = 203.

Alois P. Heinz, Table of n, a(n) for n = 0..50
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A000129, A099927, A003150, A099929, A256831, A256832.

p:= n-> (<<2|1>, <1|0>>^n)[1, 2]:
a:= n-> mul(p(i), i=n+2..2*n)/mul(p(i), i=1..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 10 2015

Pell[m_]:=Expand[((1+Sqrt[2])^m-(1-Sqrt[2])^m)/(2*Sqrt[2])]; Table[Product[Pell[k],{k,1,2*n}]/(Product[Pell[k],{k,1,n}])^2 / Pell[n+1],{n,0,15}] (* Vaclav Kotesovec, Apr 10 2015 *)

P=[lucas_number1(n, 2, -1) for n in [0..30]]
[1/P[n+1]*prod(P[1:2*n+1])/(prod(P[1:n+1]))^2 for n in [0..14]]

a(n) = Pell(2n)Pell(2n-1)...Pell(n+2)/Pell(n)Pell(n-1)...Pell(1) = A099927(2*n,n)/Pell(n+1) = A099929(n)/Pell(n+1), where Pell(k) = A000129(k).

a(n) ~ 2^(3/2) * (1+sqrt(2))^(n^2-n-1) / c, where c = A256831 = 1.141982569667791206028... . - Vaclav Kotesovec, Apr 10 2015

cp's OEIS Frontend

A270617 Primes p such that A256832(p) is divisible by p.

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A270834 Numbers n such that A256832(n)/A000129(n-1) is not divisible by n.

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A270474 Integers k such that A256832(k) is not divisible by k*(k+1)/2.

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A270491 a(n) = A256832(n) mod A003266(n).

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A099927 Pellonomial triangle P(k,n) read by rows.

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A256831 Decimal expansion of Pell factorial constant.

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A256799 Catalan number analogs for A099927, the generalized binomial coefficients for Pell numbers (A000129).

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