cp's OEIS Frontend

Note that when starting from a(n)^2, equality holds between series of first n+1 and next n consecutive squares: a(n)^2 + (a(n) + 1)^2 + ... + (a(n) + n)^2 = (a(n) + n + 1)^2 + (a(n) + n + 2)^2 + ... + (a(n) + 2*n)^2; e.g., 10^2 + 11^2 + 12^2 = 13^2 + 14^2. - Henry Bottomley, Jan 22 2001; with typos fixed by Zak Seidov, Sep 10 2015

a(n) = sum of second set of n consecutive even numbers - sum of the first set of n consecutive odd numbers: a(1) = 4-1, a(3) = (8+10+12) - (1+3+5) = 21. - Amarnath Murthy, Nov 07 2002

Partial sums of odd numbers 3 mod 4, that is, 3, 3+7, 3+7+11, ... See A001107. - Jon Perry, Dec 18 2004

If Y is a fixed 3-subset of a (2n+1)-set X then a(n) is the number of (2n-1)-subsets of X intersecting Y. - Milan Janjic, Oct 28 2007

More generally (see the first comment), for n > 0, let b(n,k) = a(n) + k*(4*n + 1). Then b(n,k)^2 + (b(n,k) + 1)^2 + ... + (b(n,k) + n)^2 = (b(n,k) + n + 1 + 2*k)^2 + ... + (b(n,k) + 2*n + 2*k)^2 + k^2; e.g., if n = 3 and k = 2, then b(n,k) = 47 and 47^2 + ... + 50^2 = 55^2 + ... + 57^2 + 2^2. - Charlie Marion, Jan 01 2011

Sequence found by reading the line from 0, in the direction 0, 10, ..., and the line from 3, in the direction 3, 21, ..., in the square spiral whose vertices are the triangular numbers A000217. - Omar E. Pol, Nov 09 2011

a(n) is the number of positions of a domino in a pyramidal board with base 2n+1. - César Eliud Lozada, Sep 26 2012

Differences of row sums of two consecutive rows of triangle A120070, i.e., first differences of A016061. - J. M. Bergot, Jun 14 2013 [In other words, the partial sums of this sequence give A016061. - Leo Tavares, Nov 23 2021]

a(n)*Pi is the total length of half circle spiral after n rotations. See illustration in links. - Kival Ngaokrajang, Nov 05 2013

For corresponding sums in first comment by Henry Bottomley, see A059255. - Zak Seidov, Sep 10 2015

a(n) also gives the dimension of the simple Lie algebras B_n (n >= 2) and C_n (n >= 3). - Wolfdieter Lang, Oct 21 2015

With T_(i+1,i)=a(i+1) and all other elements of the lower triangular matrix T zero, T is the infinitesimal generator for unsigned A130757, analogous to A132440 for the Pascal matrix. - Tom Copeland, Dec 13 2015

Partial sums of squares with alternating signs, ending in an even term: a(n) = 0^2 - 1^2 +- ... + (2*n)^2, cf. Example & Formula from Berselli, 2013. - M. F. Hasler, Jul 03 2018

Also numbers k with the property that in the symmetric representation of sigma(k) the smallest Dyck path has a central peak and the largest Dyck path has a central valley, n > 0. (Cf. A237593.) - Omar E. Pol, Aug 28 2018

a(n) is the area of a triangle with vertices at (0,0), (2*n+1, 2*n), and ((2*n+1)^2, 4*n^2). - Art Baker, Dec 12 2018

This sequence is the largest subsequence of A000217 such that gcd(a(n), 2*n) = a(n) mod (2*n) = n, n > 0 up to a given value of n. It is the interleave of A033585 (a(n) is even) and A033567 (a(n) is odd). - Torlach Rush, Sep 09 2019

A generalization of Hasler's Comment (Jul 03 2018) follows. Let P(k,n) be the n-th k-gonal number. Then for k > 1, partial sums of {P(k,n)} with alternating signs, ending in an even term, = n*((k-2)*n + 1). - Charlie Marion, Mar 02 2021

Let U_n(H) = {A in M_n(H): A*A^H = I_n} be the group of n X n unitary matrices over the quaternions (A^H is the conjugate transpose of A. Note that over the quaternions we still have A*A^H = I_n <=> A^H*A = I_n by mapping A and A^H to (2n) X (2n) complex matrices), then a(n) is the dimension of its Lie algebra u_n(H) = {A in M_n(H): A + A^H = 0} as a real vector space. A basis is given by {(E_{st}-E_{ts}), i*(E_{st}+E_{ts}), j*(E_{st}+E_{ts}), k*(E_{st}+E_{ts}): 1 <= s < t <= n} U {i*E_{tt}, j*E_{tt}, k*E_{tt}: t = 1..n}, where E_{st} is the matrix with all entries zero except that its (st)-entry is 1. - Jianing Song, Apr 05 2021

Denominators are given by the simple periodic sequence [1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, ...] (= A014695) thus we get an average of 1/2, 3/2, 3, 5, 15/2, 21/2, 14, 18, etc. swappings required to bubble sort a string of 2, 3, 4, 5, 6, ... letters.

Also numbers that cannot be expressed as the sum of two 4th powers. - Cino Hilliard, Nov 23 2003

The sequence lists the indices of the multiples of 5 in A033567. - Bruno Berselli, Jan 05 2018

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Enumeration table T(n,k). The order of the list:

T(1,1)=1;

T(1,3), T(1,2), T(2,1), T(2,2), T(3,1);

...

T(1,n), T(1,n-1), T(2,n-2), T(2,n-1), T(3,n-2), T(3,n-3)...T(n,1);

...

Descent by snake along two adjacent antidiagonal - step to the west, step to the southwest, step to the east, step to the southwest and so on. The length of each step is 1.

Table contains:

row 1 is alternation of elements A130883 and A033816,

row 2 accommodates elements A100037 in odd places;

column 1 is alternation of elements A000384 and A091823,

column 2 is alternation of elements A071355 and A014106,

column 3 accommodates elements A130861 in even places;

main diagonal accommodates elements A188135 in odd places,

diagonal 1, located above the main diagonal, is alternation of elements A033567 and A033566,

diagonal 2, located above the main diagonal, is alternation of elements A139271 and A033585.

Sequence found by reading the line from 5, in the direction 5, 29, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Jul 18 2012

Odd numbers (A005408) written clockwise as a square spiral:

.

41--43--45--47--49--51

| |

39 13--15--17--19 53

| | | |

37 11 1---3 21 55

| | | | |

35 9---7---5 23 57

| | |

33--31--29--27--25 59

|

71--69--67--65--63--61

.

Walking in straight lines away from the center:

1, 17, 49, ... = A069129(n+1) = 1 - 8*n + 8*n^2,

1, 3, 21, ... = A033567(n) = 1 - 6*n + 8*n^2,

1, 15, 45, ... = A014634(n) = 1 + 6*n + 8*n^2,

1, 5, 25, ... = A080856(n) = 1 - 4*n + 8*n^2,

1, 13, 41, ... = A102083(n) = 1 + 4*n + 8*n^2,

1, 7, 29, ... = a(n) = 1 - 2*n + 8*n^2,

1, 11, 37, ... = A188135(n) = 1 + 2*n + 8*n^2,

1, 9, 33, ... = A081585(n) = 1 + 8*n^2,

5, 29, 69, ... = A108928(n+1) = -3 + 8*n^2,

7, 31, 71, ... = A157914(n+1) = -1 + 8*n^2,

9, 35, 77, ... = A033566(n+1) = -1 + 2*n + 8*n^2.

All are quadrisections of sequences in A181407(n) (example: A014634(n) and A033567(n) in A064038(n+1)) or of this family (?): a(n) is a quadrisection of f(n) = 1,1,1,1,2,7,11,8,11,29,37,23,28,67,79,46,... f(n) is just before A064038(n+1) (fifth vertical) in A181407(n). The companion to a(n) is A188135(n), another quadrisection of f(n). Two last quadrisections of f(n) are A054552(n) and A033951(n).

For n >= 1, bisection of A193867. - Omar E. Pol, Aug 16 2011

Also the sequence may be obtained by starting with the segment (1, 7) followed by the line from 7 in the direction 7, 29, ... in the square spiral whose vertices are the generalized hexagonal numbers (A000217). - Omar E. Pol, Aug 01 2016

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Enumeration table T(n,k). The order of the list:

T(1,1) = 1;

T(1,3), T(2,2), T(1,2), T(2,1), T(3,1);

. . .

T(1,n), T(2,n-1), T(1,n-1), T(2,n-2), T(3,n-2), T(4,n-3)...T(n,1);

...

Descent by snake along two adjacent antidiagonals - step to the southwest, step to the north, step to the southwest, step to the south and so on. The length of each step is 1. Phase four steps is rotated 90 degrees counterclockwise and the mirror of the phase A211377.

Table contains the following:

row 1 is alternation of elements A130883 and A100037,

row 2 accommodates elements A033816 in even places;

column 1 is alternation of elements A000384 and A014106,

column 2 is alternation of elements A091823 and A071355,

column 4 accommodates elements A130861 in odd places;

main diagonal is alternation of elements A188135 and A033567,

diagonal 1, located above the main diagonal, accommodates elements A033585 in even places,

diagonal 2, located above the main diagonal, accommodates elements A139271 in odd places,

diagonal 3, located above the main diagonal, is alternation of elements A033566 and A194431.

Permutation of the natural numbers.

a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

Enumeration table T(n,k). The order of the list:

T(1,1)=1;

T(1,3), T(2,2), T(2,1), T(1,2), T(3,1);

. . .

T(1,2*n+1), T(2,2*n), T(2,2*n-1), T(1,2*n), ...T(2*n-1,3), T(2*n,2), T(2*n,1), T(2*n-1,2), T(2*n+1,1);

. . .

Movement along two adjacent antidiagonals - step to the southwest, step to the west, step to the northeast, 2 steps to the south, step to the west and so on. The length of each step is 1.

Table contains:

row 1 accommodates elements A130883 in odd places,

row 2 is alternation of elements A100037 and A033816;

column 1 is alternation of elements A000384 and A091823,

column 2 is alternation of elements A014106 and A071355,

column 3 accommodates elements A130861 in even places;

main diagonal is alternation of elements A188135 and A033567,

diagonal 1, located above the main diagonal accommodates elements A033566 in even places,

diagonal 2, located above the main diagonal is alternation of elements A139271 and A024847,

diagonal 3, located above the main diagonal accommodates of elements A033585.

For n > 0, first differences of A304487.

All the terms of this sequence are odd numbers.