Sequences

392,541 sequences

a(n) = j if n = [ j*sqrt(5) ], else a(n) = k if n = [ (k/4)*(5 + sqrt(5)) ].A026261
a(n) = j if n = [ j*sqrt(5) ], else a(n) = k if n = [ (k/4)*(5 + sqrt(5)) ].
Beginning with the natural numbers, swap [ k*sqrt(5) ] and [ (k/4)*(5 + sqrt(5)) ], for all k >= 1.A026262
Beginning with the natural numbers, swap [ k*sqrt(5) ] and [ (k/4)*(5 + sqrt(5)) ], for all k >= 1.
a(n) = |s(n) - n|, where s = A026262.A026263
a(n) = |s(n) - n|, where s = A026262.
(1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026262.A026264
(1/2)*(s(n) + 1), where s(n) is the n-th odd number in A026262.
a(n) = position of n in A026264.A026265
a(n) = position of n in A026264.
a(n) = (1/2)*s(n), where s = A026262.A026266
a(n) = (1/2)*s(n), where s = A026262.
a(n) = position of n in A026266.A026267
a(n) = position of n in A026266.
Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.A026268
Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.A026269
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0 = s(n), s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n) and a(n) = Sum{T(k,k-1)}, k = 1,2,...,n, where T is array in A026268.
Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-1), where T is the array in A026268.A026270
Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1 = s(n), |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-1), where T is the array in A026268.
a(n) = sum of the numbers between the two n's in A026242.A026271
a(n) = sum of the numbers between the two n's in A026242.
a(n) = smallest k such that k=a(n-k-1) is the only appearance of k so far; if there is no such k, then a(n) = least positive integer that has not yet appeared.A026272
a(n) = smallest k such that k=a(n-k-1) is the only appearance of k so far; if there is no such k, then a(n) = least positive integer that has not yet appeared.
a(n) = least k such that s(k) = n, where s = A026272.A026273
a(n) = least k such that s(k) = n, where s = A026272.
Greatest k such that s(k) = n, where s = A026272.A026274
Greatest k such that s(k) = n, where s = A026272.
Sum of numbers between the two n's in A026272.A026275
Sum of numbers between the two n's in A026272.
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 2, else a(n) = least positive integer that has not yet occurred.A026276
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 2, else a(n) = least positive integer that has not yet occurred.
a(n) = least k such that s(k) = n, where s = A026276.A026277
a(n) = least k such that s(k) = n, where s = A026276.
a(n) = greatest k such that s(k) = n, where s = A026276.A026278
a(n) = greatest k such that s(k) = n, where s = A026276.
a(n) = sum of the numbers between the two n's in A026276.A026279
a(n) = sum of the numbers between the two n's in A026276.
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 3, else a(n) = least positive integer that has not yet occurred.A026280
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 3, else a(n) = least positive integer that has not yet occurred.
a(n) = least k such that s(k) = n, where s = A026280.A026281
a(n) = least k such that s(k) = n, where s = A026280.
a(n) = greatest k such that s(k) = n, where s = A026280.A026282
a(n) = greatest k such that s(k) = n, where s = A026280.
a(n) = sum of the numbers between the two n's in A026280.A026283
a(n) = sum of the numbers between the two n's in A026280.
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 4, else a(n) = least positive integer that has not yet occurred.A026284
a(n) = a(m) if a(m) has already occurred exactly once and n = a(m) + m + 4, else a(n) = least positive integer that has not yet occurred.
a(n) = least k such that s(k) = n, where s = A026284.A026285
a(n) = least k such that s(k) = n, where s = A026284.
a(n) = greatest k such that s(k) = n, where s = A026284.A026286
a(n) = greatest k such that s(k) = n, where s = A026284.
a(n) = sum of the numbers between the two n's in A026284.A026287
a(n) = sum of the numbers between the two n's in A026284.
Number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 2, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-2), where T is the array in A026268.A026288
Number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 2, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also T(n,n-2), where T is the array in A026268.
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-3), where T is the array in A026268.A026289
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 3, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-3), where T is the array in A026268.
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 4, |s(i) - s(i-1)| <= 1 for i >=2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-4), where T is the array in A026268.A026290
a(n) = number of (s(0), s(1), ..., s(n)) such that every s(i) is an integer, s(0) = 0, s(1) = 1, s(n) = 4, |s(i) - s(i-1)| <= 1 for i >=2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also a(n) = T(n,n-4), where T is the array in A026268.
a(n) = T(2n-1,n), where T is the array in A026268.A026291
a(n) = T(2n-1,n), where T is the array in A026268.
a(n) = T(2n,n), where T is the array in A026268.A026292
a(n) = T(2n,n), where T is the array in A026268.
a(n) = T(3n,n), where T is the array in A026268.A026293
a(n) = T(3n,n), where T is the array in A026268.
T(4n,n), where T is the array in A026268.A026294
T(4n,n), where T is the array in A026268.
a(n) = T(2n,n-1), where T is the array in A026268.A026295
a(n) = T(2n,n-1), where T is the array in A026268.
a(n) = T(2n,n+1), where T is the array in A026268.A026296
a(n) = T(2n,n+1), where T is the array in A026268.
a(n) = T(n,[ n/2 ]), where T is the array in A026268.A026297
a(n) = T(n,[ n/2 ]), where T is the array in A026268.
Number of polyhexes of class PF2.A026298
Number of polyhexes of class PF2.
Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also sum of numbers in row n+1 of the array T in A026268.A026299
Number of (s(0), s(1), ..., s(n)) such that every s(i) is a nonnegative integer, s(0) = 0, s(1) = 1, |s(i) - s(i-1)| <= 1 for i >= 2, |s(2) - s(1)| = 1, |s(3) - s(2)| = 1 if s(2) = 1. Also sum of numbers in row n+1 of the array T in A026268.
Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).A026300
Motzkin triangle, T, read by rows; T(0,0) = T(1,0) = T(1,1) = 1; for n >= 2, T(n,0) = 1, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k = 1,2,...,n-1 and T(n,n) = T(n-1,n-2) + T(n-1,n-1).
Expansion of g.f. sqrt(V(x)), where V(x) = Sum_{n >= 0} A065409(n)*x^n.A026301
Expansion of g.f. sqrt(V(x)), where V(x) = Sum_{n >= 0} A065409(n)*x^n.
a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(2n) = n. Also a(n) = T(2n,n), where T is the array in A026300.A026302
a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(2n) = n. Also a(n) = T(2n,n), where T is the array in A026300.
a(n) = T(3n,n), where T is the array in A026300.A026303
a(n) = T(3n,n), where T is the array in A026300.
a(n) = T(4n,n), where T is the array in A026300.A026304
a(n) = T(4n,n), where T is the array in A026300.
a(n) = T(2n,n-1), where T is the array in A026300.A026305
a(n) = T(2n,n-1), where T is the array in A026300.
a(n) = T(2n,n+1), where T is the array in A026300.A026306
a(n) = T(2n,n+1), where T is the array in A026300.
a(n) = T(n,[ n/2 ]), where T is the array in A026300.A026307
a(n) = T(n,[ n/2 ]), where T is the array in A026300.
Expansion of 1/((1-2*x)*(1-5*x)*(1-11*x)*(1-12*x)).A026308
Expansion of 1/((1-2*x)*(1-5*x)*(1-11*x)*(1-12*x)).
a(n) = n-th nonnegative integer k satisfying sin(k) < sin(k+1).A026309
a(n) = n-th nonnegative integer k satisfying sin(k) < sin(k+1).
sin(n) > cos(n+1).A026310
sin(n) > cos(n+1).