A133558 -id:A133558 - cp's OEIS frontend

A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

1, 2, 2, 1, 2, 3, 2, 5, 1, 2, 7, 4, 2, 9, 9, 1, 2, 11, 16, 5, 2, 13, 25, 14, 1, 2, 15, 36, 30, 6, 2, 17, 49, 55, 20, 1, 2, 19, 64, 91, 50, 7, 2, 21, 81, 140, 105, 27, 1, 2, 23, 100, 204, 196, 77, 8, 2, 25, 121, 285, 336, 182, 35, 1, 2, 27, 144, 385, 540, 378, 112, 9, 2, 29, 169, 506

Offset: 0

Emeric Deutsch, May 12 2007

Also number of Fibonacci binary words of length n and having k 10 subwords.
Row n has 1+floor(n/2) terms.
Row sums are the Fibonacci numbers (A000045).
T(n,0)=2 for n >= 1.
Sum_{k>=0} k*T(n,k) = A023610(n-2).
Triangle, with zeros omitted, 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. - Philippe Deléham, Jan 14 2012
Riordan array ((1+x)/(1-x), x^2/(1-x)), zeros omitted. - Philippe Deléham, Jan 14 2012

			T(5,2)=4 because we have 10101, 01101, 01010 and 01011.
Triangle starts:
  1;
  2;
  2, 1;
  2, 3;
  2, 5, 1;
  2, 7, 4;
  2, 9, 9, 1;
Triangle (2, -1, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) begins:
  1;
  2, 0;
  2, 1, 0;
  2, 3, 0, 0;
  2, 5, 1, 0, 0;
  2, 7, 4, 0, 0, 0;
  2, 9, 9, 1, 0, 0, 0;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 200, flattened.)
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A000045, A023610.
Cf. A029635, A029653.
Columns: A040000, A005408, A000290, A000330, A002415, A005585, A040977, A050486, A053347, A054333, A054334, A057788.

T:=proc(n,k) if n=0 and k=0 then 1 elif k<=floor(n/2) then binomial(n-k,k)+binomial(n-k-1,k) else 0 fi end: for n from 0 to 18 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

MapAt[# - 1 &, #, 1] &@ Table[Binomial[n - k, k] + Binomial[n - k - 1, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 15 2019 *)

T(n,k) = binomial(n-k,k) + binomial(n-k-1,k) for n >= 1 and 0 <= k <= floor(n/2).
G.f. = G(t,z) = (1+z)/(1-z-tz^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A078050(n), A057079(n), A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively. - Philippe Deléham, Jan 14 2012
T(n,k) = T(n-1,k) + T(n-2,k-1) with T(0,0)=1, T(1,0)=2, T(1,1)=0 and T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Jan 14 2012

A133562 Numbers which are the sum of the squares of seven consecutive primes.

Original entry on oeis.org

666, 1023, 1543, 2359, 3271, 4519, 6031, 7591, 9439, 11719, 14359, 17119, 20239, 23599, 27079, 31111, 35191, 39631, 45319, 51031, 56599, 62719, 68359, 74239, 82447, 90199, 98767, 107479, 118231, 129151, 141031, 151471, 162199, 173359

Offset: 1

Artur Jasinski, Sep 16 2007

nonn

For primes in this sequence see A133560.
For sum of squares of two consecutive primes only 2^2 + 3^2 = 13 is prime.
For sum of squares of three consecutive primes A133529 it seems that only 83 is a prime (checked for all n < 1000000).
Sums of squares of four (and all even number) of consecutive primes are even numbers with exception n=1 but 2^2 + 3^2 + 5^2 + 7^2 = 87 = 3*29 is not prime.
For primes that are sums of squares of five consecutive primes see A133559.

			a(6) = 13^2 + 17^2 + 19^2 + 23^2 + 29^2 + 31^2 + 37^2 = 4519.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A133529, A133538, A133558, A133559, A133560, A133561.

seq(add(ithprime(n+k)^2,k=0..6),n=1..35); # Muniru A Asiru, Jul 08 2018

b = {}; a = 2; Do[k = Prime[n]^a + Prime[n + 1]^a + Prime[n + 2]^a + Prime[n + 3]^a + Prime[n + 4]^a + Prime[n + 5]^a + Prime[n + 6]^a; AppendTo[b, k], {n, 1, 100}]; b
Total/@Partition[Prime[Range[40]]^2,7,1] (* Harvey P. Dale, Jan 01 2025 *)

Edited by Michel Marcus, Jul 08 2018

A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 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, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 10 2012

A skew version of A122075.

			Triangle begins :
  1
  2, 0
  3, 1, 0
  5, 3, 0, 0
  8, 7, 1, 0, 0
  13, 15, 4, 0, 0, 0
  21, 30, 12, 1, 0, 0, 0
  34, 58, 31, 5, 0, 0, 0, 0
  55, 109, 73, 18, 1, 0, 0, 0, 0
  89, 201, 162, 54, 6, 0, 0, 0, 0, 0
  144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A122075, A122950, A000045, A023610, A129707

T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

G.f.: (1+x)/(1-x-(1+y)*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

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A129710 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 01 subwords (0 <= k <= floor(n/2)). A Fibonacci binary word is a binary word having no 00 subword.

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A133562 Numbers which are the sum of the squares of seven consecutive primes.

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A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 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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