cp's OEIS Frontend

i values are A053719 and j values are A053720.

sqrt((Sum_{k=0..n} 2*a(k)) + 1) = A056220(n+2). - Doug Bell, Mar 09 2009

Second leg of Pythagorean triangles with hypotenuse a square: A057769(n)^2 + a(n-1)^2 = A007204(n)^2. - Martin Renner, Nov 12 2011

Numbers which are both the sum of 2*n + 4 consecutive odd integers and the sum of the 2*n + 2 immediately higher consecutive odd integers. In general, let f(k,n) = 3*k^3*A000330(n). Then f(k,n) is both the sum of k*n + k consecutive terms from the arithmetic progression with first term A000217(k) and constant difference k and the immediately higher k*n terms from the same progression. When k = 1, f(k,n) = A059270(n). - Charlie Marion, Aug 23 2021

Also a(n) = lcm(1,...,(2n+2))/12. - Paul Barry, Jun 09 2006. Proof that this is the same sequence, from Martin Fuller, May 06 2007: k, k+1, 2k+1 are coprime so their lcm is the same as their product. Hence a(n) = lcm{k, k+1, 2k+1 | k=1..n}/6. {k, k+1, 2k+1 | k=1..n} = {1..2n+2 excluding even numbers >n+1}. Adding the higher even numbers to the set doubles the LCM. Hence lcm{k, k+1, 2k+1 | k=1..n}/6 = lcm{1..2n+2}/12.

Gives an identity for sigma(n). Alternating sum of row n equals A000203(n), the sum of the divisors of n.

Row n has length A003056(n) hence column k starts in row A000217(k).

Column 1 is A000716, but here the offset is 1 not 0.

The 1st element of column k is A000330(k).

The 2nd element of column k is A059270(k).

The 3rd element of column k is A220443(k).

The partial sums of column k give the k-th column of A249120.

This triangle has been constructed after Mircea Merca's formula for A000203.

From Omar E. Pol, May 05 2022: (Start)

In the Honda-Yoda paper see "3. String theory and Riemann hypothesis". The coefficients that are mentioned in 3.11 are the first 16 terms of A000716, the coefficients that are mentioned in 3.12 are the first 5 terms of A000330, and the coefficients that are mentioned in 3.13 are the first 16 terms of A000203.

Another triangle with the same row lengths and whose alternating row sums give A000203 is A196020. (End)

j values are A053720 and k values are A053721

From Robert Dawson, Apr 04 2018: (Start)

This sequence is the union of the following twelve subsequences.

Terms in have fewer than d digits: they are pyramorphic, and always appear elsewhere, as an earlier term in the same sequence or in a related sequence. Dashes replace solutions to the congruences for which the inequalities, or other conditions proving pyramorphicity, are not satisfied; these are not part of the subsequences.

(i) a(d) := 4 * 10^(d-1) for d >= 2:

(-, 40,400,4000,40000,400000,...)

(ii) 2a(d) for d >= 2:

(-, 80,800,8000,80000,800000,...)

(iii) b(d) such that 2^(d+1)|b(d), 5^d|b(d)-1, b(d) < 10^d:

(-,-,-,9376,-,-,7109376,-,...)

(iv) c(d) such that 2^(d+1)|c(d), 5^(d-1)|2c(d)+5, c(d) < 4*10^(d-1):

(0,<0>,160,2560,26560,226560,<226560>,12226560,...)

(v) c(d) + a(d) for d >= 2:

(-,40,560,6560,66560,626560,42265609,41226560,...)

(vi) c(d) + 2a(d) for d >= 2, when this is less than 10^d:

(-, 80,960,-,-,-,8226560,81226560,...)

(vii) c'(d) such that 2^(d+1)|c'(d)-1, 5^(d-1)|2c'(d)+5, c'(d) < 4*10^(d-1):

(1,25,385,1185,37185,317185,1117185,25117185,...)

(viii)c'(d) + a(d) for d >= 2:

(-,65,785,5185,77185,717185,5117185,65117185,...)

(ix) c'(d) + 2a(d) for d >= 2, when this is less than 10^d:

(-,-,-,9185,-,-,9117185,-,...)

(x) c"(d) such that 2^(d+1)|c"(d)-1, 5^(d-1)|c"(d), c"(d) < 4*10^(d-1):

(5,25,225,2625,10625,<90625>,<890625>,12890625,...)

(xi) c"(d) + a(d) for d >= 2:

(-,65,625,6625,50625,490625,4890625,52890626,...)

(xii) c"(d) + 2a(d) for d >= 2, when this is less than 10^d:

(-,-,-,-,90625,890625,8890625,92890625,...)

For d >= 3 the d-th terms of these sequences are always distinct.

For d > 3 there are at least eight and at most eleven square pyramorphic numbers with d digits (not including leading zeros). The minimum is first achieved for d=6; the maximum is first achieved for d=49.

(End)